An Intuitive Introduction to Limits

7/23/2019 An Intuitive Introduction to Limits

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An Intuitive Introduction To Limits

by Kalid Azad · 41 comments

Limits, the Foundations Of Calculus, seem so articial and weasely: “Letx approach 0, but not get there, yet we’ll act lie it’s there! " #gh$

%ere’s how & learned to en'oy them:

• What is a limit? Our best (rediction of a (oint we didn’t obser)e$• How do we make a prediction? *oom into the neighboring (oints$ &f

our (rediction is always in+between neighboring (oints, no matter howmuch we oom, that’s our estimate$

• Why do we need limits? -ath has “blac hole" scenarios .di)iding byero, going to innity/, and limits gi)e us an estimate when we can’tcom(ute a result directly$

• How do we know we’re right? e don’t$ Our (rediction, the limit, isn’tre1uired to match reality$ 2ut for most natural (henomena, it sure seemsto$

Limits let us as “hat if3"$ &f we can directly obser)e a function at a )alue.lie x40, or x growing innitely/, we don’t need a (rediction$ 5he limitwonders, “&f you can see e)erything except a single )alue, what do you thin isthere3"$

hen our (rediction is consistent and improves the closer we look , we

feel condent in it$ 6nd if the function beha)es smoothly, lie most real+worldfunctions do, the limit is where the missing (oint must be$

Key Analogy: Predicting A Soccer Ball

7retend you’re watching a soccer game$ #nfortunately, the connection ischo((y:

http://www.youtube.com/watch?v=2vJn5XxWg9U

http://www.youtube.com/watch?v=2vJn5XxWg9U



6c8 e missed what ha((ened at 9:00$ )en so, what’s your (rediction forthe ball’s (osition3

asy$ ;ust grab the neighboring instants .<:=> and 9:0?/ and (redict the ball tobe somewhere in+between$

6nd! it wors8 @eal+world ob'ects don’t tele(ortA they mo)e throughintermediate (ositions along their (ath from 6 to 2$ Our (rediction is “6t 9:00,the ball was between its (osition at <:=> and 9:0?"$ Bot bad$

ith a slow+motion camera, we might e)en say “6t 9:00, the ball was betweenits (ositions at <:=>$>>> and 9:00$00?"$

Our (rediction is feeling solid$ Can we articulate why3

• The predictions agree at increasing zoom levels$ &magine the <:=>+9:0? range was >$>+?0$? meters, but after ooming into <:=>$>>>+9:00$00?, the range widened to >+? meters$ #h oh8 *oomingshould narrow our estimate, not mae it worse8 Bot e)ery oom le)elneeds to be accurate .imagine seeing the game e)ery = minutes/, but tofeel condent, there must be some threshold where subse1uent oomsonly strengthen our range estimate$

• The before-and-after agree. &magine at <:=> the ball was at ?0meters, rolling right, and at 9:0? it was at =0 meters, rolling left$ hatha((ened3 e had a sudden 'um( .a camera change3/ and now we can’t



(in down the ball’s (osition$ hich one had the ball at 9:003 5hisambiguity shatters our ability to mae a condent (rediction$

ith these re1uirements in (lace, we might say “6t 9:00, the ball was at ?0meters$ 5his estimate is conrmed by our initial oom .<:=>+9:0?, whichestimates >$> to ?0$? meters/ and the following one .<:=>$>>>+9:00$00?, whichestimates >$>>> to ?0$00? meters/"$

Limits are a strategy for maing condent (redictions$

Exploring The Intuition

Let’s not bring out the math denitions 'ust yet$ hat things, in the real world,do we want an accurate (rediction for but can’t easily measure3

What’s the circumference of a circle?

Finding (i “ex(erimentally" is tough: bust out a string and a ruler3

e can’t measure a sha(e with seemingly innite sides, but we can wonder “&sthere a (redicted )alue for (i that is always accurate as we ee( increasing thesides3"

6rchimedes gured out that (i had a range of

using a (rocess lie this:

&t was the (recursor to calculus: he determined that (i was a number thatstayed between his e)er+shrining boundaries$ Bowadays, we ha)emodern limit denitions of (i$

What does perfectly continuous growth look like?

e, one of my fa)orite numbers, can be dened lie this:

http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

http://functions.wolfram.com/Constants/Pi/09/

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/


http://functions.wolfram.com/Constants/Pi/09/




e can’t easily measure the result of innitely+com(ounded growth$ 2ut, ifwe could make a prediction, is there a single rate that is e)er+accurate3 &t

seems to be around $D?EE!an we use simple shapes to measure comple! ones?

Circles and cur)es are tough to measure, but rectangles are easy$ &fwe could use an innite number of rectangles to simulate cur)ed area, can weget a result that withstands innite scrutiny3 .-aybe we can nd the area of acircle$/

an we "nd the speed at an instant?

(eed is funny: it needs a before+and+after measurement .distance tra)eled Gtime taen/, but can’t we ha)e a s(eed at indi)idual instants3 %rm$

http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/






Limits hel( answer this conundrum: (redict your s(eed when tra)eling to aneighboring instant$ 5hen as the “im(ossible 1uestion": what’s your (redicteds(eed when the ga( to the neighboring instant is ero3

Bote: 5he limit isn’t a magic cure+all$ e can’t assume one exists, and theremay not be an answer to e)ery 1uestion$ For exam(le: &s the number ofintegers e)en or odd3 5he 1uantity is innite, and neither the “e)en" nor “odd"(rediction stays accurate as we count higher$ Bo well+su((orted (rediction

exists$

For (i, e, and the foundations of calculus, smart minds did the (roofs todetermine that “Hes, our (redicted )alues get more accurate the closer weloo$" Bow & see why limits are so im(ortant: they’re a stam( of a((ro)al onour (redictions$

The Math: The ormal !e"nition #$ A Limit

Limits are well+su((orted (redictions$ %ere’s the oIcial denition:

means for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x− c| < δ, we have |f(x) − L| < ε

Let’s mae this readable:

Math English Human English

means

When we “strongly predict” that f(c) = L, we mean

for all real ε > 0 for any error margin we want (+! "# meters)there e$ists a real % > 0 there is a &oom le'el (+! "# seconds)

sch that for all $ with 0 *$ c* %, we ha'e *f($)

L* ε

where the prediction stays accrate to within the

error margin

5here’s a few subtleties here:

• 5he oom le)el .delta, J/ is the function in(ut, i$e$ the time in the )ideo• 5he error margin .e(silon, K/ is the most the function out(ut .the ball’s

(osition/ can dier from our (rediction throughout the entire oom le)el• 5he absolute )alue condition .0 M Nx cN M J/ means (ositi)e and

negati)e osets must wor, and we’re si((ing the blac hole itself.when Nx P cN 4 0/$

e can’t e)aluate the blac hole in(ut, but we can say “xce(t for the missing(oint, the entire oom le)el conrms the (rediction f.c/ 4 L$" 6nd because f.c/4 L holds for any error margin we can nd, we feel condent$

Could we ha)e multi(le (redictions3 &magine we (redicted L? and L for f.c/$ 5here’s some dierence between them .call it $?/, therefore there’s some errormargin .$0?/ that would re)eal the more accurate one$ )ery function out(ut inthe range can’t be within $0? of both (redictions$ e either ha)e a single,innitely+accurate (rediction, or we don’t$



Hes, we can get cute and as for the “left hand limit" .(rediction from beforethe e)ent/ and the “right hand limit" .(rediction from after the e)ent/, but weonly ha)e a real limit when they agree$

6 function is continuous when it always matches the (redicted )alue .anddiscontinuous if not/:

Calculus ty(ically studies continuous functions, (laying the game “e’remaing (redictions, but only because we now they’ll be correct$"

The Math: Sho%ing The Limit Exists

e ha)e the re1uirements for a solid (rediction$ Questions asing you to“7ro)e the limit exists" as you to 'ustify your estimate$

For exam(le: 7ro)e the limit at x4 exists for

5he rst chec: do we e)en need a limit3 #nfortunately, we do: 'ust (lugging in“x4" means we ha)e a di)ision by ero$ Rrats$

2ut intuiti)ely, we see the same “ero" .x P / could be cancelled from the to(and bottom$ %ere’s how to dance this dangerous tango:

• 6ssume x is anywhere except .&t must be8 e’re maing a (redictionfrom the outside$/

• e can then cancel .x P / from the to( and bottom, since it isn’t ero$• e’re left with f.x/ 4 x S ?$ 5his function can be used outside the blac

hole$• hat does this sim(ler function (redict3 5hat f./ 4 T S ? 4 =$

o f./ 4 = is our (rediction$ 2ut did you see the sneainess3 e (retended xwasn’t Uto di)ide out .x+/V, then (lugged in after that troublesome item

was gone8 5hin of it this way: we used the sim(le beha)ior from outside theevent to (redict the gnarly beha)ior at the event $

e can (ro)e these shenanigans gi)e a solid (rediction, and that f./ 4 = isinnitely accurate$

For any accuracy threshold .K/, we need to find the “oom range" .J/ where westay within the gi)en accuracy$ For exam(le, can we ee( the estimatebetween SG+ ?$03

ure$ e need to nd out where

so



&n other words, x must stay within 0$= of to maintain the initial accuracyre1uirement of ?$0$ &ndeed, when x is between ?$= and $=, f.x/ goes fromf.?$=/ 4 9 to and f.$=/ 4 W, staying SG+ ?$0 from our (redicted )alue of =$

e can generalie to any error tolerance .K/ by (lugging it in for ?$0 abo)e$ eget:

&f our oom le)el is “J 4 0$= T K", we’ll stay within the original error$ &f our erroris ?$0 we need to oom to $=A if it’s 0$?, we need to oom to 0$0=$

5his sim(le function was a con)enient exam(le$ 5he idea is to start with theinitial constraint .Nf.x/ P LN M K/, (lug in f.x/ and L, and sol)e for the distanceaway from the blac+hole (oint .Nx P cN M 3/$ &t’s often an exercise in algebra$

ometimes you’re ased to sim(ly nd the limit .(lug in and get f./ 4 =/,other times you’re ased to (ro)e a limit exists, i$e$ cran through the e(silon+delta algebra$

lipping &ero and In"nity &nnity, when used in a limit, means “grows without sto((ing"$ 5he symbol Xis no more a number than the sentence “grows without sto((ing" or “mysu((ly of under(ants is dwindling"$ 5hey are conce(ts, not numbers .for ourle)el of math, 6le(h me alone/$

hen using X in a limit, we’re asing: “6s x grows without sto((ing, can wemae a (rediction that remains accurate3"$ &f there is a limit, it means the(redicted )alue is always conrmed, no matter how far out we loo$

2ut, & still don’t lie innity because & can’t see it$ 2ut & can see ero$ ithlimits, you can rewrite

as

Hou can get sneay and dene y 4 ?Gx, re(lace items in your formula, and thenuse

http://en.wikipedia.org/wiki/Aleph_number

http://en.wikipedia.org/wiki/Aleph_number



so it loos lie a normal (roblem again8 .Bote from 5im in the comments: thelimit is coming from the right, since x was going to (ositi)e innity/$ & (referthis arrangement, because & can see the location we’re narrowing in on .we’realways running out of (a(er when charting the innite )ersion/$

'hy Aren(t Limits )sed More #$ten*

&magine a id who gured out that “7utting a ero on the end" made a number?0x larger$ %a)e =3 rite down “=" then “0" or =0$ %a)e ?003 -ae it ?000$6nd so on$

%e didn’t gure out why multi(lication wors, why this rule is 'ustied! but,you’)e gotta admit, he sure can multi(ly by ?0$ ure, there are some edgecases .ould 0 become “00"3/, but it wors (retty well$

5he rules of calculus were disco)ered informally .by modern standards/$

Bewton deduced that “5he deri)ati)e of x3is 3x2" without rigorous 'ustication$

Het engines whirl and air(lanes Yy based on his unoIcial results$

5he calculus (edagogy mistae is creating a roadbloc lie “Hou must nowLimitsZ before a((reciating calculus", when it’s clear the in)entors of calculusdidn’t$ &’d (refer this (rogression:

• Calculus ass seemingly im(ossible 1uestions: hen can rectanglesmeasure a cur)e3 Can we detect instantaneous change3

• Limits gi)e a strategy for answering “im(ossible" 1uestions .“&f you canmae a (rediction that withstands innite scrutiny, we’ll say it’s o$"/

• 5hey’re a great tag+team: Calculus ex(lores, limits )erify$ e memorie

shortcuts for the results we )eried with limits .ddxx3=3x2/, 'ust lie we

memorie shortcuts for the rules we )eried with multi(lication .adding aero means times ?0/$ 2ut it’s still nice to now why the shortcuts are 'ustied$

Limits aren’t the only tool for checing the answers to im(ossible 1uestionsAinnitesimals wor too$ 5he ey is understanding what we’re trying to(redict, then learning the rules of maing (redictions$

%a((y math$



Why Do We Need Limits and Infnitesimals?


o many math courses 'um( into limits, innitesimals and [ery mall Bumbers .5-/without any context$ 2ut why do we care3

-ath hel(s us model the world$ e can brea a com(lex idea .a wiggly cur)e/ intosim(ler (arts .rectangles/:

2ut, we want an accurate model$ 5he thinner the rectangles, the more accurate the

model$ 5he sim(ler model, built from rectangles, is easier to analye than dealingwith the com(lex, amor(hous blob directly$

5he tricy (art is maing a decent model$ Limits and innitesimals hel( us createmodels that are sim(le to use, yet share the same (ro(erties as the original item.length, area, etc$/$

The Paradox o Zero

2reaing a cur)e into rectangles has a (roblem: %ow do we get slices so thin wedon’t notice them, but large enough to “exist"3

&f the slices are too small to notice .ero width/, then the model a((ears identical tothe original sha(e .we don’t see any rectangles8/$ Bow there’s no benet \ the]sim(le’ model is 'ust as com(lex as the original8 6dditionally, adding u( ero+widthslices won’t get us anywhere$

&f the slices are tiny but measurable, the illusion )anishes$ e see that our model is a 'agged a((roximation, and won’t be accurate$ hat’s a mathematician to do3

e want the best of both: slices so thin we can’t see them .for an accurate model/

and slices thic enough to create a sim(ler, easier+to+analye model$ 6 dilemma is athand8

The Solution: Zero is Relative



5he notion of ero is biased by our ex(ectations$ &s “0 S i", a (urely imaginarynumber, the same as ero3

ell, “i" sure loos lie ero when we’re on the real number line: the “real (art" of i,@e.i/, is indeed 0$ here else would a (urely imaginary number go3 .%ow far ast isdue Borth3/

%ere’s a dierent brain bender: did your weight change by ero (ounds while reading

this sentence3 Hes, by any scale you ha)e nearby$ 2ut an atomic measurement wouldshow somemass change due to sweat e)a(oration, exhalation, etc$

Hou see, there are two answers .so far8/ to the “be ero and not ero" (aradox:

• #llow another dimension: Bumbers measured to be ero in our dimensionmight actually be small but nonero in another dimension .innitesimala((roach \ a dimension infnitely smaller than the one we deal with/

• #ccept imperfection: Bumbers measured to be ero are (robably nonero at

a greater le)el of accuracyA saying something is “ero" really means “it’s 0 SG+our measurement error" .limit a((roach/

5hese a((roaches bridge the ga( between “ero to us" and “nonero at a greaterle)el of accuracy"$

vervie! o Limits " Infnitesimals

Let’s see how each a((roach would brea a cur)e into rectangles:

• $imits% “î)e me your error margin .& now you ha)e one, you limited,

im(erfect human8/, and &’ll draw you a cur)e$ hat’s the smallest unit on yourruler3 &nches3 Fine, &’ll draw you a staircasey cur)e at the millimeter le)el andyou’ll ne)er now$ Oh, you ha)e a millimeter ruler, do you3 &’ll draw the cur)ein nanometers$ hate)er your accuracy, &’m better$ Hou’ll ne)er see thestaircase$"

• &n"nitesimals% “Forget accuracy: there’s an entire infnitely small

dimension where &’ll mae the cur)e$ 5he (recision is totally beyond your reach\ &’m at the sub+atomic le)el, and you’re a ca)eman who can barely wal andchew gum$ &t’s lie getting to the imaginary (lane from the real one \ you 'ust

can’t do it$ 5o you, the rectangular sha(e & made at the sub+atomic le)el is themost (erfect cur)e you’)e e)er seen$"

Limits stay in our dimension, but with ]'ust enough’ accuracy to maintain the illusionof a (erfect model$ &nnitesimals build the model in another dimension, and it loos(erfectly accurate in ours$

5he tric to both a((roaches is that the sim(ler model was built beyond our le)el ofaccuracy$ e might know the model is 'agged, but we can’t tell the dierence \ anytest we do shows the model and the real item as the same$

That tri#$ doesn%t !or$& does it?

Oh, but it does$ e’re triced by “im(erfect but useful" models all the time:

http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/






• 6udio les don’t contain all the information of the original signal$ 2ut can youtell the dierence between a high+1uality m(< and a (erson taling in the otherroom3

• Com(uter (rintouts are made from indi)idual dots too small to see$ Can you tella handwritten note from a high+1uality (rintout of the same3

• [ideo shows still images at 9 times (er second$ 5his “im(erfect" model is fast

enough to tric our brain into seeing Yuid motion$

On and on it goes$ e resist because of our articial need for (recision$ 2ut audio and)ideo engineers now they don’t need a (erfect re(roduction, 'ust 1uality good

enough to tric us into thining it’s the original$

Calculus lets us mae these technically im(erfect but “accurate enough" models inmath$

Wor$in' In (nother Dimension

e need to be careful when reasoning with the sim(lied model$ e need to “do ourwor" at the le)el of higher accuracy, and bring the fnal result bac to our world$e’ll lose information if we don’t$

u((ose an imaginary number .i/ )isits the real number line$ )eryone thins he’sero: after all, @e.i/ 4 0$ 2ut i does a tric8 “1uare me8" he says, and they do: “i T i 4+?_ and the other numbers are astonished$

5o the real numbers, it a((eared that “0 T 0 4 +?_, a giant (aradox$

2ut their confusion arose from their (ers(ecti)e \ they only thought it was “0 T 0 4+?_$ Hes, @e.i/ T @e.i/ 4 0, but that wasn’t the o(eration8 e want @e.i T i/, which isdierent entirely8 e s1uare i in its own dimension, and bring that result bac toours$ e need to s1uare i, the imaginary number, and not 0, our idea of what i was$

2eware similar mistaes in calculus: we deal with tiny numbers that look like zero tous, but we can’t do math assuming they are .'ust lie treating i lie 0/$ Bo, we need to“do the math" in the other dimension and con)ert the results bac$

Limits and innitesimals ha)e dierent (ers(ecti)es on how this con)ersion is done:

• $imits% “Ro the math" at a le)el of (recision 'ust beyond your detection.millimeters/, and bring it bac to numbers on your scale .inches/

• &n"nitesimals% “Ro the math" in a dierent dimension, and bring it bac tothe “standard" one .'ust lie taing the real (art of a com(lex numberA you taethe “standard" (art of a hy(erreal number \ more later/

Bobody e)er told me: Calculus lets you wor at a better le)el of accuracy, with asim(ler model, and bring the results bac to our world$

( Real )xam*le: sin+x, - x

Let’s try a conce(tual exam(le$ u((ose we want to now what ha((ens to sin.x/ G xat ero$ Bow, if we 'ust (lug in x 4 0 we get a nonsensical result: sin.0/ 4 0, so weget 0 G 0 which could be anything$

http://betterexplained.com/articles/learning-calculus-overcoming-our-artifical-need-for-precision/




Let’s ste( bac: what does “x 4 0_ mean in our world3 ell, if we’re allowing theexistence of a greater le)el of accuracy, we now this:

• 5hings that appear to be ero may be nonero in a dierent dimension .'ust lie i mighta((ear to be 0 to us, but isn’t/

e’re going to say that x can be really, really close to ero at this greater le)el ofaccuracy, but not “true ero"$ &ntuiti)ely, you can thin of x as 0$0000!0000?, wherethe “!" is enough eros for you to no longer detect the number$

.&n limit terms, we say x 4 0 S d .delta, a small change that ee(s us within our errormargin/ and in innitesimal terms, we say x 4 0 S h, where h is a tiny hy(errealnumber, nown as an innitesimal/

O, we ha)e x at “ero to us, but not really"$ Bow we need a sim(ler model of sin.x/$hy3 ell, sine is a cray re(eating cur)e, and it’s hard to now what’s ha((ening$2ut it turns out that a straight line is a darn good model of a cur)e o)er shortdistances:

;ust lie we can brea a lled sha(e into tiny rectangles to mae it sim(ler, we candissect a cur)e into a series of line segments$ 6round 0, sin.x/ loos lie the line “x"$

o, we switch sin.x/ with the line “x"$ hat’s the new ratio3

ell, “xGx" is ?$ @emember, we aren’t really di)iding by ero because in this su(er+accurate world: x is tiny but non+ero .0 S d, or 0 S h/$ hen we “tae the limit or“tae the standard (art" it means we do the math .x G x 4 ?/ and then nd theclosest number in our world .? goes to ?/$

o, ? is what we get when sin.x/ G x a((roaches ero \ that is, we mae x as small as(ossible so it becomes 0 to us$ &f x became (ure, true ero, then the ratio would beundened .and it is at the innitesimal le)el8/$ 2ut we’re ne)er sure if we’re at(erfect ero \ something lie 0$0000!000? loos lie ero to us$

http://www.wolframalpha.com/input/?i=Plot%5B%7BSin%5Bx%5D%2C+x%7D%2C+%7Bx%2C+-5.0%2C+5.0%7D%5D






o, “sin.x/Gx" loos lie “xGx 4 ?" as far as we can tell$ &ntuiti)ely, the result maessense once we read about radians/$

.isuali/in' The Pro#ess

5oday’s goal isn’t to sol)e limit (roblems, it’s to understand the (rocess of sol)ingthem$ 5o sol)e this exam(le:

• @ealie x40 is not reachable from our accuracyA a “small but nonero" x is always

a)ailable at a greater le)el of accuracy• @e(lace sin.x/ by a straight line as a sim(ler model• “Ro the math" with the sim(ler model .x G x 4 ?/• 2ring the result .?/ bac into our accuracy .stays ?/

%ere’s how & see the (rocess:

&n later articles, we’ll learn the details of setting u( and sol)ing the models$

0aveats: The Tri#$ Doesn%t (l!ays Wor$

ome functions are really “'um(y" \ and they might dier on an innitesimal+by+innitesimal le)el$ 5hat means we can’t reliably bring them bac to our world$ &t looslie the function is unstable at microsco(ic le)el and doesn’t beha)e “smoothly"$

5he rigorous (art of limits is guring out which functions beha)e well enough thatsim(le yet accurate models can be made$ Fortunately, most of the natural functionsin the world .x, x, sin, ex/ beha)e nicely and can be modeled with calculus$

Limits r Infnitesimals?

Logically, both a((roaches sol)e the (roblem of “ero and nonero"$ & lieinnitesimals because they allow “another dimension" which seems a cleaner

se(aration than “always 'ust outside your reach"$ &nnitesimals were the foundationof the intuition of calculus, and a((ear inside (hysics and other sub'ects that use it$

5his isn’t an analysis class, but the math robots can be assured that innitesimalsha)e a rigorous foundation$ & use them because they clic for me$

http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/

http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/



Summary

7hew8 ome of these ideas are tricy, and & feel lie &’m taling from both sides of mymouth: we want to be sim(ler, yet still (erfectly accurate3

5his famous dilemma about “being ero sometimes, and non+ero others" is a famouscriti1ue of calculus$ &t was mostly ignored since the results wored out, but in the?E00s limits were introduced to really resol)e the dilemma$ e learn limits today, butwithout understanding the nature of the (roblem they were trying to sol)e8

%ere are the ey conce(ts:

• *ero is relati)e: something can be ero to us, and non+ero somewhere else• &nnitesimals .“another dimension"/ and limits .“beyond our accuracy"/ resol)e the

dilemma of “ero and nonero"• e create sim(ler models in the more accurate dimension, do the math, and bring the

result to our world• 5he nal result is (erfectly accurate for us

-y goal isn’t to do math, it’s to understand it$ 6nd a huge (art of groing calculus isrealiing that sim(le models created beyond our accuracy can loo “'ust ne" in ourdimension$ Later on we’ll learn the rules to build and use these models$ %a((y math$

ther Posts In This Series

?$ 6 êntle &ntroduction 5o Learning Calculus$ %ow 5o #nderstand Reri)ati)es: 5he 7roduct, 7ower ` Chain @ules<$ %ow 5o #nderstand Reri)ati)es: 5he Quotient @ule, x(onents, and Logarithms9$ 6n &ntuiti)e &ntroduction 5o Limits=$ hy Ro e Beed Limits and &nnitesimals3W$ Learning Calculus: O)ercoming Our 6rticial Beed for 7recisionD$ 7rehistoric Calculus: Risco)ering 7i

E$ 6 Calculus 6nalogy: &ntegrals as -ulti(lication>$ Calculus: 2uilding &ntuition for the Reri)ati)e?0$#nderstanding Calculus ith 6 2an 6ccount -eta(hor??$6 Friendly Chat 6bout hether 0$>>>$$$ 4 ?

7osted in Calculus, -ath

questions and insights for the article. Thanks

12 #omments

?$ amilo 'artin says:

%i alid8 4/ onderful to ha)e those insights as always8

;ust one )ery unim(ortant correctionA you said:“[ideo shows still images at 9 times (er second"%owe)er the correct would be to say that Film shows the frames at 9timesGsecond$ )en when con)erted to digital, it is s(ed+u( to = F7.76L video standart/ or “telecined" to >$>D F7 .B5C standart, #6’s

)ideo/Bote: &t’s not so sim(le, there are other standarts and )ariants, & wasrefering to R )ideo .not %R, which is commonly the double F7/ andalso: digital )ideo on a com(uter can ha)e any F7, e)en Yoating


http://betterexplained.com/articles/derivatives-product-power-chain/

http://betterexplained.com/articles/how-to-understand-derivatives-the-quotient-rule-exponents-and-logarithms/

http://betterexplained.com/articles/an-intuitive-introduction-to-limits/



http://betterexplained.com/articles/a-calculus-analogy-integrals-as-multiplication/

http://betterexplained.com/articles/calculus-building-intuition-for-the-derivative/

http://betterexplained.com/articles/understanding-calculus-with-a-bank-account-metaphor/

http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/

http://betterexplained.com/articles/category/math/calculus/

http://betterexplained.com/articles/category/math/



http://betterexplained.com/articles/how-to-understand-derivatives-the-quotient-rule-exponents-and-logarithms/

http://betterexplained.com/articles/an-intuitive-introduction-to-limits/





http://betterexplained.com/articles/understanding-calculus-with-a-bank-account-metaphor/

http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/

http://betterexplained.com/articles/category/math/calculus/

http://betterexplained.com/articles/category/math/



numbers and )ariable frame+rate .a totally com(lex thing to handle,easier to get in than to get out of it/

6nd as &’m taling about )ideo, &’d lie to suggest an idea for an article:how digital )ideo and image wors$ &t’s a fabulous bunch of 6%68moments when you get the conce(t behind all of those blocy artifactsand color schemes other than ye olde @^2$

6nd & lo)ed to thin on innitesimals in a new way8 5hans8 4R

$ (arag )hah says:

%i alid,

5hans for the wonderful (ost$ & ha)e totally forgotten all my math andha)e been thining of re+learning it .es(ecially from a com(uter science(ers(ecti)e/$

& found your (ost )ery useful, and & thin it will also gi)e that little (ush &needed to get started$

25, & too share your (assion for hel(ing others learn$ & ha)e aggregated)arious o(en com(uter science course )ideos on my website$

<$ 'urugesh (rabhu says:

%i!$5his one is as good as the (re)ious (osts8 & a((reciate ur enthusiasm

in (romoting the interest in -ath among young readers8 & en'oyed e)erybit of the article man!$5han you )ery much!$

9$ *alid says:

Camilo: 6h, than you for the clarication8 &’ll change ])ideo’ to ]lm’ :/$

5hat’d be a really cool article \ & don’t now too much about the )ideoformats, but now that -7^ has some really neat technology to mae itcom(ress well$

7arag: ^reat, glad you en'oyed the article8 Checing out your site now,thans for collecting all those lins \ &’m ho(ing to go bac and refresh alot of my cs nowledge also :/$

-urugesh: 5hans for the su((ort, &’)e had a lot of fun trying to get mybrain around these conce(ts again, but being able to as “ait, whatdoes it really mean to me3"$ ^lad it was useful for you8

=$ #nonymous says:

& found a few ty(os:

http://www.adaptivelearningonline.net/

http://www.adaptivelearningonline.net/



7aradox of ero: “slices so thin we can’t seem them" .see/ummary: “& feel lie &’m taling from boths ides of my mouth" .bothsides/ummary: “%ere’s the ey conce(ts" .%ere are/

5hans for another great article8

W$ *alid says:

6nonymous: Hou’re welcome, and thans for the corrections8 & 'ustmade them now$

D$ nanoturkiye says:

6gain & started reading and did not realied that & nished a long article8&t was )ery entertaining$ 5han you )ery much for your eorts$

E$*alid

says:nanoturiye: Hou’re welcome8 ^lad you en'oyed the article$

>$ #rbie )among says:

%i halid, (ro(s for this great series which & 'ust found out recently andreading your (osts has been a daily habit for me$

;ust one confusion in this to(ic, can you elaborate more on this:

\\6round 0, sin.x/ loos lie the line “x"$\\

& thin of x as the x+axis in the (lane that was demonstrated$ &t could alsobe 'ust the )ariable in the e1uation$ 2ut neither maes sense$ & now xGxis ?, but how come sin.x/ is x3

thans8 and more (ower8

?0$ mcml!!!vi says:

%ello, alid,[ery well+written and descri(ti)e$ 5han you for gi)ing me a good and(leasant read on things (ast and nearly forgotten8

& could only wish that more (eo(le lie you were teaching in high schoolsand uni)ersities$ 6round here, the tutors are often silled in their eld,but regularly and gra)ely fail to con)ey the meaning behind the

denitions, theorems and (roofs they teach P only the items themsel)esAand the educational (rocess (lummets$

6rbie:\\

http://orfornano.wordpress.com/

http://orfornano.wordpress.com/



6round 0, sin.x/ loos lie the line “x"$\\& belie)e this means the line “y 4 x"$ 5hus y? 4 sin.x/, y 4 x and y?4 y for x + 0$

??$ *alid says:

6rbie: ow, & lie that functional re(resentation of it8 Hes, integrations

are a general “a((lying" one function to another, )s$ some staticmulti(lication 'ust to nd area .area 'ust limits our creati)ityGintuition &thin/$

6h, & should be more clear about that! the & meant the line “y 4 x", thatis, a 9= degree line extending from the origin$ o the e1uation y 4 sin.x/loos )ery similar to y 4 x for )ery small numbers .sin.x/ extends 9=degrees from the origin when it rst starts o/$

%o(e this hel(s8

?$ *alid says:

mcmlxxx)i: ^lad you en'oyed it, and thans for the comment$ & toowish there was more em(hasis on true understanding )s$ the “let’s learnenough to (ass the next test" mentality$ Learning the intuition may taea bit longer than memoriing in the short term, but in the long run itgi)es you a more Yexible set of nowledge, and not to mention it’s waymore fun$ & sometimes see grades as a curse because rather than beingan indication of nowledge, they become an end in itself )s$ the learningit should re(resent$ &t’s )ery hard to test intuition \ it’s a gutchec youneed to as yourself$ 2ut with no grades there’s no “incenti)e" .carrot orstic/ \ & don’t now the answer, but & too wish there was another way$

?<$ #nonymous says:

5his (a(er oers similar )iews about mathematics education as well as acriticism of the cultural o(inion of mathematics that you might

lie$htt(:GGmaadotorgGde)linGde)lin0<0E$html

?9$ *alid says:

6nonymous: 5han you \ &’)e seen the essay and really lie it :/$

?=$ asdf says:

mooth &nnitesimal 6nalysis handles innitesimals better than Bon+tandard 6nalysis:

htt(:GGen$wii(edia$orgGwiiGmoothinnitesimalanalysis

http://maadotorg/devlin/devlin_03_08.html

http://betterexplained.com/

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

http://maadotorg/devlin/devlin_03_08.html

http://betterexplained.com/

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis



&n intuitionistic math, the law of excluded middle is re'ected .i$e$ not not6 doesn’t im(ly 6/ so you must (ro)ide an algorithm for constructing allyour ob'ects$

5here is no general (rocedure for detecting whether or not ob'ects aree1ual$ Hou must ex(licitly (ro)ide an algorithm for showing ob'ects aree1ual$

5he trichotomy law .a b+ a , b doesn’t hold in general.

6ll functions are continuous$ 7iecewise functions are nonsensical$

&n other words, the continuum is unbreaable into (oints$ Functionstransform the continuum onto the continuum$

ith this as our basis, mooth &nnitesimal 6nalysis introduces an ob'ectcalled e(silon$

5here is no algorithm to tell whether or not e(silon 84 0 or e(silon 4 0$ 5his a)oids the rst (roblem entirely$

e(silon 4 0 though which gi)es us a way to get rid of them from ourformulas$

o & )iew innitesimals as the glue that maes the continuumunbreaable and there is no algorithm to decide if the ex(ression“e(silon 4 0 or e(silon 84 0" is true .see why we ha)e to re'ect the law of

excluded middle to mae this wor3/$

?W$ *alid says:

asdf: ow, really interesting stu8 & lie that insight of innitesimals asthe “glue" that maes the continuum unbreaable$ ^reat analogy$

?D$ ave says:

%ey, alid, &’)e 'ust got a 1uic 1uestion to as$

&f you learn calculus )ia the use of innitesimals, is it (ossible to thenmae the lea( o)er to using limits3 hile & doubt it would ha((en, &’d lieto be an amateur mathematician in the )ein of Fermat some time andde)elo( (roofs .more as a beauty thing, to be honest/, but writing in afashion that is contrary to the norm is rather lie handing out (anish(am(hlets in an nglish neighborhood+ they might understand, but theywon’t lie it$

o, yeah, can you 'um( from innitesimals o)er to limits3 From what & cantell, limits are mainly used because they’re easily to rigorously dene anto ee( the constructi)ist cam( from yelling at you$

?E$ *alid says:



Ra)e: ^reat 1uestion$ & can’t say &’m com(letely comfortable withlimits, but & thin you can 'um( bac and forth .the eisler Calculus boohas some exam(les lie this & belie)e/$ & thin the bigger goal is to gureout what is being said, i$e$ “hat does this e1uation e1ual, within somele)el of tolerance3"$ Limits and innitesimals are two ways to dene thattolerance threshold, but innitesimals are “easier" in that it’s built in.and you don’t need to ex(licitly dene e(silon, delta, etc$/$

?>$ werterber says:

%ello, i ha)e silly 1uestion$ %ow intuiti)ely ex(lain that cos xGx isundend3 5here is graf htt(:GGwww$wolframal(ha$comGin(utG3i47lot=2 cos=2x=RCSxCSxCS+?$0CS?$0=R

thx

0$ *alid says:

werterber: Bot a silly 1uestion at all8 &n my head, it’s saying “what’s theratio of width Ucos.x/V to distance tra)eled .x/"$

6s our distance tra)eled goes to 0 .we aren’t mo)ing from the starting(oint/, cos.x/ tends towards ? \ we’re (retty much at the same width$ oit becomes “? G 0" in my head$

?$ *ostya says:

5his comes down to this: we can’t (ossibly describe what we can’t(ossibly imagine$ 5hat’s why it must always be “small enoughrectangles" of a sort!

&nterestingly, 2rian ^reene in his “legant #ni)erse" gi)es to understand,that the “su(erstring theory", along with ex(ected resolution of somefundamental (roblems, must bring about radical change in mathematicalmodes, so that you can’t decrease the sie of those “small rectangles"

down to innity, but that it must ha)e its limit somewhere around thele)el 7lan constant ?0+<9$ 6fter which further decrease will actuallymean increase$Bow e)ery theory ser)es for some con)enience$ 5herefore, aren’t we freeto tae such a((roximation with those rectangles, as will ser)e our(ur(ose the best3 6nd not bother any more than we can hel(3 Causethat’s what we do anyway$

$ #nonymous says:

%ey, alid ! Hou hold a mar)elous sca(e )al)e from the montains ofunintuiti)e theorems and corolaries contained in e)ery text+boo$Outside,our memory rests in (eace, and the big (icture awaes our dee( (assionsabout math$Oh, (recious and full of insight sca(e )al)e$

http://www.wolframalpha.com/input/?i=Plot%5B

http://www.wolframalpha.com/input/?i=Plot%5B



<$ kalid says:

6non: 5hans :/$

9$ Tue /guyen says:

5hans alid, Hour articles did hel( me a lot$

2y the way, what software do you use to illustrate exam(les in yourarticles .lie this one/3 5hans

=$ kalid says:

5hans 5ue \ & use 7ower7oint 00D to mae the diagrams$

W$ skrsccrfrk says:

%ey halid (l i am getting a doubt 8 Hou said that innitesimal are the )alues which we cannot measure 8 -y1uestion is can we imgaine innitesimal 33 6ccording to me , humans canthin only of nite )alues !$so whene)er we try to assign a )alue toinnitesimal it woud be of nite digits and tat would be against thedenation of innitesimal !! o according to me due to the limitedsco(e of human brain we can ne)er thin of what )alue wud be ofinnitesimal !! 6m i correct (l 3333

D$ skrsccrfrk says:

%i halid (l i ha) a doubt 33-y 1uestion is can we thin about wat number would be innitesimal 336ccording to me we humans ha) a limited horion of thining and so wecan 'ust thin of nit numbers!!$ o e)en if we assign any )alue toinnitesimal it would be some ite )alue and a )alue smaller than it willstill exist!!$$ o is the limit which we are taling about is the limit of ourbrains to com(rehend such small amounts 333 7l hel( 33

E$ andy012 says:

alid alid

%i alid (l i ha) a doubt 33 -y 1uestion is can we thin about watnumber would be innitesimal 33 6ccording to me we humans ha) alimited horion of thining and so we can 'ust thin of nit numbers!!$o e)en if we assign any )alue to innitesimal it would be some ite

)alue and a )alue smaller than it will still exist!!$$ o is the limit whichwe are taling about is the limit of our brains to com(rehend such smallamounts 333 7l hel( 33

>$ prashant sharma says:



)ery nice, i lo)ed the way, you taught us$ [ery interesting8

<0$ 3ric 4 says:

Ra)e(ost ?D

@egarding your 1uestion, “&f you learn calculus )ia the use of

innitesimals, is it (ossible to then mae the lea( o)er to using limits3", &su((ose it is (ossible for & ha)e .in a way/ done it, though & ne)er new &was learning innitesimals$

& must admit that (rior to reading this (ost & ha)e ne)er e)en heardabout the dened mathematic conce(t of ]innitesimal’$ & also ne)er tooa formal Calculus course$ & originally learned Calc in my 67 (hysics classin high school$ Our teacher .one of the few who truly lo)ed the craft ofteaching and had a (assion for what she did/ had both the constraint of

(utting her 7hysics class on hold to teach Calc to those who ha)e ne)erseen it, and also the freedom that bre)ity (ro)idedA she was free to teachthe idea of calculus without the strict (rocedural rigor that a formal classdrags its (u(il through$ e learned the basic idea of the integral beforethe deri)ati)e, heresy in Calc?0?$ %ere it is ? years later and & can stillhear her )oice saying ]5aing the integral 'ust means add u( a wholebunch of things, and ]taing a dierential element of’ 'ust means cut thething into really teenie weenie chuns$" e learned the idea of aderi)ati)e as slo(e of a function without being gi)en (oints, 'ust one

(oint and an inter)al to the next$ 6fter seeing what ha((ened as theinter)al got smaller we nally )isualied ]slo(e at a (oint’$ Only afterwardwere we shown the ]oIcial’ formula with a limit in it$ & saw it as a(erfectly nice (iece of legal+ee that made the rest of the world ha((y forme to ha)e learned the ]right way’, and & was enormously grateful ourteacher taught us the intuiti)e way$

<?$ 3ric 4 says:

Fascinating article alid8

5his is something new for me$ 6fter reading this (ost & started someresearch on innitesimals, and 1uicly re+aIrmed how )aluable yourcommon sense a((roach is by com(arison to an army of e1uations,lemmas, and theorems$

-y great ]a+ha’ moment was your descri(tion of innitesimals as anotherdimension, similar to the way imaginary numbers are another dimensionto reals$ &n a strange way, that may not be ob)ious at rst, it reminded

me of a conundrum & faced learning the history of (hysics$ &t seems thate)ery time we dene what an ]element’ is +the smallest indi)isiblecom(onent of a thing+ some cle)er lad comes along later and gures outway to brea that ]element’ into something smaller$ 5his means, ofcourse, that the old thing ne)er was a true element, we 'ust thought it



was$ 2ut then what about this new ]element’, how can we now it is thesmallest thing36 re)elation came when & realied that in order to be an ]element’ wedon’t really need it to be true that you can’t brea it a(art, it 'ust meansthat if you do brea it down further then it is no longer the same stu$ 5hus the element is really 'ust the smallest (ossible (iece of a thing%&C% can still be the same thing$ $g$ an ]element’ of water .%O/ canbe broen down, but it is no longer water, 'ust hydrogen and oxygenatoms$ 6n atom can be broen down into (rotons, neutrons, andelectrons, but it is no longer the same stu as the original atom$ 6 littlechun of matter .a su(erstring exhibiting one class of )ibration in ?0dimensions/ can indeed be broen down, it is 'ust no longer matter$ &t isalso not exactly energy, but when the ]stu’ comes bac together in adierent (attern .the su(erstring ha)ing the same )ibration 'ust in adierent dimension/ it a((ears to us as a little chun of energy$&t seems natural to me to tae a cue from the (hysical world tocom(rehend numbers$ hen we loo at an element and it a((ears we’)e]hit the limit’ in terms of breaing it u(, but we can go further it 'ustmeans we ha)e to )iew it in a dierent dimension$ hy then could wenot do the same with numbers3 %ere’s a rational number you can onlybrea it a(art but to a certain extent and no smaller$ & now you mayob'ect and say ]tae that number and di)ide by , it is smaller and stillrational’$ 2ut tae notice of the irrationals, lie s1rt./$ &t does exist,sitting there staring us in the face$ &t is in between rationals$ o how doesthere exist any s(ace between rationals3 %ow can the rationals bebroen down ner than it is (ossible to brea them down3 &magine

thining you understand that atoms are elementary (articles, then thisclown @utherford comes along and ex(erimentally identies this ob'ect.nucleus/ in the middle of an atom$& say the best way forward is to tae as true those things that must betrue and re+e)aluate our (reconcei)ed notions that ha)e (igeon+holed usinto an a((arent (aradox$ &t is diIcult and un+ner)ing$ Hou can beguaranteed you’ll get it wrong a few times before you mae some(rogress, but some (rogress is far better than the certainty of smallerminds$

<$ kalid says:

5hans ric, that’s a really thought+(ro)oing comment$

& thin the element analogy is a(t, we’re able to function at a certainle)el .water molecules/ and while we TcanT go to a dee(er le)el.indi)idual subatomic (articles/ those details (resumably don’t changethe measurements we’re maing at the macro le)el$ &n the same way,innitesimals can bounce around in funny ways but not eect thenumbers one le)el u($ .&$e$, when we switch domains, the innitesimal(art goes to 0$/

5rying to “ll in" the number line with rationals is another great exam(le$e ha)e a smooth continuum on the number line, but the rationals are



so s(arse they’ll ne)er com(lete it8 5here must be another way to get tothose in+between numbers, and it isn’t by di)iding the ones we ha)e intosmaller bits$

<<$ 5ohn 6riggs says:

alidLeibnit and Bewton originated calculus in the ?Dth century, long before

imaginary numbers were around$ Can’t we 'ust say that limits are(aradoxical but they wor and lea)e it at that3



( 0al#ulus (nalo'y: Inte'rals as 3ulti*li#ation


&ntegrals are often described as nding the area under a cur)e$ 5hisdescri(tion is too narrow: itjs lie saying multi(lication exists to nd the area of rectangles$ Finding area is a useful application, but not the (ur(ose$ &ntegrals

hel( us combine numbers when multi(lication canjt$& wish & had a minute with myself in high school calculus:

k7sst8 &ntegrals let us jmulti(lyj changing numbers$ ejre used to k< x 9 4 ?k,but what if one 1uantity is changing3 e canjt multi(ly changing numbers, sowe integrate$

Houjll hear a lot of tal about area ++ area is 'ust one way to )isualiemulti(lication$ 5he ey isnjt the area, itjs the idea of combining 1uantities into

a new result$ e can integrate .kmulti(lyk/ length and width to get (lain oldarea, sure$ 2ut we can integrate s(eed and time to get distance, or length,width and height to get )olume$

hen we want to use regular multi(lication, but canjt, we bring out the bigguns and integrate$ 6rea is 'ust a visualization technique, donjt get too caughtu( in it$ Bow go learn calculus8k

5hatjs my aha moment: integration is a kbetter multi(licationk that wors onthings that change$ Letjs learn to see integrals in this light$

4nderstandin' 3ulti*li#ation

Our understanding of multi(lication changed o)er time:

• ith integers .< x 9/, multi(lication is repeated addition

• ith real numbers .<$? x s1rt.//, multi(lication is scaling

• ith negati)e numbers .+$< T 9$</, multi(lication is ipping and scaling• ith com(lex numbers .< T <i/, multi(lication is rotating and scaling

ejre e)ol)ing towards a general notion of ka((lyingk one number to another,and the (ro(erties we a((ly .re(eated counting, scaling, Yi((ing or rotating/can )ary$ &ntegration is another ste( along this (ath$

4nderstandin' (rea

6rea is a nuanced to(ic$ For today, letjs see area as a visual representation o

o multiplication:





ith each count on a dierent axis, we can ka((ly themk .< a((lied to 9/ andget a result .? s1uare units/$ 5he (ro(erties of each in(ut .length and length/were transferred to the result .s1uare units/$

im(le, right3 ell, it gets tricy$ -ulti(lication can result in knegati)e areak .<x .+9/ 4 +?/, which doesnjt exist$

e understand the gra(h is a representation of multi(lication, and use theanalogy as it ser)es us$ &f e)eryone were blind and we had no diagrams, wecould still multi(ly 'ust ne$ 6rea is 'ust an inter(retation$

3ulti*li#ation Pie#e 5y Pie#e

Bow letjs multi(ly < x 9$=:

hatjs ha((ening3 ell, 9$= isnjt a count, but we can use a k(iece by (iecek

o(eration$ &f <x9 4 < S < S < S <, then< x 9$= 4 < S < S < S < S <x0$= 4 < S < S < S < S ?$= 4 ?<$=

ejre taing < .the )alue/ 9$= times$ 5hat is, we combined < with 9 wholesegments .< x 9 4 ?/ and one (artial segment .< x 0$= 4 ?$=/$



ejre so used to multi(lication that we forget how well it wors$ e can breaa number into units .whole and (artial/, multi(ly each (iece, and add u( theresults$ Botice how we dealt with a fractional (art3 5his is the beginning ofintegration$

The Pro6lem With Num6ers

Bumbers donjt always stay still for us to tally u($ cenarios lie kHou dri)e

<0m(h for < hoursk are for con)enience, not realism$

Formulas lie kdistance 4 s(eed T timek 'ust mas the (roblemA we still need to(lug in static numbers and multi(ly$ o how do we nd the distance we wentwhen our s(eed is changing o)er time3

Des#ri6in' 0han'e

Our rst challenge is describing a changing number$ e canjt 'ust say k-ys(eed changed from 0 to <0m(hk$ &tjs not s(ecic enough: how fast is it

changing3 &s it smooth3

Bow letjs get s(ecic: e)ery second, &jm going twice that in m(h$ 6t ? second,&jm going m(h$ 6t seconds, 9m(h$ < seconds is Wm(h, and so on:

Bow this is a good descri(tion, detailed enough to now my s(eed at anymoment$ 5he formal descri(tion is ks(eed is a function of timek, and means wecan (lug in any time .t/ and nd our s(eed at that moment .ktk m(h/$

.5his doesnjt say why s(eed and time are related$ & could be s(eeding u(because of gra)ity, or a llama (ulling me$ ejre 'ust saying that as timechanges, our s(eed does too$/

o, our multi(lication of kdistance 4 s(eed T timek is (erha(s better written:

where s(eed.t/ is the s(eed at any instant$ &n our case, s(eed.t/ 4 t, so wewrite:



2ut this e1uation still loos weird8 ktk still loos lie a single instant we need to(ic .such as t4< seconds/, which means s(eed.t/ will tae on a single )alue.Wm(h/$ 5hatjs no good$

ith regular multi(lication, we can tae one s(eed and assume it holds for theentire rectangle$ 2ut a changing s(eed re1uires us to combine s(eed and time

(iece+by+(iece .second+by+second/$ 6fter all, each instant could be dierent$

5his is a big (ers(ecti)e shift:

• @egular multi(lication .rectangular/: 5ae the amount of distance mo)edin one second, assume itjs the same for all seconds, and kscale it u(k$

• &ntegration .(iece+by+(iece/: ee time as a series of instants, each withits own s(eed$ 6dd u( the distance mo)ed on a second+by+second basis$

e see that regular multi(lication is a special case of integration, when the

1uantities arenjt changing$

7o! lar'e is a 8*ie#e8?

%ow large is a k(iecek when going (iece by (iece3 6 second3 6 millisecond3 6nanosecond3

Quic answer: mall enough where the )alue loos the same for the entireduration$ e donjt need (erfect (recision$

5he longer answer: Conce(ts lie limits were in)ented to hel( us do (iecewisemulti(lication$ hile useful, they are a solution to a problem and can distractfrom the insight of kcombining thingsk$ &t bothers me that limits are introducedin the )ery start of calculus, before we understand the (roblem they werecreated to address .lie showing someone a seatbelt before theyj)e e)en seena car/$ 5heyjre a useful idea, sure, but Bewton seemed to understand calculus(retty well without them$

What a6out the start and end?

Letjs say wejre looing at an inter)al from < seconds to 9 seconds$

5he s(eed at the start .<x 4 Wm(h/ is dierent from the s(eed at the end.9x 4 Em(h/$ o what )alue do we use when doing ks(eed T timek3

5he answer is that we brea our (ieces into small enough chuns .<$00000 to<$0000? seconds/ until the dierence in s(eed from the start and end of theinter)al doesnjt matter to us$ 6gain, this is a longer discussion, but ktrust mekthat therejs a time (eriod which maes the dierence meaningless$

On a gra(h, imagine each inter)al as a single (oint on the line$ Hou can draw astraight line u( to each s(eed, and your kareak is a collection of lines whichmeasure the multi(lication$

Where is the 8*ie#e8 and !hat is its value?





e(arating a piece from its value was a struggle for me$

6 k(iecek is the inter)al wejre considering .? second, ? millisecond, ?nanosecond/$ 5he k(ositionk is where that second, millisecond, or nanosecondinter)al begins$ 5he )alue is our s(eed at that (osition$

For exam(le, consider the inter)al <$0 to 9$0 seconds:

• kidthk of the (iece of time is ?$0 seconds• 5he (osition .starting time/ is <$0• 5he )alue .s(eed.t// is s(eed.<$0/ 4 W$0m(h

6gain, calculus lets us shrin down the inter)al until we canjt tell the dierencein s(eed from the beginning and end of the inter)al$ ee( your eye on thebigger (icture: we are multi(lying a collection of (ieces$

4nderstandin' Inte'ral Notation

e ha)e a decent idea of k(iecewise multi(licationk but canjt really ex(ress it$

kRistance 4 s(eed.t/ T tk still loos lie a regular e1uation, where t ands(eed.t/ tae on a single )alue$

&n calculus, we write the relationshi( lie this:

• 5he integral sign .s+sha(ed cur)e/ means wejre multi(lying things (iece+

by+(iece and adding them together$

• dt re(resents the (articular k(iecek of time wejre considering$ 5his is

called kdelta tk, and is not kd times tk$

• t re(resents the (osition of dt .if dt is the s(an from <$0+9$0, t is <$0/$

• s(eed.t/ re(resents the )alue wejre multi(lying by .s(eed.<$0/ 4 W$0//

& ha)e a few gri(es with this notation:

• 5he way the letters are used is confusing$ kdtk loos lie kd times tk incontrast with e)ery e1uation youj)e seen (re)iously$

• e write s(eed.t/ T dt, instead of s(eed.tdt/ T dt$ 5he latter maes itclear we are examining ktk at our (articular (iece kdtk, and not someglobal ktk

• Houjll often see , with an implicit dt$ 5his maes it easy to forgetwejre doing a (iece+by+(iece multi(lication of two elements$

&tjs too late to change how integrals are written$ ;ust remember the higher+le)el conce(t of jmulti(lyingj something that changes$

Readin' In 9our 7ead

hen & see



& thin kRistance e1uals s(eed times timek .reading the left+hand side rst/ orkcombine s(eed and time to get distancek .reading the right+hand side rst/$

& mentally translate ks(eed.t/k into s(eed and kdtk into time and it becomes amulti(lication, remembering that s(eed is allowed to change$ 6bstracting

integration lie this hel(s me focus on whatjs ha((ening .kejre combinings(eed and time to get distance8k/ instead of the details of the o(eration$

5onus: ollo!;u* Ideas

&ntegrals are a dee( idea, 'ust lie multi(lication$ Hou might ha)e some follow+u( 1uestions based on this analogy:

• &f integrals multi(ly changing 1uantities, is there something to di)idethem3 .Hes ++ deri)ati)es/

• 6nd do integrals .multi(lication/ and deri)ati)es .di)ision/ cancel3 .Hes,with some ca)eats/$

• Can we re+arrange e1uations from kdistance 4 s(eed T timek to ks(eed 4distanceGtimek3 .Hes$/

• Can we combine se)eral things that change3 .Hes ++ itjs called multi(leintegration/

• Roes the order we combine se)eral things matter3 .#sually not/Once you see integrals as kbetter multi(licationk, youjre on the looout forconce(ts lie kbetter di)isionk, kre(eated integrationk and so on$ ticing with

karea under the cur)ek maes these to(ics seem disconnected$ .5o the mathnerds, seeing karea under the cur)ek and kslo(ek as in)erses ass a lot of astudent/$

Readin' inte'rals

&ntegrals ha)e many uses$ One is to ex(lain that two things are kmulti(liedktogether to (roduce a result$

%erejs how to ex(ress the area of a circle:

ejd lo)e to tae the area of a circle with multi(lication$ 2ut we canjt ++ theheight changes as we go along$ &f we kunrollk the circle, we can see the areacontributed by each (ortion of radius is kradius T circumferencek$ e can writethis relationshi( using the integral abo)e$ .ee the introduction to calculus formore details/$

6nd herejs the integral ex(ressing the idea kmass 4 density T )olumek:





hatjs it saying3 @ho: is the density function ++ telling us how dense a materialis at a certain (osition, r$ d) is the bit of )olume wejre looing at$ o we

multi(ly a little (iece of )olume .d)/ by the density at that (osition and addthem all u( to get mass$

ejd lo)e to multi(ly density and )olume, but if density changes, we need tointegrate$ 5he subscri(t [ means is a shortcut for k)olume integralk, which isreally a tri(le integral for length, width, and height8 5he integral in)ol)es fourkmulti(licationsk: < to nd )olume, and another to multi(ly by density$

e might not sol)e these e1uations, but we can understand what theyjreex(ressing$

n!ard an u*!ard

5odayjs goal isnjt to rigorously understand calculus$ &tjs to ex(and our mentalmodel, and realie therejs another way to combine things: we can add,subtract, multi(ly, di)ide$$$ and integrate$

ee integrals as a better way to multi(ly: calculus will become easier, andyoujll antici(ate conce(ts lie multi(le integrals and the deri)ati)e$ %a((ymath$

1. Matt sas!

Fran:&t might mae more sense if you imagine di)iding u( the area between

the x+axis and the function y4x into many )ertical rectangles, and addingu( their areas$ 5he more rectangles you use, the better thea((roximation of the area$ 5he idea behind integration is that if & di)ideu( the area into innitely many rectangles with innitely small width, nomatter how far you “oom in", you’ll ne)er see the dierence betweenthe “real" sha(e .which is triangular/ and my “a((roximated" sha(e.which is com(osed of many rectangles/$ o it’s reasonable to say thatthe area is in fact the same$ Bow how exactly do we add u( innitelymany innitely small things to get a real number3 #h!alid383

". Kalid sas!

7eter: Cool, &’ll chec it out8

-att: 5hans for the comment8 One of the hardest (arts is getting myhead around the idea of “accurate enough"$

%ere’s how & thin about it$ &n real life, we hit this all the time: 6 screenimage is a grid of (ixels, yet we can see (erfectly smooth sha(es liecur)es, circles, faces, etc$ imilarly, in'et (rinters s(ray a matrix of dots

on a (a(er, but to us it loos lie a smooth unbroen image or line$

5he ey is realiing that the a((roximation is only an a((roximation atthat higher le)el of accuracy \ at the le)el that we wor at, it a((earsindistinguishable from the real thing$ Calculus hel(s formalie some of



these ideas with limits .informally, two numbers that ha)e a dierenceless than our error margin a((ear the same to us/$

#nfortunately, we don’t really tal about this much, and we sometimessay numbers are e1ual, and sometimes say they aren’t$ 5here’s a notionof innitely small numbers which maes this clearer, and is used in(hysics$ 5hat is, you can tal about how innitely small numbers interactwith each other, and with innity, to gi)e numbers we can detect$ 6 (oor

analogy but it may wor: 6 ca)eman could (robably not concei)e of anindi)idual atom, or the gargantuan 6)ogadro’s number .W x ?0</, butwhen this tiny (article and huge number combine we can get somethingwe can detect$

5he ey is writing this idea down in the language of math: numbers thatare too small and too large for us to detect can interact to gi)e usnumbers we can wor with$

#. ram sas!

%i alid,

your ex(lanations of the underlying conce(ts of mathematics do bringthe sub'ect at a democratic la)el, a le)el on which (eo(le communicate,collaborate and wor towards maing the sub'ect useful for greaternumber of (eo(le$

Bow coming to the sub'ect, cud i say dat, dierentiation is in)erse ofintegration3

and going by dat if i ha)e to a((ly dierentiation, let’s say on theexam(le of circle, all i ha)e to do is to run a (laybac, i$e$ to (eel of allthose tiny rings .or, in other words/ thus di)ide the circle into the tiniest(ossible rings$Once i m done (eeling! & would measure this ring, to see the result ofdierentiation a((lication, which should be T(iTr$25, would not & b a((lying multi(lication again, to measure that tiniestring, i$e$ nding the area of that ring, (iTr3

$. Arbie Samong sas!

One way & understood the basic integral notation is with my crudeunderstanding of sets and functional (rogramming$ #sing the gi)enexam(le abo)e .s(eed and time/:

5here’s an im(lied set of )alues of time, and we tae a (iece of it or amember of that set$ 5hat becomes the slice of time$ e then a((ly it to afunction of time that is s(eed$ 5his results in another set whose membersare results from each function result using the said function gi)en a slice.or element of im(lied set of time/ as in(ut$ Finally, we a((ly the

]integrate’ o(erator, or (robably a ]ma(’ to the ]integrate’ functionA or, to(ut it sim(ly, use the integrate function on all members of the resultingset to return the integrated )alue$

http://phektus.multiply.com/

http://phektus.multiply.com/



or something lie:ma(.]integrate’, .get(eed.t/ N t M4 timeslices//

%. bill sas!

note to extend idea by alid abo)e:

circumference of a cirle .a ?d distance/ 4 (i r

area of a circle .a d area/ 4 (i r

.note the integral Gderi)ati)e of each other$/

surface area of a s(here 4 9 (i r )olume of a s(here 4 9G< (i r <

try this with s1uares and cubes! hint, base it on the shortest distancefrom the centre to a side$

how cool is that8



0al#ulus: 5uildin' Intuition or the Derivative


%ow do you wish the deri)ati)e was ex(lained to you3 %erejs my tae$

7sst8 5he deri)ati)e is the heart of calculus, buried inside this denition:

2ut what does it mean3

Letjs say & ga)e you a magic news(a(er that listed the daily stoc maretchanges for the next few years .S? -onday, + 5uesday$$$/$ hat could youdo3

ell, youjd a((ly the changes one+by+one, (lot out future (rices, and buy low Gsell high to build your em(ire$ Hou could e)en hire away the moneys who

currently throw darts at news(a(ers$

Others call the deri)ati)e kthe slo(e of a functionk ++ itjs so bland8 Lie ha)ingthe magic news(a(er, the deri)ati)e is a crystal ball that lets you see how a(attern will (lay out$ Hou can (lot the (astG(resentGfuture, ndminimumsGmaximums, and yes, sta your simian worforce to (ic stocs$

te( away from the gnarly e1uation$ 1uations exist to con)ey ideas:understand the idea, not the grammar$

erivatives create a perfect model of change from an imperfect

guess.

5his result came o)er thousands of years of thining, from 6rchimedes toBewton$ Letjs loo at the analogies behind it$

We all live in a shiny #ontinuum

&nnity is a constant source of (aradoxes .kheadachesk/:

• 6 line is made u( of (oints3 Sure.

• o therejs an innite number of (oints on a line3 ep.

• %ow do you cross a room when therejs an innite number of (oints to)isit3 !"ee# thanks $eno %.

6nd yet, we mo)e$ -y intuition is to ght innity with innity$ ure, therejsinnity (oints between 0 and ?$ 2ut & mo)e two infnities of (oints (er second.somehow8/ and & cross the ga( in half a second$

Ristance has innite (oints, motion is (ossible, therefore motion is in terms of

kinnities of (oints (er secondk$

&nstead of thining of dierences .k%ow far to the next (oint3k/ we cancom(are rates .k%ow fast are you mo)ing through this continuum3k/$

http://en.wikipedia.org/wiki/Zeno's_paradoxes

http://en.wikipedia.org/wiki/Zeno's_paradoxes



&tjs strange, but you can see ?0G= as k& need to tra)el ?0 jinnitiesj in =segments of time$ 5o do this, & tra)el jinnitiesj for each unit of timek$

#nalogy% )ee division as a rate of motion through a continuum of

points

What<s ater /ero?

6nother brain+buster: hat number comes after ero3 $0?3 $000?3

%rm$ 6nything you can name, & can name smaller .&jll 'ust hal)e your number$$$nyah8/$

)en though we canjt calculate the number after ero, it must be there, right3Lie demons of yore, itjs the knumber that cannot be written, lest ye besmittenk$

Call the ga( to the next number kdxk$ & donjt now exactly how big it is, but itjs

there8

#nalogy% d! is a 78ump7 to the ne!t number in the continuum.

3easurements de*end on the instrument

5he deri)ati)e (redicts change$ O, how do we measure s(eed .change indistance/3

OIcer: Ro you now how fast you were going3

Rri)er: & ha)e no idea$

OIcer: >= miles (er hour$

Rri)er: 2ut & ha)enjt been dri)ing for an hour8

e clearly donjt need a kfull hourk to measure your s(eed$ e can tae abefore+and+after measurement .o)er ? second, letjs say/ and get yourinstantaneous s(eed$ &f you mo)ed ?90 feet in one second, youjre going

>=m(h$ im(le, right3

Bot exactly$ &magine a )ideo camera (ointed at Clar ent .u(ermanjs alter+ego/$ 5he camera records 9 (icturesGsec .90ms (er (hoto/ and Clar seemsstill$ On a second+by+second basis, hejs not mo)ing, and his s(eed is 0m(h$

rong again8 2etween each (hoto, within that 90ms, Clar changes tou(erman, sol)es crimes, and returns to his chair for a nice (hoto$ emeasured 0m(h but hejs really mo)ing ++ he goes too fast for our instruments8

#nalogy% $ike a camera watching )uperman+ the speed we measure

depends on the instrument9

Runnin' the Treadmill



ejre nearing the chewy, slightly tangy center of the deri)ati)e$ e needbefore+and+after measurements to detect change, but our measurementscould be Yawed$

&magine a shirtless anta on a treadmill .go on, &jll wait/$ ejre going tomeasure his heart rate in a stress test: we attach doens of hea)y, coldelectrodes and get him 'ogging$

anta hus, he (us, and his heart rate shoots to ?>0 beats (er minute$ 5hatmust be his kunder stressk heart rate, correct3

Bo(e$ ee, the )ery (resence of stern scientists and cold electrodes increasedhis heart rate8 e measured ?>0b(m, but who nows what wejd see if theelectrodes werenjt there8 Of course, if the electrodes werenjt there, wewouldnjt ha)e a measurement$

hat to do3 ell, loo at the system:

• measurement 4 actual amount S measurement eect6h$ 6fter lots of studies, we may nd kOh, each electrode adds ?0b(m to theheartratek$ e mae the measurement .im(erfect guess of ?>0/ and remo)ethe eect of electrodes .k(erfect estimatek/$

#nalogy% :emove the 7electrode e;ect7 after making your

measurement

2y the way, the kelectrode eectk shows u( e)erywhere$ @esearch studies

ha)e the%awthorne ect where (eo(le change their beha)ior because theyare being studied$ êe, it seems e)eryone we scrutinie stics to their diet8

4nderstandin' the derivative

6rmed with these insights, we can see how the deri)ati)e models change:

tart with some system to study, f.x/:

?$ Change by the smallest amount (ossible .dx/$ êt the before+and+after dierence: f.x S dx/ + f.x/

http://en.wikipedia.org/wiki/Hawthorne_effect

http://en.wikipedia.org/wiki/Hawthorne_effect



<$ e donjt now exactly how small kdxk is, and we donjt care: get the rateof motionthrough the continuum: Uf.x S dx/ + f.x/V G dx

9$ 5his rate, howe)er small, has some error .our cameras are too slow8/$7redict what ha((ens if the measurement were (erfect, if dx wasnjtthere$

5he magicjs in the nal ste(: how do we remo)e the electrodes3 e ha)e twoa((roaches:

• Limits: what ha((ens when dx shrins to nothingness, beyond any errormargin3

• &nnitesimals: hat if dx is a tiny number, undetectable in our numbersystem3

2oth are ways to formalie the notion of k%ow do we throw away dx when itjsnot needed3k$

-y (et (ee)e: Limits are a modern formalism, they didnjt exist in Bewtonjstime$ 5hey hel( mae dx disa((ear kcleanlyk$ 2ut teaching them before the

deri)ati)e is lie showing a steering wheel without a car8 &tjs a tool to hel( thederi)ati)e wor, not something to be studied in a )acuum$

(n )xam*le: +x, = x>

Letjs shae loose the cobwebs with an exam(le$ %ow does the function f.x/ 4x change as we mo)e through the continuum3

Bote the dierence in the last e1uations:

• One has the error built in .dx/• 5he other has the ktruek change, where dx 4 0 .we assume our

measurements ha)e no eect on the outcome/ 5ime for real numbers$ %erejs the )alues for f.x/ 4 x, with inter)als of dx 4?:

• ?, 9, >, ?W, =, <W, 9>, W9$$$ 5he absolute change between each result is:

•

?, <, =, D, >, ??, ?<, ?=$$$.%ere, the absolute change is the ks(eedk between each ste(, where theinter)al is ?/

Consider the 'um( from x4 to x4< .< + 4 =/$ hat is k=k made of3

http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/






• -easured rate 4 6ctual @ate S rror• = 4 x S dx• = 4 ./ S ?

ure, we measured a k= units mo)ed (er secondk because we went from 9 to >in one inter)al$ 2ut our instruments tric us8 9 units of s(eed came from thereal change, and ? unit was due to shoddy instruments .?$0 is a large 'um(,no3/$

&f we restrict oursel)es to integers, = is the (erfect s(eed measurement from 9to >$ 5herejs no kerrork in assuming dx 4 ? because thatjs the true inter)albetween neighboring (oints$

2ut in the real world, measurements e)ery ?$0 seconds is too slow$ hat if ourdx was 0$?3 hat s(eed would we measure at x43

ell, we examine the change from x4 to x4$?:

• $? + 4 0$9?@emember, 0$9? is what we changed in an inter)al of 0$?$ Our s(eed+(er+unitis 0$9? G $? 4 9$?$ 6nd again we ha)e:

• -easured rate 4 6ctual @ate S rror• 9$? 4 x S dx

&nteresting$ ith dx40$?, the measured and actual rates are close .9$? to 9,$= error/$ hen dx4?, the rates are (retty dierent .= to 9, = error/$

Following the (attern, we see that throwing out the electrodes .letting dx40/re)eals the true rate of x$

&n (lain nglish: e analyed how f.x/ 4 x changes, found an kim(erfectkmeasurement of x S dx, and deduced a k(erfectk model of change as x$

The derivative as 8#ontinuous division8

& see the integral as better multi(lication, where you can a((ly a changing1uantity to another$

5he deri)ati)e is kbetter di)isionk, where you get the s(eed through thecontinuum at e)ery instant$ omething lie ?0G= 4 says kyou ha)e a constants(eed of through the continuumk$

hen your s(eed changes as you go, you need to describe your s(eed at eachinstant$ 5hatjs the deri)ati)e$

&f you a((ly this changing s(eed to each instant .tae the integral of thederi)ati)e/, you recreate the original beha)ior, 'ust lie a((lying the daily stoc

maret changes to recreate the full (rice history$ 2ut this is a big to(ic foranother day$

@ot#ha: The 3any meanin's o 8Derivative8

Houjll see kderi)ati)ek in many contexts:





• k5he deri)ati)e of x is xk means k6t e)ery (oint, we are changing by

a s(eed of x .twice the current x+(osition/k$ .êneral formula forchange/

• k5he deri)ati)e is 99k means k6t our current location, our rate of changeis 99$k hen f.x/ 4 x, at x4 wejre changing at 99 .(ecic rate ofchange/$

• k5he deri)ati)e is dxk may refer to the tiny, hy(othetical 'um( to the next

(osition$ 5echnically, dx is the kdierentialk but the terms get mixed u($ometimes (eo(le will say kderi)ati)e of xk and mean dx$

@ot#ha: ur models may not 6e *ere#t

e found the k(erfectk model by maing a measurement and im(ro)ing it$ometimes, this isnjt good enough ++ wejre (redicting what would ha((en if dxwasnjt there, but added dx to get our initial guess8

ome ill+beha)ed functions defy the (rediction: therejs a dierence betweenremo)ing dx with the limit and what actually ha((ens at that instant$ 5heseare called kdiscontinuousk functions, which is essentially kcannot be modeledwith limitsk$ 6s you can guess, the deri)ati)e doesnjt wor on them becausewe canjt actually (redict their beha)ior$

Riscontinuous functions are rare in (ractice, and often exist as kôtcha8k test1uestions .kOh, you tried to tae the deri)ati)e of a discontinuous function,you failk/$ @ealie the theoretical limitation of deri)ati)es, and then realie

their (ractical use in measuring e)ery natural (henomena$ Bearly e)eryfunction youjll see .sine, cosine, e, (olynomials, etc$/ is continuous$

@ot#ha: Inte'ration doesn<t really exist

5he relationshi( between deri)ati)es, integrals and anti+deri)ati)es is nuanced.and & got it wrong originally/$ %erejs a meta(hor$ tart with a (late, yourfunction to examine:

• Rierentiation is breaing the (late into shards$ 5here is a s(ecic

(rocedure: tae a dierence, nd a rate of change, then assume dx isnjtthere$

• &ntegration is weighing the shards: your original function was kthisk big$ 5herejs a (rocedure, cumulati)e addition, but it doesnjt tell you what the plate looked like$

• 6nti+dierentiation is guring out the original sha(e of the (late from the(ile of shards$

5herejs no algorithm to nd the anti+deri)ati)eA we ha)e to guess$ e mae aloou( table with a bunch of nown deri)ati)es .original (late 4 (ile of

shards/ and loo at our existing (ile to see if itjs similar$ kLetjs nd the integralof ?0x$ ell, it loos lie x is the deri)ati)e of x$ o$$$ scribble scribble$$$?0x is the deri)ati)e of =x$k$



Finding deri)ati)es is mechanicsA nding anti+deri)ati)es is an art$ ometimeswe get stuc: we tae the changes, a((ly them (iece by (iece, andmechanically reconstruct a (attern$ &t might not be the krealk original (late,but is good enough to wor with$

6nother subtlety: arenjt the integral and anti+deri)ati)e the same3 .5hatjs what& originally thought/

Hes, but this isnjt ob)ious: itjs the fundamental theorem of calculus8 .&tjs liesaying k6renjt a S b and c the same3 Hes, but this isnjt ob)ious: itjsthe 7ythagorean theorem8k/$ 5hans to ;oshua *ucer for hel(ing sort me out$

Readin' math

-ath is a language, and & want to kreadk calculus .not krecitek calculus, i$e$ liewe can recite medie)al êrman hymns/$ & need the message behind thedenitions$

-y biggest aha8 was realiing the transient role of dx: it maes ameasurement, and is remo)ed to mae a (erfect model$ LimitsGinnitesimalsare a formalism, we canjt get caught u( in them$ Bewton seemed to do owithout them$

6rmed with these analogies, other math 1uestions become interesting:

• %ow do we measure dierent sies of innity3 .&n some sense theyjre allkinnitek, in other senses the range .0,?/ is smaller than .0,//

•

hat are the real rules about maing kdx go awayk3 .%ow doinnitesimals and limits really wor3/• %ow do we describe numbers without writing them down3 k5he next

number after 0k is the beginnings of analysis .which & want to learn/$• AK sas!

& 'ust wanted to let you now that & really a((reciate the eort you (utinto this$ & only disco)ered this website a few days ago, and &’)e beenha)ing a blast reading all those intuiti)e a((roaches88

Hou should consider writing an elementary and highschool boo ofmathematics, as well as teaching on hansacademy

7lease ee( this Yowing and if there’s any way we, theaudience, can su((ort you, (lease do mention how8

• kalid sas!

6: 5hans for the comment \ really a((reciate the su((ort8 &’mactually looing at ways to hel( ta( into the community \ one idea isgetting a little section after each (ost to share the analogies that wored.or 1uestions that are still outstanding/$ &’d lo)e certain articles .lie the



one on e, for exam(le/ to become a li)ing reference about “hat actuallymade it clic"$ ii(edia is great for strict denitions, han and othersfor detailed tutorials G (ractice (roblems, and &’d lie to contribute aha8moments .i$e$ the last ste( that turned the light bulb on/$ Renitelysomething &’m looing to de)elo(, &’ll be (osting on this soon 4/$

• Pat Shaughnessy sas!

hat a great ex(lanation8 &t too me bac to my days as a 7hysics ma'or

in college P only & wish & had this ex(lanation bac then

• zaine_ridling sas!

ow, now that’s (ower$ 5aes a brilliant mind to brea com(lexity down,maing this one of the best sites online8

• kalid sas!

7at: 5hans, glad you lied it8 Oh man, how & wish & could go bac intime and gi)e myself some tutorials :/$

*aine: 5hans, & really a((reciate it8

• kalid sas!

;oshua *ucer emailed me after the comment form ate his re(ly, (astingbelow:

6((arently my long comment on your recent (ost got eaten somewherealong the line$ Rarn$

6nyway, my (oint was that you really misre(resent integrals$ 5hey’reeasier than deri)ati)es, not harder$ &t’s antideri)ati)es that aretough, and although the fundamental theorem says they’re the same asintegrals, the whole (oint of the theorem is that there’s something

meaningful to say there8 ell, actually, antideri)ati)es aren’treally tough, it’s 'ust that we’re (icy about wanting to write themin terms of certain inds of functions, which is your “brea lots of (lates" analogy$ e now exactly how the (ieces were made, so we can 'ust glue them bac together$ 5he hard (art is recogniing the brandname of the (late when we’re done, not reassembling the (late$

Hou also seem inconsistent about saying in your intro that you can usethe rate of change to reconstruct the future (rices, and then latersaying that (utting the (ieces bac together is hard$ &ntegrals are,as you say “better multi(lication" \ you 'ust ha)e to multi(ly andadd$

5here is lots and lots of good stu in the (ost too, of course8 &(articularly lo)e the idea of the deri)ati)e as an inference of what

http://patshaughnessy.net/


https://plus.google.com/108871877301789098084/posts


https://plus.google.com/108871877301789098084/posts



the (erfect tool would measure, from a((roximations using im(erfecttools$ & don’t thin & e)er thought of it as a tool in 1uite thatsense, and it’s a useful thing$ & mean, & ha)e thought of thederi)ati)e at a (oint as a local (ro(erty, and the deri)ati)e as ano(erator that ma(s functions to functions, but this feels more lie acali(er that is o(en to some nite amount and then you’re reducingthat amount to see what’s going onA it ca(tures more of the limit(rocess in there$

Oh, one more note: Oddly, &’m totally comfortable with the idea of dx4 the next number right after 0, or the 'um( between “ad'acent" realnumbers, but & am really bothered by the analogy of di)iding =innities by innities of (oints to get =G$

4444

%i ;oshua,

^reat feedbac \ & thin the nuances of integrals )s$ anti+deri)ati)eswere (re)iously lost on me :/$ 6fter a little reading.htt(:GGmathforum$orgGlibraryGdrmathG)iewG=<D==$html/ & thin &’m u( tos(eed:

T &ntegration is literally the (rocess of gluing the (ieces together.mechanical, nding the sum of many (roducts/T 6nti+deri)ati)es are the function whose deri)ati)e is f .i$e$, the “brand"as you say/

5he essence of the F5OC .which &’)e (re)iously missed/ is that &ntegralsare Tcom(utableT from anti+deri)ati)es, which is (retty amaing$ Literallygluing (ieces isn’t hard, but saying “this reconstructed (late is an &eaFur'en" is the tricy (art .realiing what function, easily dened, wouldcreate such an integral/$

“!the idea of the deri)ati)e as an inference of what the (erfect toolwould measure, from a((roximations using im(erfect tools" \ & lo)e thisconcise descri(tion, that’s exactly it$ Hes, in this context it’s lie a littlecali(er which is (rodding, only to disa((ear again to hel( gure out a

greater result$ 5he o(erator and local (ro(erty G slo(e inter(retations areother ones to switch between$ hen writing this article, & was ruminatingon the (ur(ose of limits, which always bothered me because they wereignored so often in engineering classes .e)en though the deri)ati)ewasn’t8/$ &n this case, limits were mathematical scaolding$

5he = )s innities doesn’t 1uite sit right with me either \ it’s my gutscreaming for there to be “some" way to mo)e through an innitude of(oints$ -y analysis nowledge is )ery limited, but (erha(s something liea Lebesgue measure could ca(ture this notion .that 0+= is a larger innite

range than 0+/3 .htt(:GGen$wii(edia$orgGwiiGLebesguemeasure/$

@eally a((reciate the discussion, & lo)e rening these thoughts8 &’llu(date the article soon, as & get my intuitions in order$

http://mathforum.org/library/drmath/view/53755.html

http://en.wikipedia.org/wiki/Lebesgue_measure

http://mathforum.org/library/drmath/view/53755.html

http://en.wikipedia.org/wiki/Lebesgue_measure



444

;osh: & thin a better analogy is this:

&ntegration is (iling all the shards on a scale and reading the total$6ntidierentiation is (utting the shards carefully bac together inexactly the right order and recogniing the (late$

• Ogbuka chukwuma sas!

& dont understand cumulati)e fren1uency so well$(ls hel(

• BASSMAN sas!

i need more details on how to sol)e the (artial fractions and integrations$

• Asmaul Houe sas!

&t is realy interesting$ & ha)e en'oy it $$nd lear a lot$ 5oday & unmderstoodhat is Reri)ati)e 3 6ctually & am searching this but gi)e us$ 5han you somuch$ 7lease gi)e the this o((ortunity to learn math$

• !ohn !ordan sas!

halid,

Long+time lurer, rst+time (oster$ Firstly, 'ust wanted to say congrats onall your wor here, really im(ressi)e$ 5his is my fa)ourite maths site onthe webA & see the seeds of an educational re)olution here$ @eminds meof the time & got a weighty boo "6((lying -aths in the Chemical and2iological ciences"$$& was ho(ing for an interesting no)el, what & got wasalmost (ure grammar, i$e$ & was looing for semantics but all & got wassyntax$ Hour articles ex(lain the meaning, i$e$ utility, of these abstractnotions$ Hour com(lex numbers article hel(ed sol)ed the riddle of how"imaginary numbers" could be use in the real world, so thans8

Lie the .modied8/ analogy for the distinction of integral and anti+deri)ati)e, which was yet another one of those esoteric relationshi(s thatwas ne)er ex(lored in high schoolA are you going to amend the originalarticle3

@egards,

;ohn

• kalid sas!

2assman, Ogbua: &’ll tae those as suggestions for future to(ics,thans$

6smaul: ^lad it was hel(eful8

http://facebook.com/oguama.chinedu

http://facebook.com/oguama.chinedu



;ohn: 5hans for the note, really a((reciate it8 & hear you, so manymath ex(lanations 'ust focus on the grammar, lie the lifeless languageclasses that nobody e)er seems to learn from .contrasted with learning alanguage by actually being immersed in it and s(eaing it, )s$ trying tocrunch through the rules lie a com(uter/$

&’m going to u(date the article right now with the new integralGanti+deri)ati)e analogy$ 5hans again for (osting8

• Anonymous sas!

5his came at a (retty good time for me since its (ublication coincidedwith my own autodidactic 'ourney through math8 & was fresh intocalcGderri)ati)es when this came and & simmed through, initially gettingabout half of it$ 5hen while waling my dogs today & got dee( intothining about really understanding derri)ati)es after a few (lug andchug sessions, and & begun recalling what you had written .es(eciallyregarding the “actual rateSerror" (art/ and the su(erman analogy$

&n retros(ect it was a good thing & was waling in the barren woodsbecause the unconcious “OOOOOOOOOO%%%8" of my aha moment wasso loud$ -y dogs didn’t seem to care though, they were busy (oo(ingand such$

5han you, than you, than you8

• Kalid sas!

6nonymous: 6wesome, &’m glad the aha8 came :/$ &’m (lanning onmaing some changes to the site to hel( share and discuss the indi)idualaha8 moments, really a((reciate the note8

• Sebastian Maruez sas!

halid,

5his is great8 Reri)ati)es were always out of focus to me but this ishel(ing clear things u($

ebastian

• kalid sas!

ebastian: 5hans, glad it hel(ed :/$

• "ust a kid sas!

%ey alid, another great article82ut & noticed somethingA couldn’t you 'ust, instead of e)en doing all theother math, 'ust tae the ex(onent of the original number, multi(ly thenumber in front of it and then minus one from the ex(onent3 if you didn’tget that, here’s what & mean: the deri)ati)e of x4T?.x/.+?/, whiche1uates to x$ &t also wors in the re)erse of nding the original number

http://studiamirabilium.com/

http://studiamirabilium.com/



using the deri)ati)e: ?0x?A?0x.?S?/4?0xA .?0G/x4=x$hould & ha)e (ut this here, or on your new aha moents and F6Q thingy3 5han you, ;ust a id$

• kalid sas!

'ust a id: 5hans for the comment8 For (osting, either method is ne8

5he aha8GF6Q thingy is a way to ha)e longer discussions, since regularword(ress comments don’t ha)e threading .and the discussions could gethard to follow/$

Hour shortcut denitely wors .tae the ex(onent, decrease by one/$ &t’sneat to see why this wors: if we’re taing the deri)ati)e of xn .x raisedto some (ower/, we mae a model lie this:

U.x S dx/n P xn V G dx

4 U.xn S omething T x.n+?/ T dx S omething T x.n+/ T dx S!/ P xn V G dx

4 omething T x.n+?/ S omething T x.n+/ T dx

-ost of the other terms go away because we want dx to be ero .i$e$,assume a (erfect model/$ e’re left with

omething T x.n+?/

6nd what is the “omething3 ell, it’s the number “n" .this is due to the2inomial 5heorem/, more detailshere: htt(:GGbetterex(lained$comGarticlesGhow+to+understand+combinations+using+multi(licationG

2ut ye(, you got it \ there’s a shortcut to gure out how the deri)ati)eof a regular (olynomial .xn/ will beha)e :/$

• wm tanksley sas!

5his is a fair ex(lanation of the theory behind deri)ati)esA but & lie howilberger ex(lains and moti)ates tangent cur)es .which are directly andsim(ly related to deri)ati)es/$ Bot only does he BO5 use the idea of “dx".which doesn’t actually exist in any system of numbers beyond theintegers, since there is no uni1ue number that is closest to ero/, but hewinds u( dening the theory so that it wors on arbitrary algebraiccur)es .not only functions/$

Chec it out \ loo at his .n'wilberger’s/ -ath Foundations series on Hou5ube$ -ost (eo(le reading here will be able to si( to something lie

the e(isode on doing calculus on the unit circle, but don’t ex(ect tounderstand [@H5%&B^ if you do that$ 5he interesting thing is that hedenes this without using limits at allA the essential (oint is that he uses“the nth degree (olynomial that best a((roximates the surface at that(oint" .of course, this is the 5aylor ex(ansion at that (oint/$

http://betterexplained.com/articles/how-to-understand-combinations-using-multiplication/






+m

• kalid sas!

wm: 5hans for the (ointer8 &’ll chec it out$

• S#$%% %&A'N$N( sas!

@eally great stu$ -athematics is the foundation of all science andscience is the com(ass to hel( us na)igate the uni)erse$ ee( u( thegood wor$ [ery much a((reciated$

• kalid sas!

till learning: 5hans \ really a((reciate the encouragement8

• Sudar sas!

%i halid,^reat article$ & ha)e always been fascinated by calculus and alwayswanted to deci(her the true meaning of deri)ati)e$ Hour article gi)es mea great insight$ %owe)er & would beg you to clarify the followingconfusion that has arisen$e all now that deri)ati)e of H 4 is x$ when you calculate )aluesof y for x4 and <, you get y 4 9 and > res(ecti)ely$ 5he change in yhere is >+9 4 =$ %owe)er if & substitute x4 in the deri)ati)e functiondyGdx it gi)es me x 4 9$ you showed us why this dierence exists$ &t is

because of the dx factor .hoddy instrument/$ 2ut the reality is that ychanged by = units when x changed from to <$ 6re you saying thatdyGdx or deri)ati)e is not here to calculate rate of change for such largechanges and if you use it for large changes results are inaccurate$ Roesthat mean that dyGdx can only be used to calculate )ery small changes$arlier & thought if you want to nd how a function f.x/ is changing w$r$t xbetween )alues without substituting the )alues, 'ust calculate thederi)ati)e and substitute x but it seems & was wrong3

6lso & didn’t understand when you say

5he deri)ati)e is 99_ means “6t our current location, our rate of change is99$"Change is a relati)e term$ %ow can there be a change at a currentlocation$ &t has always got to be between two locations$

• here sas!

%eya, & 'ust ho((ed o)er to your web+site through tumble#(on$ Botsomthing & would ty(ically browse, but & en'oyed your thoughts none theless$ 5han you for maing some thing worth reading through$


udar, & understand your confusion$

http://www.yahoo.com/d7ffa485dc26aeb419f767a5e1a90c5d48b8ad2f

http://www.yahoo.com/d7ffa485dc26aeb419f767a5e1a90c5d48b8ad2f



Hour last (aragra(h is the most im(ortant$ 5he dierential -&^%5be understood as the rate of change at a single (oint, but it also might beconfusing if you thin of it lie that$ &t’s im(ortant that you see that thedierential is not the same thing as thedierence $ 5he dierence re1uires twodierent (ointA the dierential taes only one (oint$

6nother way to thin of the dierential is that it’s the slo(e of the line

that best a((roximates the cur)e at that (oint$ 5his denition gi)es yousome sur(rising algebraic (ower \ and it also suggests some othero(erations, such as the “linear subderi)ati)e", which is the line thatbest a((roximates the cur)e, and from that the “1uadratic subderi)ati)e".and so on/$ 5hese are )ery cleanly dened o(erations on algebraiccur)es, and re1uire only algebra, no analysis or limits$

+m

• Nikhil Panikkar sas!

5he deri)ati)e is a conce(t that relates a continuous (ro(erty. a)eragechange / to a discrete one .instantaneous change/$ )en if one had a(erfect instrument to measure instantaneous change, one wouldn’t beable to P because of our conce(tion .and conse1uent denition/ of s(eed$

5o (ro(erly understand a deri)ati)e you would need the conce(t of alimit$ Limits are to calculus what de 2roglie’s wa)elength is to 1uantum(hysics .it bridges the ga( between wa)e and (article (ro(erties Pbetween discrete and continuous/

6lso limit is not a way to mae the deri)ati)e wor$ &t is 'ust onea((lication of limit$&n (hysics and signal theory certain functions are so com(licated that youha)e to use limits to dene them P we call them generalied functions$

5he deri)ati)e can be taught without limits . since the deri)ati)e dealswith rate of change / but if you are introducing intesimals then i thinyou could ha)e introduced the limit too$


Bihil, you do not need limits or innitesimals to (ro(erly understand thederi)ati)e$ 5he deri)ati)e is suIciently understood as the slo(e of theline tangent to a cur)e at a (oint$ 5his geometric understanding does notin)oe limits or innitesimals$ Hou can add in limits to this denition tohandle (iecewise continuous cur)es, but as+is this denition can handlearbitrary cur)es, rather than being limited to functions$

&f one is learning general calculus then innitesimals are essentialA but ifone is learning the deri)ati)e they are not, and therefore no limits areneeded$ &)erson actually wrote a Calculus text without using limits, andhe only used innitesimals informally$ &t’s a)ailable onlineat htt(:GGwww$'software$comG'wiiG2oos $ 6side from that oddity, the textis notable for its com(utational focus and for its treatment of some

http://n.a/

http://www.jsoftware.com/jwiki/Books

http://n.a/

http://www.jsoftware.com/jwiki/Books



ad)anced theoretical to(ics such as fractional integrals .ii(edia callsthis the “dierintegral"/$

+m


" Bihil, you do not need limits or innitesimals to (ro(erly understand

the deri)ati)e$ “

& ne)er said that you need limits to understand the deri)ati)e$ @ead thelast (aragra(h of my comment$

&n your ex(lanation, you tal about innity and the continuum$ hat &was saying is P that the conce(tual lea( from there to that of a limit is)ery small$ o there is no need to a)oid the conce(t of a limit$

One doesn’t need the e(silon P delta denition to introduce the conce(t

of a limit$


5ansley , you say that the conce(t of a limit restricts the denition of aderi)ati)e to functions and maes it ina((licable to arbitrary cur)es$ & didnot follow this$ Could you elaborate 3


Bihail, you said “5o (ro(erly understand a deri)ati)e you would needthe conce(t of a limit$" 5hat’s the sentence & was seeing to correct$ Hourlast (aragra(h claims that you don’t need limits but then im(lies that youneed innitesimals, and this is also something & dis(uted \ but assumingyour (ost is not self+contradictory, your claim would im(ly that you needinnitesimals in order to understand deri)ati)es im(ro(erly, and if youadd limits you can understand them (ro(erly, and there’s no other wayto e)en begin to understand deri)ati)es$

& contradicted this claim by saying that there is another way ofunderstanding the deri)ati)e: the geometric denition$ &t re1uires nolimits, no innitesimals, no continuum$ &t wors not only on smoothfunctions, but also on arbitrary smooth cur)es$ .&’ll ex(lain in my nextcomment$/

Hou said that & mentioned innity and the continuum$ & didn’t mentioneitherA the only (lace & can nd those conce(ts is in the original (ost$ &would also disagree entirely with the original (ost’s tae on themA forexam(le, there is not only one innitesimal, rather, there are an unnownnumber of them, so you cannot iterate through the continuum by adding 'ust any innitesimal to a number .if you do this, you’ll miss (oints on thecontinuum/$

On the other hand, & do agree that one does not need e(silon+delta tointroduce limitsA one can introduce limits for other (ur(oses$ Or one can

http://n.a/

http://n.a/

http://n.a/

http://n.a/



introduce limits for their own sae$ 2ut this has nothing to do with theto(ic of understanding the deri)ati)e$

+m


“Hou said that & mentioned innity and the continuum$ & didn’t mention

eitherA the only (lace & can nd those conce(ts is in the original (ost$"

Hes & was referring to the original (ost$ & 'ust stumbled u(on this articlewhile doing a google search$ and assumed that you were its author$ Bow &ha)e ex(lored the site, and disco)ered it was alid$

& was thining about your comment P “5he deri)ati)e is suIcientlyunderstood as the slo(e of the line tangent to a cur)e at a (oint$"$ & ha)esome doubts regarding this denition$ 2ut & will rst wait for yourcomment on the a((lication of the deri)ati)e to arbitrary cur)es and how

the limit restricts this a((licability$


.Bote: & ho(e the La5e below wors$ & wish there were a (re)iewmode!/

& claimed that using limits and innitesimals to dene the deri)ati)e ledto restricting oursel)es to functions, while using the geometric denitionof the deri)ati)e allowed arbitrary cur)es rather than only functions$

.5here are other ad)antagesA for exam(le, using the geometric denitionallows you to reason about deri)ati)es of cur)es o)er arbitrary eldsrather than only the continuum$/

@ecall that the geometric denition of the deri)ati)e is the slo(e of theline tangent to the cur)e at any (oint on the cur)e$ First let medistinguish a “function" from a “cur)e"$ )ery function is a cur)e, but afunction has at most one )alue (er in(ut, while a cur)e can ha)e anynumber of )alues$ e can consider the subset of general cur)es calledthe “algebraic cur)es", consisting of the Cartesian gra(hs of the(olynomials of the a((ro(riate number of )ariables for the dimensionwe’re examiningA analytic cur)es are also amenable to this analysis, orcur)es on other coordinate systems$

6nd a sim(le exam(le of that is the classical unit circle$ &n order to ndthe deri)ati)e of the unit circle using limits, one has to s(lit the circle intou((er and lower hal)es$ &f one uses the geometric denition, howe)er,there is only one cur)e, and com(uting a formula for its tangent line issim(le algebra$ 5he result is a formula for the tangent line to the circle ate)ery (oint on the (lane .sometimes called the “rst ordersemideri)ati)e"/, and it’s easy to see how to extract the slo(e of thatline$

5he algebra one (erforms in order to extract this is to e)aluate the cur)eat , where r and s are )ariables re(resenting arbitrary

http://n.a/

http://n.a/



numbers, then ex(ress the result in terms of (owers of x and y, andnally e)aluate that at , thereby gi)ing a net eect of addingand subtracting ero and rewriting the ex(ression in terms of (owersof and $ .5his action substitutes for adding and subtracting aninnitesimal, but we need no assum(tion that innitesimals exist$/ &f theoriginal cur)e was algebraic it will also be analytic, and so the rewrittenresult will be a 5aylor ex(ansion$

Bow, to nd the slo(e of the tangent line, one needs only to see that thee1uation of the tangent line is the e1uation setting all the eroth and rstorder terms in the 5aylor ex(ansion to ero .and discarding all the higherorder terms/A and the e1uation of the slo(e of that line is sim(ly thecoeIcient of di)ided by the coeIcient of $

o, let’s com(ute the rst order semideri)ati)e of the unit circle$

5he cur)e is $ )aluating at , we get the translated

cur)e , which ex(ands

to $ 5he 5aylor ex(ansion is

therefore $

5o nd the e1uation of the tangent line .the rst+order semideri)ati)ewith res(ect to x and y/, we set the eroth and rst order terms of the

5aylor ex(ansion e1ual to ero: $7utting this in the standard y4mxSb line, we

get as the e1uation of the line tangent to the

unit circle at .r,s/$ 5herefore, the deri)ati)e of the unit circle cur)e at the(oint on the circle is for all (oints where $

5his follows directly for all algebraic cur)es, and can be conrmed for allanalytic cur)es$ For non+analytic cur)es, it can be shown that we cana((roximate the deri)ati)e as closely as desired$

+m


orry about the La5ex$ #gly$


Bihil said: “2ut & will rst wait for your comment on the a((lication ofthe deri)ati)e to arbitrary cur)es and how the limit restricts thisa((licability$"

5han you for reminding me that & said that \ & forgot to ex(lain that

(art$

& 'ust ex(lained how to a((ly the geometric denition of the deri)ati)e toarbitrary algebraic cur)es$ -ore com(lex cur)es are also a)ailable, and



there are (roofs that the geometric denition yields both exact solutionsand a sim(le method for deri)ing a((roximations$

& also ex(lained one ob)ious way in which the geometric denition issu(erior, in that it allows deri)ati)es of cur)es that aren’t sim(lefunctions$ 2ut & didn’t ex(lain in what as(ects the innitesimal denitionof the deri)ati)e is inade1uate$ Botice that &’m not trying to say that it’sbad or wrong, or that it’s 6L6H inade1uateA rather, &’m (ointing out

some s(ecic (roblems that hinder certain uses$ 6lso notice that &’m notcom(laining about limitsA &’m taling s(ecically about the use ofinnitesimals in the denition of the deri)ati)e$ Limits may still be useful.for exam(le, & mentioned (iecewise smooth functions, whose deri)ati)esre1uire limits/$

5he most interesting (roblem is that innitesimals re1uire the use of thecontinuum, and not all numbers are embedded in a continuum$ 5herationals are )ery useful for most (ur(osesA and Yoating (ointcom(utation is a use of a s(ecial ty(e of rational number$ 5here are other

innite elds as well, and ob)iously the nite elds cannot bea((roached with limits at all .but are 1uite easily a((roached withgeometry/$ 6nd yes, the denition of “algebraic cur)e" a((lies o)er anyeld, nite or innite, so this method will nd its deri)ati)e$ Com(lexnumbers are reachable as well \ in fact, you can (robably see that thee1uation & deri)ed for the tangent line has )alues o)er the entire (lane,not 'ust on the unit circle, and in fact those )alues are geometricallymeaningful$

5here are more interesting results as well$ 5he tangent line is interesting

and useful, but there are also tangent conics, cubics, and so on$

+m


5han you 5ansley, for your ex(lanation$ 2ut & ha)e to admit, there is alot in the abo)e ex(lanation that & am not familiar with. lie the rst ordersemideri)ati)e / , so &’ll ha)e to go through it ste( by ste($ & ho(e you’llstay on the site to clarify my doubts8

&n the mean time, can we discuss your earlier comment “5he deri)ati)e issuIciently understood as the slo(e of the line tangent to a cur)e at a(oint$" 3

Let’s say we want draw a tangent to a cur)e$ 5his raises the 1uestionwhat is a tangent$

?$ Let’s say the tangent a (oint is a line that best a((roximates the cur)eat the (oint$ 5his raises the 1uestion what is meant by besta((roximation 3

$ 6 sim(lied answer to this 1uestion would be that it should ha)e thesame )alue at the (oint as the cur)e$

http://n.a/

http://n.a/



o if your function is y 4 x$ 5he then at x 4 , y 4 9$ 2ut you can drawany number of lines through the (oint .,9/$ o how do you go fromthere3

For a circle or a conic you can draw the a line from the centre or the fociiand then dene the tangent as the line that is (er(endicular to this line$2ut how would you draw a tangent to an arbitrary cur)e .one that has nocentre or focii/3

& would also lie to now if my statements ? and are correct, or do theyneed some mathematical renement$


5ansley, & went through your last (ost again and & thin & am beginningto understand the denition of the deri)ati)e as the slo(e of the tangent$

tatement ? should be P the tangent is the best rst order a((roximation

to the cur)e at a (oint ie it should ha)e the same )alue as the originalcur)e and also the sam rate of change at that (oint$

o, if my function is x<, and & want to draw a tangent at

change in y 4 .x S a/< P x<

4<a x S <a x S a<

5he eroeth order a((roximation is found by setting the rst and second

degree terms to ero$ 5his would be y0 4 a<$5his has the same )alue asthe function at x4a$

5he rst order a((roximation is found by setting the second degreeterms to ero$ 5his would be y? 4 < a x S a<$ 5he slo(e of this linehas the same )alue as the deri)ati)e of the function at x 4 a ie < a$

&ntuiti)ely too, this maes sense$Let’s consider a body starting from rest.at t 4 0/ and undergoing uniform acceleration of ?m (er seconds1uared$ hen & say this body has an instantaneous )elocity of Em (er

second at t 4 E, what it means is that the body has a (otential to tra)elEm (er second, if it were mo)ing at a constant )elocity of Em (er second,in either direction$ .2ut this doesn’t ha((en, because by the time the tbecomes >, the body has already accelerated through ?m (er seconds1uared$ o the distance tra)elled between t 4 E and t 4 > is not Em$/

&s my understanding correct, or is there something that &’)e missed 3


5here’s a mistae in my abo)e (ost, & said “tatement ? should be P thetangent is the best rst order a((roximation to the cur)e at a (oint ie itshould ha)e the same )alue as the original cur)e and also the sam rateof change at that (oint$"

http://n.a/

http://n.a/

http://n.a/

http://n.a/



5his is what one would ex(ect gi)en the traditional denition of thetangent$+ ie the tangent line to a (lane cur)e at a gi)en (oint is thestraight line that 'ust touches the cur)e at that (oint$

2ut if you loo at the e1uation to the tangent line deri)ed in my last (ost,y? 4 <ax S a<,at x 4 a, y 4 9 a <$ 5he (oint .a, 9 a </, does not lie on the cur)e y 4x<$

o is there something wrong with the denition, or is there something&’)e missed 3


&’m really sorry, but &’m 'ust not able to get the time to re(ly thisweeend$ Hou’re on the right trac in general .in fact, &’m 1uiteim(ressed, gi)en the tiny bit of ex(lanation &’)e been able to gi)e/A butthere’s more to do$

&f you don’t mind, &’m going to (oint you to a Hou5ube )ideo where a fairlycom(lex cur)e is analyed according to these rules$

htt(:GGwww$youtube$comGwatch3)4i>o0Of)QHm6

#nfortunately, he uses some unusual terms while doing this \ forexam(le, he denotes the cur)e using a “(olynumber", which he writes asan array of integers$ Hou may be able to gure how a (olynumber is lie a(olynomial without ex(licitly written )ariablesA if you need a better

ex(lanation the (re)ious )ideos in his series will ex(lain com(letely$ eethe entire (laylist at:

htt(:GGwww$youtube$comG(laylist3list47L=6D?9C>9R90<>62`feature4(lc(

+m


O 5ansley, &’ll chec out the )ideos, and then &’ll (ost what &’)eunderstood$ 2ut since & am unfamiliar with a lot of what is beingdiscussed here, &’ll need your conrmation to be sure what & understoodis correct$ &’ll wait for your comment$

6nd than you for (ointing me to these )ideos$ &t’s a new a((roach forme P integrating algebra, geometry and calculus$ 5he only hindrance ismy own less rigorous math bacground$ o &’ll ha)e to go through it ste(by ste($ & ho(e you will stay on the site to comment on my (rogress$


Bo (roblem, &’ll be here$

http://www.youtube.com/watch?v=i9o0OfvQYmA

http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=plcp


http://n.a/

http://www.youtube.com/watch?v=i9o0OfvQYmA



http://n.a/



6nd if it hel(s any, the (laylist & (ointed you to is “-athFundamentals",so it’s no (roblem not nowing math$ %e starts at counting with tallies, ifyou want to start at the beginning$

&f you wanted an ad)anced (laylist, he’ got one on uni)ersal hy(erbolicgeometry and another on algebraic to(ology$ hew8

+m

• Ale) sas!

&’m uncertain about calling the deri)ati)e a “better di)ision", although it’sbetter than “continuous di)ision"$ &’d (robably call it “generaliseddi)ision"$ &t does follow the (attern .established with integrals/ that thederi)ati)e is about changing 1uantities$ & belie)e the (roblematic as(ectof the deri)ati)e, is that it is a number at a s(ecic (oint ]a’, fj.a/, but afunction at a generalised (oint, fj.x/$

& do lie the 9+ste( (rocedure: .?/ Choose an inter)alA ./ Find the rawchangeA .</ nd the rate of changeA and .9/ -ae your model (erfect$ 2utthe limit is not only about maing your model ](erfect’, because it is alsoused to Tsim(lifyT a (roblem by neglecting the contribution of a certaincom(onent$

5hat last ste( “-ae your model (erfect" seems to be what the changefrom %y(erreal numbers to @eal numbers .by taing the standard (art/ isall about$ Or at least that was something that immediately s(rung tomind$

6rgh$ omebody mentioned the e(silon P delta denition$ &t’s not somuch the denition itself .that basically relates in(ut error to out(uterror/, but most ex(lanation are 'ust so!$ugh$

5hans

• robin sas!

Rear alid, than you )ery much for your eorts and the time that you

(ut in to write such excellent ex(lanations of basic math conce(ts$ & am astudent of (sychology$ & am using this to learn more about calculus andmaths$ 6 note: & belie)e mathematics (edagogy in schools all o)er theworld has to radically change$ &t is essential and benecial to use < Ranimations and other )isual techni1ues to im(art mathematical ideas$#ntil that ha((ens there would always be (otential mathematicians whowould ne)er go on to do set theory or matrices or calculus$ 6lso withbetter < R )isual re(resentations and animations of mathematical ideas,fundamental conce(ts lie calculus, gra(h theory and e)en dynamicalsystems could be im(arted to students at an earlier age$

• kalid sas!

5hans @obin, & really a((reciate the note$ & lo)e it when (eo(le in otherelds are able to tae away some insights$ & denitely thin math



(edagogy needs to change, to use other techni1ues, but really, to 'ustas “6re we actually learning here3"$ & feel there’s a giant em(eror’sclothes (roblem where nobody wants to admit “%ey, this conce(t we’resu((osed to be teaching! it’s not clicing at an intuiti)e le)el, and itshould$" <d )isualiations and other tools can hel( get ideas to really sinin$ 6((reciate the note8

• kishore sas!

one thing that always bothers me is the chicen egg (roblem$ whichcomes rst, dierential e1uation or the function$ Let me gi)e andexam(le$ 5ae for instance decay laws which is stated in dierentiale1uations$ 2ut when you carry out (ractical ex(eriments, we would (lot agra(h and would a((roximate the gra(h to a function through cur)etting techni1ues$ Bow where is dierential e1uation tting in$2ecause ican mae all the (redictions through a function$ hat is the (oint inre(resenting e)ent through dierential e1uations if my function could doall the 'ob$


ishore, the dierential e1uation doesn’t come from a function$ &t comesfrom a model that (redicts the obser)ed )alues$ 5he model ha((ens toim(ly certain relationshi(s between (hysical measurements, which .whenstated mathematically, using nown laws of (hysics, and ex(ressed withthe smallest number of inde(endent )ariables/ often winds u( ha)ingintegrals and dierentials embedded in it \ hence it’s a dierentiale1uation$

• simo sas!

Roes not dx . in our case dx4?/ re(resent acceleration in s(eed ininter)al x4, x4<3

2etween (oints x4$E and x4$> is s(eed

=$D 4 =$W S 0$?

• *el+andian sas!

xcellent8888& ho(e there are more (eo(le who can ex(lain maths lie this, @emo)esthe fear of maths and (uts the 'oy of learning it$

• themytho, sas!

& nd many (luses and minuses with these ty(es of a((roaches to to(ics$

&t’s good because some (eo(le nd it more a((roachable$ & feel it’s bad if they are not able to con)ert it to a logical mathematical understandingusing mathematical language$

5his is one of the most harmful as(ects of math education today$)eryone is focused on (ushing standardied testing and standardied



testing destroys the nurturing of (roblem sol)ing and logicalunderstanding of conce(ts because there is no time and e)eryone has to(lay the ]rat race’ within education to get the ]golden ticet’ to a niceex(ensi)e college$ 5his is what you get when you turn education into a](roduct’ and students into ]consumers’$ 5hose students who ha)eexcellent (roblem sol)ing sills ne)er ha)e time to nurture them ande)en wind u( ha)ing those sills stunted$

-y rant aside, no, the slo(e of the tangent line is not a bland descri(tionat all$ 5he (roblem here usually is that students don’t ha)e a solidfoundation in algebra rst, which is a must and then a )ery goodfoundation in (re+calculus$

5o see a whole to(ic on deri)ati)es without a single gra(h is doing adisser)ice$

ôod to see students get something here but calculus needs a unieda((roach and the understanding of the deri)ati)e begins with a strong

foundation in algebra .coordinate geometry/ and (re+calculus$

• Har+reet Singh sas!

& thin currently your a((roach is better than others$& don’t now why (eo(les want to sandwich the “6ha8 moments" with fasttrac academies$ee( it u(8 halid you ha)e been blessed to reduce com(lexity to bring insim(licity$ 5hans$

• Soham -howdhury sas!

& really dig your articles and &’m going through some of them .mostlybecause & lie math, and .also/ because of the fact that & can ha)e someeighth+grade swag at nowing calculus :R/$%a)e you e)er considered teaching3

On another note, why don’t you use -ath;ax for your e1uations3 &t’s somuch better$

• (aura* M #ulsiani sas!

5hans a ton halid$ & stumbled on betterex(lained while searching forsome limits ex(lainations on the web$ Hou are doing an awesome 'ob byreminding (eo(le im(ortance of intuition ` beauty of maths$ 6s of nowbefore starting any new to(ic & go through better ex(lained to now what& am going to do$

•

kalid sas!

oham: 5hans for the comment8 & ha)en’t thought about in+(ersonteaching that much, but it might be something in the future$ &’d lie tointegrate -ath;ax as well, the only (roblem is it doesn’t wor in @



feedsGemail$ & might nd a way to use -ath;ax on the website and fallbac to the images on the other (laces$

âura): 6wesome, glad you’re en'oying the site :/$

• ansh choubey sas!

6ha moment came near reading your fab articles must write a high

school boo soon & wanna read more and more$ ,$ 7l do write on(hysics too$

• kalid sas!

%i 6nsh, glad you en'oyed it8

• ansh choubey sas!

6ha moment came near reading your fab articles must write a high

school boo soon & wanna read more and more$ ,$ 7l do write on(hysics too$2oom

• Ademilson sas!

Congratulations888 it’s really a )ery intuiti)e ex(lanation86nalogy is the ey8 “6s abo)e so below"Bamaste8

• Math enthusiast. Northwestern Student sas!

5his was absolutely great!$well done$ what an excellent ex(lanation ofcalculus! also & would lie to add this$

& was doing a lot of research and thining and came to the conclusionthat lie you mentioned the integral in some res(ects is not directlyrelated to dierentiation$ -ore (recisely, the denite integral is unrelatedto dierentiation, and anti+direntiation is the im(erfect

re)ersal.o((osite o(eration/ of dierentiation .)ery intuiti)e/$ 5he reasonwhy is sim(le: the denite integral com(utes the signed area under acur)e and the change in (osition of the original function .i$e .dxGdt/ times.dt/ e1uals dx/! which is com(letely useless if you are trying to nd theoriginal function! the antideri)ati)e, howe)er, is useful for that as longas the constant is dened!$ the indenite integral is )irtually the sameas an anti+deri)ati)e exce(t it’s syntax actually means nothing innature! ha)e you e)er wondered why there is a dx .or the a((ro(riatedierential/ at the end of the integrand e)en though there are no boundsof integration!333 dx would re(resent an innitesimally small width but

since there are no bounds of integration the dx means nothing! it’s adummy )ariable as some would say!

great stu





• Adrienne sas!

& LO[ 5%&$ %el(s me a((reciate math so much more$ Hou’re awesomealid$

• elisen sas!

thans

• elisen sas!

Rear halid,& 'ust wanted to say than you again and the fact that you are )ery cle)erand how your dream lies in hel(ing other (eo(le understand is great$& ha)en’t been listening to my maths classes recently and therefore &need to do a lot of wor$ Hour website had made me more condent$

indeed the idea of a rate of change at a (oint is )ery confusing$

& want to as whether your (assion for maths or any learning stems fromyour curiousity, whether you ha)e read history on your maths$

and i ho(e you chec this out$this guy is 1uite smart and uses analogies too$i want to be able to do ame analogies myself so i can understand andrelate ideas so i can a((ly them to life and mae use of them$ because alllearning is (recious$

because i am a (erson who needs to understand,

therefore you ahas hel( .my mother on the other hand are the ones whoremember and dont 1uestion haha, but indeed there are dierent (eo(le,and their way of learning and how their brain functions, their beha)iour,their attitude to learning a((roaches is dierent/

htt(:GGwww$scotthyoung$comGblogG00DG0<G=Ghow+to+ace+your+nals+without+studyingG

i want to than you again and how you ha)e left a Question (art showsyour dedication$ Hours &ncerelylisen

• mahendra sas!

[oila outstanding ,udos what a lucid ex(lanation ,ee( the great worYowingcheers

• kalid sas!

mahendra: 5hans8

http://www.scotthyoung.com/blog/2007/03/25/how-to-ace-your-finals-without-studying/






lisen: @eally a((reciate the note, than you$ .cott is a friend and &really lie how he breas down his methods8/$

-y (assion for math .or learning in general/ came when & realied howmuch sim(ler an idea could be if we looed at it the right way$ omethingwhich was once confusion becomes sim(le with the right a((roach .thinabout how diIcult multi(lication is with @oman numerals, but how easyit is with decimal numbers/$ & had this belief that any idea could be made

sim(le, and it’s what ee(s me going$ &f something seems diIcult, it’s o\ it 'ust means & ha)en’t found the sim(le )ersion of it yet$

@eally glad the site has been hel(ing :/$



7o! To 4nderstand Derivatives: The Produ#t& Po!er " 0hain Rules

by Kalid Azad · 4! comments

5he 'umble of rules for taing deri)ati)es ne)er truly cliced for me$ 5headdition rule, (roduct rule, 1uotient rule \ how do they t together3 hat arewe e)en trying to do3

%ere’s my tae on deri)ati)es:

• e ha)e a system to analye, our function f • 5he deri)ati)e f’ .aa dfGdx/ is the moment+by+moment beha)ior• &t turns out f is (art of a bigger system .h 4 f S g/• #sing the beha)ior of the (arts, can we gure out the beha)ior of the

whole3 Hes$ 3very part has a <point of view= about how much change it

added. ombine every point of view to get the overall behavior. achderi)ati)e rule is an exam(le of merging )arious (oints of )iew$

6nd why don’t we analye the entire system at once3 For the same reason youdon’t eat a hamburger in one bite: small (arts are easier to wra( your headaround$

&nstead of memoriing se(arate rules, let’s see how they t together:

5he goal is to really gro the notion of “combining (ers(ecti)es"$ 5hisinstallment co)ers addition, multi(lication, (owers and the chain rule$ Onward8

un#tions: (nythin'& (nythin' 5ut @ra*hs

5he default calculus ex(lanation writes “f.x/ 4 x" and sho)es a gra(h inyour face$ Roes this really hel( our intuition3





Bot for me$ ^ra(hs s1uash in(ut and out(ut into a single cur)e, and hide themachinery that turns one into the other$ 2ut the deri)ati)e rules are about themachinery, so let’s see it8

& )isualie a function as the (rocess “in(ut.x/ 4 f 4 out(ut.y/"$

&t’s not 'ust me$ Chec out this incredible, mechanical targetting com(uter.beginning of youtube series/$

5he machine com(utes functions lie addition and multi(lication with gears \you can see the mechanics unfolding8

5hin of function f as a machine with an in(ut le)er “x" and an out(ut le)er“y"$ 6s we ad'ust x, f sets the height for y$ 6nother analogy: x is the in(utsignal, f recei)es it, does some magic, and s(its out signal y$ #sewhate)er analogy hel(s it clic$

Wi''le Wi''le Wi''le

5he deri)ati)e is the “moment+by+moment" beha)ior of the function$ hatdoes that mean3 .6nd don’t mindlessly mumble “5he deri)ati)e is theslo(e"$ See any graphs around these parts# ella&/

5he deri)ati)e is how much we wiggle$ 5he le)er is at x, we “wiggle" it, andsee how y changes$ “Oh, we mo)ed the in(ut le)er ?mm, and the out(utmo)ed =mm$ &nteresting$"

5he result can be written “out(ut wiggle (er in(ut wiggle" or “dyGdx" .=mm G?mm 4 =, in our case/$ 5his is usually a formula, not a static )alue, because itcan de(end on your current in(ut setting$

For exam(le, when f.x/ 4 x, the deri)ati)e is x$ He(, you’)e memoriedthat$ hat does it mean3

&f our in(ut le)er is at x 4 ?0 and we wiggle it slightly .mo)ing it by dx40$? to?0$?/, the out(ut should change by dy$ %ow much, exactly3

http://www.youtube.com/watch?v=mpkTHyfr0pM

http://betterexplained.com/articles/learning-to-learn-embrace-analogies/

http://www.youtube.com/watch?v=mpkTHyfr0pM

http://betterexplained.com/articles/learning-to-learn-embrace-analogies/



• e now fj.x/ 4 dyGdx 4 T x• 6t x 4 ?0 the “out(ut wiggle (er in(ut wiggle" is 4 T ?0 4 0$ 5he

out(ut mo)es 0 units for e)ery unit of in(ut mo)ement$• &f dx 4 0$?, then dy 4 0 T dx 4 0 T $? 4

6nd indeed, the dierence between ?0 and .?0$?/ is about $ 5hederi)ati)e estimated how far the out(ut le)er would mo)e .a (erfect, innitelysmall wiggle would mo)e unitsA we mo)ed $0?/$

5he ey to understanding the deri)ati)e rules:

• et u( your system• iggle each (art of the system se(arately, see how far the out(ut mo)es• Combine the results

5he total wiggle is the sum of wiggles from each (art$

(ddition and Su6tra#tion

5ime for our rst system:

hat ha((ens when the in(ut .x/ changes3

&n my head, & thin “Function h taes a single in(ut$ &t feeds the same in(ut to f

and g and adds the out(ut le)ers$ f and g wiggle inde(endently, and don’te)en now about each other8"

Function f nows it will contribute some wiggle .df/, g nows it will contributesome wiggle .dg/, and we, the (rowling o)erseers that we are, now theirindi)idual moment+by+moment beha)iors are added:

6gain, let’s describe each “(oint of )iew":

• 5he o)erall system has beha)ior dh• From f’s (ers(ecti)e, it contributes df to the whole Uit doesn’t now about

gV

• From g’s (ers(ecti)e, it contributes dg to the whole Uit doesn’t nowabout fV

)ery change to a system is due to some (art changing .f and g/$ &f we add thecontributions from each (ossible )ariable, we’)e described the entire system$



d vs d-dx

ometimes we use df, other times dfGdx \ what gi)es3 .5his confused me for awhile/

• df is a general notion of “howe)er much f changed"• df>d! is a s(ecic notion of “howe)er much f changed, in terms of how

much x changed"

5he generic “df" hel(s us see the o)erall beha)ior$

6n analogy: &magine you’re dri)ing cross+country and want to measure the fueleIciency of your car$ Hou’d measure the distance tra)eled, chec your tan tosee how much gas you used, and nally do the di)ision to com(ute “miles (ergallon"$ Hou measured distance and gasoline se(arately \ you didn’t 'um( intothe gas tan to get the rate on the go8

&n calculus, sometimes we want to thin about the actual change, not the ratio$oring at the “df" le)el gi)es us room to thin about how the functionwiggles o)erall$ e caneventually scale it down in terms of a s(ecic in(ut$

6nd we’ll do that now$ 5he addition rule abo)e can be written, on a “(er dx"basis, as:

3ulti*li#ation +Produ#t Rule,

Bext (ule: su((ose our system multi(lies (arts “f" and g"$ %ow does itbeha)e3

%rm, tricy \ the (arts are interacting more closely$ 2ut the strategy is thesame: see how each (art contributes from its own (oint of )iew, and combinethem:

• total change in h 4 f’s contribution .from f’s (oint of )iew/ S g’s

contribution .from g’s (oint of )iew/Chec out this diagram:



hat’s going on3

• e ha)e our system: f and g are multi(lied, gi)ing h .the area of therectangle/

• &n(ut “x" changes by dx o in the distance$ f changes by some amount df .thin absolute change, not the rate8/$ imilarly, g changes by its ownamount dg$ 2ecause f and g changed, the area of the rectangle changestoo$

• hat’s the area change from f’s (oint of )iew3 ell, f nows he changedby df, but has no idea what ha((ened to g$ From f’s (ers(ecti)e, he’s theonly one who mo)ed and will add a slice of area 4 df T g

•

imilarly, g doesn’t now how f changed, but nows he’ll add as slice ofarea “dg T f" 5he o)erall change in the system .dh/ is the two slices of area:

Bow, lie our miles (er gallon exam(le, we “di)ide by dx" to write this in termsof how much x changed:

.6side: Ri)ide by dx3 ngineers will nod, mathematicians will frown$ 5echnically, dfGdx is not a fraction: it’s the entire o(eration of taing thederi)ati)e .with the limit and all that/$ 2ut innitesimal+wise, intuition+wise, weare “scaling by dx"$ &’m a smiler$/

5he ey to the (roduct rule: add two “sli)ers of area", one from each (oint of)iew$

otcha% 2ut isn’t there some eect from both f and g changingsimultaneously .df T dg/3

He($ %owe)er, this area is an innitesimal T innitesimal .a “nd+orderinnitesimal"/ and in)isible at the current le)el$ &t’s a tricy conce(t, but .df T



dg/ G dx )anishes com(ared to normal deri)ati)es lie dfGdx$ e )ary f and ginde(dendently and combine the results, and ignore results from them mo)ingtogether$

The 0hain Rule: It%s Not So 5ad

Let’s say g de(ends on f, which de(ends on x:

5he chain rule lets us “oom into" a function and see how an initial change .x/can eect the nal result down the line .g/$

&nterpretation 2% onvert the rates

6 common inter(retation is to multi(ly the rates:

x wiggles f$ 5his creates a rate of change of dfGdx, which wiggles g by dgGdf$ 5he entire wiggle is then:

5his is similar to the “factor+label" method in chemistry class:

&f your “miles (er second" rate changes, multi(ly by the con)ersion factor to

get the new “miles (er hour"$ 5he second doesn’t now about the hour directly\ it goes through the second 4 minute con)ersion$

imilarly, g doesn’t now about x directly, only f$ Function g nows it shouldscale its in(ut by dgGdf to get the out(ut$ 5he initial rate .dfGdx/ gets modiedas it mo)es u( the chain$

&nterpretation 1% onvert the wiggle

& (refer to see the chain rule on the “(er+wiggle" basis:

• x wiggles by dx, so• f wiggles by df, so• g wiggles by dg



Cool$ 2ut how are they actually related3 Oh yeah, the deri)ati)e8 .&t’s theout(ut wiggle (er in(ut wiggle/:

@emember, the deri)ati)e of f .dfGdx/ is how much to scale the initial wiggle$6nd the same ha((ens to g:

&t will scale whate)er wiggle comes along its in(ut le)er .f/ by dgGdf$ &f we writethe df wiggle in terms of dx:

e ha)e another )ersion of the chain rule: dx starts the chain, which results insome nal result dg$ &f we want the nal wiggle in terms of dx, di)ide bothsides by dx:

5he chain rule isn’t 'ust factor+label unit cancellation \ it’s the (ro(agation ofa wiggle, which gets ad'usted at each ste($

5he chain rule wors for se)eral )ariables .a de(ends on b de(ends on c/, 'ust(ro(agate the wiggle as you go$

5ry to imagine “ooming into" dierent )ariable’s (oint of )iew$ tarting fromdx and looing u(, you see the entire chain of transformations needed beforethe im(ulse reaches g$

0hain Rule: )xam*le Time

Let’s say we (ut a “s1uaring machine" in front of a “cubing machine":

in(ut.x/ 4 f:x 4 g:f< 4 out(ut.y/

f:x means f s1uares its in(ut$ g:f< means g cubes its in(ut, the )alue of f$For exam(le:

in(ut./ 4 f./ 4 g.9/ 4 out(ut:W9

tart with , f s1uares it . 4 9/, and g cubes this .9< 4 W9/$ &t’s a Wth

(ower machine:

6nd what’s the deri)ati)e3



• f changes its in(ut wiggle by dfGdx 4 x• g changes its in(ut wiggle by dgGdf 4 <f

5he nal change is:

0hain Rule: @ot#has

@unctions treat their inputs like a blob

&n the exam(le, g’s deri)ati)e .“x< 4 <x"/ doesn’t refer to the original “x", 'ust whate)er the in(ut was .foo< 4 <Tfoo/$ 5he in(ut was f, and it treats fas a single )alue$ Later on, we scurry in and rewrite f in terms of x$ 2ut g hasno in)ol)ement with that \ it doesn’t care that f can be rewritten in terms ofsmaller (ieces$

&n many e!amples+ the variable <!= is the <end of the line=.

Questions as for dfGdx, i$e$ “î)e me changes from x’s (oint of )iew"$ Bow, xcould de(end on something dee(er )ariable, but that’s not being ased for$ &t’slie saying “& want miles (er hour$ & don’t care about miles (er minute or miles(er second$ ;ust gi)e me miles (er hour"$ dfGdx means “sto( looing at in(utsonce you get to x"$

How come we multiply derivatives with the chain rule+ but add themfor the others?

5he regular rules are about combining points o view to get an o)erall (icture$hat change does f see3 hat change does g see3 6dd them u( for the total$

5he chain rule is about going dee(er into a single (art .lie f/ and seeing if it’scontrolled by another )ariable$ &t’s lie looing inside a cloc and saying “%ey,the minute hand is controlled by the second hand8"$ e’re staying inside thesame (art$

ure, e)entually this “(er+second" (ers(ecti)e of f could be added to some(ers(ecti)e from g$ ^reat$ 2ut the chain rule is about di)ing dee(er into “f’s"root causes$

Po!er Rule: t 3emori/ed& Seldom 4nderstood

hat’s the deri)ati)e of x93 9x<3 ^reat$ Hou brought down the ex(onentand subtracted one$ Bow ex(lain why8

%rm$ 5here’s a few a((roaches, but here’s my new fa)orite: x9 is really x T xT x T x$ &t’s the multi(lication of 9 “inde(endent" )ariables$ ach x doesn’tnow about the others, it might as well be x T u T ) T w$

Bow thin about the rst x’s (oint of )iew:



• &t changes from x to x S dx• 5he change in the o)erall function is U.x S dx/ P xVUu T ) T wV 4 dxUu T ) T

wV• 5he change on a “(er dx" basis is Uu T ) T wV

imilarly,

• From u’s (oint of )iew, it changes by du$ &t contributes .duGdx/TUx T ) T wVon a “(er dx" basis

• ) contributes .d)Gdx/ T Ux T u T wV• w contributes .dwGdx/ T Ux T u T )V

5he curtain is un)eiled: x, u, ), and w are the same8 5he “(oint of )iew"con)ersion factor is ? .duGdx 4 d)Gdx 4 dwGdx 4 dxGdx 4 ?/, and the totalchange is

&n a sentence: the deri)ati)e of x9 is 9x< because x9 has four identical

“(oints of )iew" which are being combined$ 2ooyeah8

Ta$e ( 5reather

& ho(e you’re seeing the deri)ati)e in a new light: we ha)e a system of (arts,we wiggle our in(ut and see how the whole thing mo)es$ &t’s about combining(ers(ecti)es: what does each (art add to the whole3

&n the follow+u( article, we’ll loo at e)en more (owerful rules .ex(onents,1uotients, and friends/$ %a((y math$

1. (oura* sas!

6wesome article$ 5he descri(tion of the (roduct rule really changed how &thin about them$

Out of curiosity, how do you thin your idea of the (ower rule extends tonegati)e, fractional and irrational (owers3 &t’s a bit harder to thin aboutsince you can’t 'ust s(lit them into linear (arts$

". kalid sas!

%i ôura), thans for the note$ ^reat 1uestion about the negati)e,fractional and irrational (owers$ 5o follow the analogy, we could use thechain ruleA su((ose we ha)e f.x/ 4 x+<$ ee x+< as shorthand for?Gx<$ e can do:

dGdx x+< 4 dGdx ?Gx< 4 dGdx ?Gu 4 +?Gu T duGdx

duGdx can be understood intuiti)ely .<x/, and we di)ide it by .x</$

e can see the x (owers ght it out as .x+?/ P x 4 +x P ? U5he .x+?/(ower is from duGdx, and +x is from ?Gu$ ith x4<, get +< P ? 4 +9 asthe (owerV$ Botice how we still brought down the “<" .which was induGdx/$ %o(e this (art made sense$



Once we get to fractional and irrational (owers, it’s (robably easiest torewrite things in terms of e: x<$9 4 eUln.x/T<$9V$ From here, we canuse the chain rule and (roduct rule and ex(onent rule .to be ex(lainednext time/ we can get the result$ ssentially, e)en a com(lex idea lie afractional ex(onent can be further broen down$ &t’s something &’d lie towrite more about \ it’s hel(ing to really test my intuition :/$

#. !isoo sas!

& found the following website useful for understanding the (roduct ruleusing what & already now$htt(:GGwoobiola$netGmathGcalcb$htm

$. !ohn Paton sas!

Bice (ost alid$ &’)e s(end the last cou(le of hours trying to de)elo( this]machine+lie’ intuition$ 6ny chance that you could also (ost someexam(les on the 1uotient rule$ &’)e been trying to wor it out on my own

but ha)en’t managed to get there$ %onestly this is slightly worrying$ & feelthat if & truly understood what you are saying then the 1uotient ruleshould be no big deal$ 5hans$

%. kalid sas!

;ohn: 5hans for the comment$ &ntuiti)ely, the 1uotient rule can beseen as a )ariation of the (roduct rule since di)ision is a )ariation ofmulti(lication .in my head, “multi(lying by a 1uantity that is gettingsmaller"/$ o, the 1uotient rule should loo a lot lie the (roduct rule .two

“slices" to tae into account/, but one of the slices is a shrining one$ &’llbe (osting a follow+u( soon$

6nd great gut+chec by the way$ &f a conce(t isn’t clicing dee( down itmeans there’s more intuition to build .and, (robably, the ex(lanation can

use some renement /$

;isoo: ^reat, & lie the sim(le diagrams8

&. Phoeni) sas!

First of all, great initiati)e and material$ Lo)ed the way u analyse things$ &read this (age a cou(le of months ago$@ecently & also read about binomial series and somehow & was able tonarrate how the (ower rule was actually deri)ed$ o here it goes$

Lets tae the sim(lest function y 4 x$ Bow what do we mean by

deri)ati)e3 &t is sim(ly the change in the out(ut when we twea the in(uta little$

Bow lets tae two number x and xS?$ Bow & want to nd out how much ychanges when we change x$

http://woobiola.net/math/calc2b.htm

http://optimizethyself.com/

http://woobiola.net/math/calc2b.htm

http://optimizethyself.com/



Change 4 .xS?/ P xCon)entional calculus tells us that it is Tx$2ut the actual )alue can be obtained by using binomial theorem$e all now .aSb/ formula, 4 a S b S TaTbBow .xS?/ 4 x S ? S TxChange 4 x S ? S Tx P x 4 Tx S ?%aha , we ha)e arri)ed at the answer$ 5he calculus )alue and the actual)alue dier by ?$ 5o remo)e that we a((ly the rule that the change in

in(ut is )ery )ery less when com(ared to the in(ut )alue$ x?$6((lying the abo)e, we can a((roximate Tx S ? as Tx$

&n the same way , & a((lied the same to x< and the dierence is ? S<TxT?.xS?/ which can be a((roximated as ? S <Tx.x/ which can bereduced to <Tx since x?$

&n general, for xn, we ha)e nS? terms in the series$ Of that we omit all(owers of n u(to n+$ e tae only xn and xn+? terms$ 5he co+eIcientof xn+? is n and hence the (ower rule is gi)en as

.dGdx/ of xn 4 nT.x.n+?//$

5hans to the admin for in)oing a interest in me to sol)e this$ %o(e ithel(s$

'. kalid sas!

7hoenix: 5hans for the comment$ He(, that’s the essence of it \ to getmore (articular, turn the ? into “dx" .the amount of change, so it’s

.xSdx//, then do the binomial theorem and throw away the dx at theend .i$e$, assume your change was “(erfect"/$

. #rae Barlow sas!

e now f’.x/ 4 dyGdx 4 T x6t x 4 ?0 the “out(ut wiggle (er in(ut wiggle" is 4 T ?0 4 0$ 5heout(utmo)es 0 units for e)ery unit of in(ut mo)ement$&f dx 4 0$?, then dy 4 0 T dx 4 0 T $? 4

#mm what3&f F.x/ 4 x and x4?0 then the result of that would be ?00, not 0$&f T?04 then the out(ut would mo)e units for e)ery unit of in(utmo)ement, not ?0, this doesn’t mae any sense at all$

. #rae Barlow sas!

5hat said, & s(ea as someone who (assed a college le)el Calc course butne)er understood a lic of anything & was doing .dunno how that’s

(ossible/$ im(ly memoriing e)erything, & 'ust remember ano)erwhelming urge to bash my brains out on my school des$ 6ctuallygetting that same urge now$

Cray ehh3



10.#rae Barlow sas!

&f our in(ut le)er is at x 4 ?0 and we wiggle it slightly .mo)ing it bydx40$? to ?0$?/, the out(ut should change by dy$ %ow much, exactly3

e now f’.x/ 4 dyGdx 4 T x6t x 4 ?0 the “out(ut wiggle (er in(ut wiggle" is 4 T ?0 4 0$ 5heout(ut mo)es 0 units for e)ery unit of in(ut mo)ement$

&f dx 4 0$?, then dy 4 0 T dx 4 0 T $? 4

5o be clear, let me ex(lain what &’m confused with$ “5he out(ut mo)es 0units for e)ery unit of in(ut mo)ement$" hat are we calling a unit3 6ninteger3 6 doen3 twenty3 .$?3 ?3 03/

5he other thing that confuses me is that you go form dyGdx 4 = to awhole dierent formulaGe1uation$ Consistency would hel( those not asliterate in mathematics out .lie myself/ 1uite a bit$

11. andy sas!

what s the (ayo in learning all of this! the ban account meta(hor wasinsightful, (ou(ulations models might ser)e as good exam(le, but alsoremember that learning is dierent is for each indi)idual de(ending onwhat resides in their subconscious!

1". kalid sas!

andy: 5he (ayo is understanding something you didn’t

before8 He(, if the analogy wors for you, it wors$

1#. Matthias sas!

5han you8 & really wanted to understand how and why these rules wor,not 'ust how to a((ly them

1$. !! sas!

alid, 5hans for your extraordinarily sim(le ex(lanations of calc8 &’m currentlya so(homore in high school, and & could ha)e 'ust waited until next yearto tae the class, but &’)e wanted to learn for too long already8 &t’samaing how sim(le and easy the math of change is8 Bow & get to maethe semi+intelligent 'uniors feel dumb for someone a grade below themnowing more about the sub'ect than they do!

1%. /ishwas sas!

5o use calculus on any changing system, is it mandatory, that the system“-#5" follow a (articular rule of change$For exam(le, when some system is changing by ratio of ?:, then one can



nd out the change as :9 or ?00:00 nally$ hat is the rule of changein calculus 3 &f not, will calculus be able to nd an accurate answer e)erytime 3

1&. kalid sas!

-atthias: ^lad it hel(ed$

;;: 6wesome, glad you’re getting a head start8 Hou got it, the math isn’tmuch more than algebra, it’s 'ust seeing how to (ut the )ariablestogether$

[ishwas: Calculus is made for instantaneous rates of change, i$e$ therate of change at a certain moment in time$ 6s you mo)e away from thatmoment, the rate of change )aries and is no longer accurate Uand youuse integration to add+u( these constantly+shifting momentsV$

1'. 0inne (illan sas!

till does not mae any sense unfortunately P including the mechanicalcom(uter )ideo and wiggles$ 7erha(s it’s ho(eless and & will ne)erunderstand calculus des(ite wanting to$ & was lost earlier$ &f f.x/ 4 xand in(ut is ?0$ iggle of 0$? gi)es wiggle of $0? and wiggle of 0$0?gi)es 0$00?$ %ow do these relate to the deri)ati)e3

1.Silrak sas!

%i, am woring through the tutorials from naught, to gather an

understanding of calculus and within 9 wees when my course will start$&s there anyone who might hel( me excel more than is (ossible on myown )ia sy(e or (hone3 2ased in -elb, 9$0?$?9$

1. kalid sas!

%i ilra, there’s a full series on calculushere: htt(:GGbetterex(lained$comGcalculusGwhich might hel($

"0. ra"u sas!

why you neglect df T dg area3if rate of change is more then error (roduces in the way of intuition,

i understand >= about (roduct rule exce(t this (art

what a intuition to (rodut rule8 i really lie it exce(t neglecting dy T dg

"1. &ric / sas!

ra'u^reat 1uestion8 5his 1uestion, and a few others )ery much lie it, ga)eme a bit of trouble until & (erformed a bit of mental 'u 'itsu to con)incemyself & understood it$ &’ll ha)e a crac at answering this, if it hel(s great,if not feel free to ignore me, & won’t be oended$

http://betterexplained.com/calculus/

http://betterexplained.com/calculus/



& can answer it with a bit of sim(le, but creati)e, algebra$ 5o do so &’mgoing to need to ex(lain two dierent ]siing’ o(erations: the integral . p /and the dierential element .d/$ & don’t want to throw a bunch ofintegration at you in trying to ex(lain deri)ati)es, but if you blur out thestrict denitions and 'ust loo at the ideas, then p and d are 'ust twodierent ideas that are a ind of com(liment to each other$

p P add u( a whole bunch of things

d P tae a thing and cut it into a bunch of ]itty bitty (ieces’, or i$b$($

5aing a dierential element of a (ia, or d.(ia/, is 'ust sha)ing o alittle bit, it’s lifting a (e((eroni and licing the bottom then re(lacing itwhile nobody’s looing$ 5he small amount of (e((eroni grease that’smissing is so small com(ared to the whole sie of the (ia that no onenotices it’s missing$

&n that )ein &’m going to tal about taing some i$b$($’s of a few things$tart with:

h.x/ 4 f.x/ T g.x/h is the full sie of h, the out(utdh is an i$b$($ of the out(ut

x is the full sie of an in(utdx is an i$b$($ of an in(ut

&’m going to tae a few liberties and do some ]algebra magic’ with1uantities lie dx, (lease understand that what &’m about to insinuate isnot technically allowed, and if we followed the ]mathematically correct’

(ath we would (erform a whole lot of weird calculus o(erations 'ust tocome u( with the same result, see alid’s ]6side’ note about thengineers nodding and the mathematician frowning$

&’m going to assume you followed alid’s logic and got to this (art:dh 4 fTdg S gTdf

dh is 'ust an i$b$($ of h, it is the little bit of rectangle you add on to the fullsie rectangle of fTg$

fTdg is the )ertical sli)er rectanglegTdh is the horiontal sli)er rectangle

o it seems your 1uestion becomes the following: if & want to describe allof that i$b$($ of h that &’m adding & can see & need to add the two sli)ers,but don’t & also need to add that little s1uare3 ure it’s teeny, but thesli)ers are teeny too, aren’t they3

%ow about we don’t disregard it, how about we add it in then see why it)anishes and the sli)ers are big enough to stay$

e should ha)e:dh 4 fTdg S gTdf but logic tells us we ha)e:dh 4 fTdg S gTdf S .dfTdg/ that (esy little s1uare



dh 4 fTdg S gTdf S .dfTdg/ TTnow di)ide both sides by dxdhGdx 4 f T dgGdx S g TdfGdx S .dfTdg/Gdx TTre+arrange ./ in <rd termdhGdx 4 f T dgGdx S g TdfGdx S df T .dgGdx/ TT(ut that <rd term right afterthe nd4 f T dgGdx S df T .dgGdx/ S g T dfGdx TTfactor out the dgGdx4 .f S df/ .dgGdx/ S g T dfGdx TTalmost there, com(are it to what weshould ha)e:

4 .f / .dgGdx/ S g T dfGdx

hat ha((ens to .f S df/ as df gets ]eensy weensy’3 &t gets really close tof$ 5he full sie of f on its own is indistinguishable from the full sie of f(lus an i$b$($ of f$ Bo one sees the little bit of (e((eroni grease missingcom(ared to the full sie of the (ie .& li)e in Bew ngland, we call (ia](ie’ u( here, we also tae ]r’s out of words and stic ’em other (lacesthey don’t belong: (ah the cah, Relter airlines/$

ith that (esy little s1uare we ha)e:

dhGdx 4 .f S df/ .dgGdx/ S g T dfGdx

ithout we ha)e:dhGdx 4 .f / .dgGdx/ S g T dfGdx

2ut these two are the same8\\\\\\\\\\\

&f & ha)en’t confused you yet maybe this will throw you o guard .& 'est, &really do wan’t you to understand/8

%ere’s another reason that little s1uare . df T dg / )anishes but the sli)ersremain$ Let’s use an analogy that all of us understand so well on anintuiti)e le)el: 2oolean 6lgebra8 .2ang head against wall now/S means O@T means 6BR7.a/ means (robability e)ent a ha((ens7.a/ S 7.b/ means (robability of a or b ha((ening7.a/ T 7.b/ means (robability of a ha((ening 6BR 5%B b ha((ening 5hat little s1uare is df T dg, its ind of lie tae a little chun of f, then alittle chun of g$ &t’s lie licing the (e((eroni, 6BR 5%B a little Yea

'um(s on your tongue and lics a little grease from your tongue$ Houdon’t notice it com(ared the ]large’ little bit of grease you got, whichitself is small com(ared to the (ie$ ;ust df on its own is you licing the(e((eroni$ ;ust dg on its own is the Yea licing your tongue$ 5he sum dfS dg is either you licing the (e((eroni or the Yea licing your tongue$2ut df Tdg is licing the (e((eroni 6BR 5%B getting liced by the Yea, itmeans com(aring the Yea’s )ery little bit of grease to the whole (ie$

%o(e this hel(s, or at least that you got the chance to laugh at Calculusfor a little while$

xcelsior,ric [

"". &ric / sas!



orry if the formatting is hard to follow$ Can’t use tab so & used a bunch of s(aces instead to ee( my 4 lined u( under each other$ 5he 7osting^nome ate my s(aces$ %e also messed u( my comment marers TT atthe end of lines$

"#.Melissa sas!

666%%% L&^%52#L2888

"$. #heo A1H1 sas!

5his )ideo hel(ed me get the Chain @ule by woring out dGdx.sin.x//com(ared to dGdx.sin.x// in a really )isual and intuiti)eway$ htt(:GGyoutu$beGbcÔ*LL?)9H

"%.Bonnie sas!

5his is denitely written for (eo(le who ha)e already taen calculus and

not understood it, )ersus someone with almost no ex(osure who is tryingto learn$ & don’t e)en thin reading o)er this would be hel(ful$ Houassume & already now all this stu8 Ro you now of any good (lace to56@5 learning calculus3 -aybe & can come bac to this site after &memorie all those rules, if &’m still confused$

"&. kalid sas!

%i 2onnie, ye(, this lesson is denitely geared for someone in the tail endof a calculus class$ &f you’re 'ust starting out, chec out:

htt(:GGbetterex(lained$comGcalculusGlesson+?

%o(e that hel(s8

http://youtu.be/bcGOZLL1v4Y

http://betterexplained.com/calculus/lesson-1

http://youtu.be/bcGOZLL1v4Y

http://betterexplained.com/calculus/lesson-1



7o! To 4nderstand Derivatives: The Auotient Rule& )x*onents& and Lo'arithms

by Kalid Azad · 1" comments

Last time we tacled deri)ati)es with a “machine" meta(hor$ Functions are amachine with an in(ut .x/ and out(ut .y/ le)er$ 5he deri)ati)e, dyGdx, is how

much “out(ut wiggle" we get when we wiggle the in(ut:

Bow, we can mae a bigger machine from smaller ones .h 4 f S g, h 4 f T g,etc$/$ 5he deri)ati)e rules .addition rule, (roduct rule/ gi)e us the “o)erall

wiggle" in terms of the (arts$ 5he chain rule is s(ecial: we can “oom into" asingle deri)ati)e and rewrite it in terms of another in(ut .lie con)erting “miles(er hour" to “miles (er minute" \ we’re con)erting the “time" in(ut/$

6nd with that reca(, let’s build our intuition for the ad)anced deri)ati)e rules$Onward8

Division +Auotient Rule,

6h, the 1uotient rule \ the one nobody remembers$ Oh, maybe you

memoried it with a song lie “Low dee high, high dee low!", but that’s notunderstanding8

&t’s time to )isualie the di)ision rule .who says “1uotient" in real life3/$ 5heey is to see di)ision as a ty(e of multi(lication:





e ha)e a rectangle, we ha)e area, but the sides are “f" and “?Gg"$ &n(ut xchanges o on the side .by dx/, so f and g change .by df and dg/! but howdoes ?Gg beha)e3

Chain rule to the rescue8 e can wra( u( ?Gg into a nice, clean )ariable andthen “oom in" to see that yes, it has a di)ision inside$

o let’s (retend ?Gg is a se(arate function, m$ &nside function m is a di)ision,

but ignore that for a minute$ e 'ust want to combine two (ers(ecti)es:

• f changes by df, contributing area df T m 4 df T .? G g/• m changes by dm, contributing area dm T f 4 3

e turned m into ?Gg easily$ Fine$ 2ut what is dm .how much ?Gg changed/ interms of dg .how much g changed/3

e want the dierence between neighboring )alues of ?Gg: ?Gg and ?.g S dg/$For exam(le:

• hat’s the dierence between ?G9 and ?G<3 ?G?• %ow about ?G= and ?G93 ?G0• %ow about ?GW and ?G=3 ?G<0

%ow does this wor3 e get the common denominator: for ?G< and ?G9, it’s?G?$ 6nd the dierence between “neighbors" .lie ?G< and ?G9/ will be ? Gcommon denominator, aa ? G .x T .x S ?//$ ee if you can wor out why8

&f we mae our deri)ati)e model (erfect, and assume there’s no dierencebetween neighbors, the S? goes away and we get:

.5his is useful as a general fact: 5he change from ?G?00 to ?G?0? 4 one tenthousandth/

5he dierence is negati)e, because the new )alue .?G9/ is smaller than theoriginal .?G</$ o what’s the actual change3

• g changes by dg, so ?Gg becomes ?G.g S dg/• 5he instant rate of change is +?Gg Uas we saw earlierV• 5he total change 4 dg T rate, or dg T .+?Gg/

6 few gut checs:

•

hy is the deri)ati)e negati)e3 6s dg increases, the denominator getslarger, the total )alue gets smaller, so we’re actually shrining .?G< to ?G9is a shrin of ?G?/$



• hy do we ha)e +?Gg T dg and not 'ust +?Gg3 .5his confused me at

rst/$ @emember, +?Gg is the chain rule conversion actor between the“g" and “?Gg" scales .lie saying ? hour 4 W0 minutes/$ Fine$ Hou stillneed to multi(ly by how far you went on the “g" scale, aa dg8 6n hourmay be W0 minutes, but how many do you want to con)ert3

• here does dm t in3 m is another name for ?Gg$ dm re(resents the total

change in ?Gg, which as we saw, was +?Gg T dg$ 5his substitution tricis used all o)er calculus to hel( s(lit u( gnarly calculations$ “Oh, it looslie we’re doing a straight multi(lication$ hoo(s, we oomed in and sawone )ariable is actually a di)ision \ change (ers(ecti)e to the inner)ariable, and multi(ly by the con)ersion factor"$

7hew$ 5o con)ert our “dg" wiggle into a “dm" wiggle we do:

6nd get:

Hay8 Bow, your o)ereager textboo may sim(lify this to:

and it burns8 &t burns8 5his “sim(lication" hides how the di)ision rule is 'ust a

)ariation of the (roduct rule$ @emember, there’s still two sli)ers of area tocombine:

• 5he “f" .numerator/ sli)er grows as ex(ected



• 5he “g" .denominator/ sli)er is negative .as g increases, the area getssmaller/

#sing your intuition, you now it’s the denominator that’s contributing thenegati)e change$

)x*onents +e>x,

e is my fa)orite number$ &t has the (ro(erty

which means, in nglish, “e changes by ?00 of its current amount" .readmore/$

5he “current amount" assumes x is the ex(onent, and we want changes fromx’s (oint of )iew .dfGdx/$ hat if u.x/4x is the ex(onent, but we still wantchanges from x’s (oint of )iew3

&t’s the chain rule again \ we want to oom into u, get to x, and see how awiggle of dx changes the whole system:

• x changes by dx•

u changes by duGdx, or d.x/Gdx 4 x• %ow does eu change3

Bow remember, eu doesn’t now we want changes from x’s (oint of )iew$ eonly nows its deri)ati)e is ?00 of the current amount, which is the ex(onentu:

5he o)erall change, on a (er+x basis is:

5his confused me at rst$ & originally thought the deri)ati)e would re1uire us tobring down “u"$ Bo \ the deri)ati)e of efoo is efoo$ Bo more$

2ut if foo is controlled by anything else, then we need to multi(ly the rate ofchange by the con)ersion factor .d.foo/Gdx/ when we 'um( into that inner (ointof )iew$

Natural Lo'arithm

5he deri)ati)e is ln.x/ is ?Gx$ &t’s usually gi)en as a matter+of+fact$

-y intuition is to see ln.x/ as the time needed to grow to x:


http://betterexplained.com/articles/developing-your-intuition-for-math/









• ln.?0/ is the time to grow from ? to ?0, assuming ?00 continuousgrowth

O, ne$ %ow long does it tae to grow to the “next" )alue, lie ??3 .x S dx,where dx 4 ?/

hen we’re at x4?0, we’re growing ex(onentially at ?0 units (er second$ &ttaes roughly ?G?0 of a second .?Gx/ to get to the next )alue$ 6nd when we’reat x4??, it taes ?G?? of a second to get to ?$ 6nd so on: the time to the next)alue is ?Gx$

5he deri)ati)e

is mainly a fact to memorie, but it maes sense with a “time to grow"inte(reration$

( 7airy )xam*le: x>x

5ime to test our intuition: what’s the deri)ati)e of xx3

5his is a bad mamma 'amma$ 5here’s two a((roaches:

#pproach 2% :ewrite everything in terms of e.

Oh e, you’re so mar)elous:

6ny ex(onent .ab/ is really 'ust e in dierent clothing: Ueln.a/Vb$ e’re 'ustasing for the deri)ati)e of efoo, where foo 4 ln.x/ T x$

2ut wait8 ince we want the deri)ati)e in terms of “x", not foo, we need to 'um( into x’s (oint of )iew and multi(ly by d.foo/Gdx:

5he deri)ati)e of “ln.x/ T x" is 'ust a 1uic a((lication of the (roduct rule$ &fh4xx, the nal result is:

e wrote eUln.x/TxV in its original notation, xx$ Hay8 5he intuition was“rewrite in terms of e and follow the chain rule"$



#pproach 1% &ndependent (oints Af 4iew

@emember, deri)iati)es assume each (art of the system wors inde(endently$@ather than seeing xx as a giant glob, assume it’s made from two interactingfunctions: u)$ e can then add their indi)idual contributions$ e’re sneaythough, u and ) are the same .u 4 ) 4 x/, but don’t let them now8

From u’s (oint of )iew, ) is 'ust a static (ower .i$e$, if )4<, then it’s u</ so we

ha)e:

6nd from )’s (oint of )iew, u is 'ust some static base .if u4=, we ha)e =)/$e rewrite into base e, and we get

e add each (oint of )iew for the total change:

6nd the re)eal: u 4 ) 4 x8 5here’s no con)ersion factor for this new )iew(oint

.duGdx 4 d)Gdx 4 dxGdx 4 ?/, and we ha)e:

&t’s the same as before8 & was (retty excited to a((roach xx from a few

dierent angles$

2y the way, use olfram 6l(ha .lie so/ to chec your wor on deri)ati)es.clic “show ste(s"/$

Buestion% &f u were more comple!+ where would we use du>d!?

&magine u was a more com(lex function lie u4x S <: where would wemulti(ly by duGdx3

Let’s thin about it: duGdx only comes into (lay from u’s (oint of )iew .when )is changing, u is a static )alue, and it doesn’t matter that u can be furtherbroen down in terms of x/$ u’s contribution is

http://www.wolframalpha.com/input/?i=d%2Fdx+x%5E3

http://www.wolframalpha.com/input/?i=d%2Fdx+x%5E3



if we wanted the “dx" (oint of )iew, we’d include duGdx here:

e’re multi(lying by the “duGdx" con)ersion factor to get things from x’s (ointof )iew$ imilarly, if ) were more com(lex, we’d ha)e a d)Gdx term whencom(uting )’s (oint of )iew$

Loo what ha((ened \ we gured out the genric dGdu and con)erted it into amore s(ecic dGdx when needed$

It%s )asier With Infnitesimals

e(arating dy from dx in dyGdx is “against the rules" of limits, but wors great

with innitesimals$ Hou can gure out the deri)ati)e rules really 1uicly:

(roduct rule%

e set “df T dg" to ero when 'um(ing out of the innitesimal world and bacto our regular number system$

5hin in terms of “%ow much did g change3 %ow much did f change3" andderi)ati)es sna( into (lace much easier$ “Ri)ide through" by dx at the end$

Summary: See the 3a#hine

Our goal is to understand calculus intuition, not memoriation$ & need a fewanalogies to get me thining:

• Functions are machines, deri)ati)es are the “wiggle" beha)ior• Reri)ati)e rules nd the “o)erall wiggle" in terms of the wiggles of each

(art• 5he chain rule ooms into a (ers(ecti)e .hours 4 minutes/• 5he (roduct rule adds area• 5he 1uotient rule adds area .but one area contribution is negati)e/• e changes by ?00 of the current amount .dGdx ex 4 ?00 T ex/• natural log is the time for ex to reach the next )alue .x unitsGsec means

?Gx to the next )alue/ith (ractice, ideas start clicing$ Ron’t worry about getting tri((ed u( \ & still

tried to o)eruse the chain+rule when woring with ex(onents$ Learning is a(rocess8

%a((y math$



(**endix: Partial Derivatives

Let’s say our function de(ends on two in(uts:

5he deri)ati)e of f can be seen from x’s (oint of )iew .how does f change withx3/ or y’s (oint of )iew .how does f change with y3/$ &t’s the same idea: weha)e two “inde(endent" (ers(ecti)es that we combine for the o)erall beha)ior.it’s lie combining the (oint of )iew of two oli(sists, who thin they’re theonly “real" (eo(le in the uni)erse/$

&f x and y de(end on the same )ariable .lie t, time/, we can write thefollowing:

&t’s a bit of the chain rule \ we’re combining two (ers(ecti)es, and for each(ers(ecti)e, we di)e into its root cause .time/$

&f x and y are otherwise inde(endent, we re(resent the deri)ati)e along eachaxis in a )ector:

5his is the gradient, a way to re(resent “From this (oint, if you tra)el in the x

or y direction, here’s how you’ll change"$ e combined our ?+dimensional“(oints of )iew" to get an understanding of the entire d system$ hoa$

1. !oe sas!

hen will -ath curriculums begin combining conce(ts in meaningfulways lie this3 Calculus classes lie to s(lit ]7ower @ule,’ ]Quotient @ule,’and ]Chain @ule’ into discrete sections, when really they’re conse1uencesof the same basic idea$ 7erha(s it’s less labor+intensi)e teaching distinctformulas to be memoried, but it’s 'ust another reason (eo(le hear]Calculus’ and immediately glae o)er$

6nd while &’m lamentingPyour mention of innitesimals brings u( anothersore s(ot of mine$ 6 Calc 56 told me how se(arating ]dyGdx’ is ]againstthe rules,’ as you say, and & too it to heart$ &magine (oor, confused me a

cou(le semesters later in Ri1: “& thought this was against the rules8" 5he limit+based a((roach to teaching Calculus needs some seriousre)ision, (articularly for non+mathematicians mo)ing into (ractical elds$

". kalid sas!

http://en.wikipedia.org/wiki/Solipsism

http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/

http://en.wikipedia.org/wiki/Solipsism

http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/



;oe: & hear you \ we slice and dice conce(ts and miss the cohesi)ewhole$ 6ll the calculus rules are 'ust exam(les of how dierent sub(artscan contribute to the whole, but &’m only seeing that now, ?0S yearsafter high school$ #gh$

6nd yeah \ there’s so much “don’t do this, & don’t now why, but don’t8"in math$ hy is it against the rules3 hat are the “rules"3 Limits are aseatbelt introduced to address theoretical concerns many, many years

after Calculus was (ut into use$ Learning about seatbelts is ne, but don’tdi)e into them before you ex(lain what a car Ui$e$, calculusV is8

1. !ackson sas!

5han you for the time you’)e (ut into these articles they’)e hel(ed me alot and &’m glad to now there are (eo(le who care about intuition andshare it, but &’m confused about your intuition of the natural log$ hy isthe deri)ati)e always (redicting the next increment by one3 hy not $=3houldn’t it be intestimally small because it is using the in(ut of the a

naturally growing function3

". kalid sas!

5hans ;acson, great 1uestion$ &ntuiti)ely, thin about taing a “singleste( forward", which is ?Tdx$ 6nother way of seeing it: when taing thederi)ati)e, we s(lit our continuous function into discrete ste(s .a singledx wide at each ste(/ and see our rate of change when we increment bythe next dx$

6n analogy: we re(resent a (hoto with indi)idual (ixels .dx/ and ste(through one (ixel at a time$ 5he (ixels are chosen at an “innitely smallretina resolution" where we don’t notice them at the macro scale$.5here’s more on limits later in this series$/

#. !ackson sas!

orry for my incon)ienience but &’m confused how you got ?Gx$ ouldn’tthe deri)ati)e be dxGx because dx would be the change and x would bethe current )alue as dx a((roach 0$&’m 'ust confused why ?4dx instead

of a((roaching 0$

$. kalid sas!

Bo worries, great 1uestion, & realie it can be unclear$ & start withscenarios where “dx 4 ?" .which is a ^&6B5 ste(/ to estimate results inmy head$ 5hen, & can set dx 4 0 .taing the limit/ to get an exact(rediction$

Let’s say & want the deri)ati)e of x$ & imagine going from ?0 to

?? .we 'um(ed from x4?0 to x4??, so dx4?/$ 5he dierence is ?, orx S dx .0 S ?/$ & can then set dx 4 0 and get the exact answer of x$.&f there was no ga( between x and the next )alue, the deri)ati)e wouldbe x$/



5he natural log is harder to com(ute: it’s the time ex needs to growfrom ? to x$ %ow does it change3

&magine going from ?0 to ?? .again, dx4?/$ %ere, we’re at ?0 and wegrow ex(onentially u( to ??$ ince ex assumes we’re growing at ?00of our current )alue, it taes ?G?0 of a unit time to get to ??$ .?0 S.?G?0/T?0 4 ??/$

Bow, this isn’t T1uiteT accurate because as we’re going to ??, we’regetting faster$ &$e$, when we’re at ?0$= we’re growing at ?0$= units (erunit time, not the ?0 we ex(ected$ @emo)ing the imaginary dx xes this.we assume there is no mid(oint between x4?0 and the next )alue, so itreally is a (erfect ?Gx amount of time we wait/$



4nderstandin' 0al#ulus With ( 5an$ (##ount 3eta*hor

by Kalid Azad · 3# comments

Calculus exam(les are boring$ k%ey ids8 )er wonder about the distance,)elocity, and acceleration of a mo)ing (article3 Bo3 ell youjre loced in herefor =0 minutes8k

& lo)e (hysics, but itjs not the best lead+in$ &t maes us wait till science class

.>th grade3/ and worse, it im(lies calculus is kmath for science classk$ Couldnjtwe introduce the themes to =th graders, and relate it to e)eryday life3

& thin so$ o herejs the goal:

• #se money, not (hysics, to introduce calculus conce(ts• x(lore how (atterns relate .ban account to salaryA salary to raises/• #se our intuition to ex(lore (otential issues .can we ee( drilling into

(atterns3/

tra( on your math helmet, time to di)e in$

3oney money money

-y fa)orite calculus exam(le is the relationshi( between your ban account,salary, and raises$

%erejs ;oe .k'i# (oek/$ Hou, the sly scoundrel you are, snea onto ;oejs com(uterand monitor his ban account each wee$ hat can you learn3

6c$ Clearly, not much ha((ened ++ ;oe isnjt earning anything$ 6nd what if yousee this3



asy enough: ;oejs maing some money$ 6nd how much3 ith a 1uicsubtraction, we can gure out his weely (aychec$ 5urns out ;oe is maing asteady q?00Gwee$

• ey idea: &f & now your ban account, & now your salary 5he ban account is dependent on the salary ++ it changes because of theweely salary$

Raise the roo

Letjs go dee(er: nowing the salary, what else can we gure out3 ell, thesalary is another (attern to analye ++ we can see if it changes8 5hat is, we cantell if ;oejs salary is changing wee by wee .is he getting a raise3/$

5he (rocess:

• Loo at ;oejs weely ban account• 5ae the dierence in ban account to get the weely salary• 5ae the dierence in salary to get the weely raise .if any/

&n the rst exam(le .q?00Gwee/, itjs clear therejs no raise .sorry, ;oe/$ 5hemain idea is to ktae the dierencek to analye the rst (attern .ban accountto salary/ and ktae the dierence againk to nd yet another (attern .salary toraise/$

Wor$in' 6a#$!ards

e 'ust went kdownk, from ban account to salary$ Roes it wor the other way:nowing the salary, can & (redict the ban account3

Houjre hesitating, & can tell$ Hes, nowing ;oe gets q?00Gwee is nice$ 2ut$$$donjt we need to now the starting account balance3

Hes8 5he changes to his account .salary/ is not enough ++ where did it start3 For

sim(licity .i$e$, what you see in homewor (roblems/ we often assume ;oestarts with q0$ 2ut, if you are actually maing a (rediction, you want to nowthe initial conditions .the kS Ck/$

( 3ore 0om*lex Pattern



Letjs say ;oejs account grows lie this: ?00, <00, W00, ?000, ?=00$$$

hatjs going on3 &s it random3 ell, we can do our wee+by+wee subtractionto get this:

&nteresting ++ ;oejs income is changing each wee$ e do another wee+by+wee dierence and get this:

6nd ye(, ;oejs getting a steady raise of q?00Gwee$ Letjs get wild and chartthem on the same gra(h:



One way to thin about it: ;oe gets a raise each wee, which changes hissalary, which changes his ban account$ 6s the raises continue to a((ear, hissalary continues to increase and his ban account rises$ Hou can almost thinof the raise k(ushing u(k the salary, which k(ushes u(k the ban account$

SoBBB Where<s the 0al#ulus?

hatjs the formula for ;oejs ban account for any wee3 ell, itjs the sum ofhis salaries u( to that (oint:

?00 S 00 S <00 S 900$$$ 4 ?00 T n T .n S ?/G

5he formula for adding u( a series of numbers .? S S < S 9$$$/ is )ery closeto nG, and gets closer as the number of ste(s increases$

5his is our rst kcalculusk relationshi(:

• 6 constant raise .q?00Gwee/ leads to a$$$• Linear increase in salary .?00, 00, <00, 900/ which leads to a$$$• Quadratic .something T n/ increase in ban account .?00, <00, W00,

?000$$$ you see it cur)e8/Bow, why is it roughly ?G T n and not n3 One intuition: 5he linear increase insalary .?00, 00, <00/ gi)es us a triangle$ 5he area of the triangle re(resentsall the (ayments so far, and the area is ?G T base T height$ 5he base is n .thenumber of wees/ and the height .income/ is ?00 T n$

êometric arguments get more diIcult in higher dimensions ++ 'ust becausewe can wor out T?00 with addition doesnjt mean itjs the easiest way$Calculus gi)es us the rules to 'um( between (atterns .taing deri)ati)es andintegrals/$

http://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/

http://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/



Points to )x*lore

Our understanding of ban accounts, salaries, and raises lets us ex(loredee(er$

ould we "gure out the total earnings between weeks 2 and 2C?

ure8 5herejs two ways: we could add u( our income for each wee .wee ?

salary S wee salary S wee < salary$$$/ or 'ust subtract the ban account.ee ?0 ban account + wee ? ban account/$ 5his idea has a beefy name:the Fundamental 5heorem of Calculus8

an we keep going 7down7 Dtaking derivatives beyond the raise?

ell, why not3 &f the raise is q?00Gwee, if we tae the dierence again we seeit dro(s to 0 .there is no kraise raisek, aa the raise is always steady/$ 2ut, wecan imagine the case where the raise itself is raising .wee? raise 4 ?00,wee raise 4 00/$ #sing our intuition: if the kraise raisek is constant, the

raise is linear .something T n/, the income is 1uadratic .something T n/ andthe ban account is cubic .something T n</$ 6nd yes, itjs true8

an derivatives go on forever?

He($ -aybe the connection is ban account 4 salary 4 raise 4 inYation4 mil out(ut of Farmer ;oejs cow 4 how much ;oe feeds the cow eachwee$ -any (atterns ksto( ha)ing deri)ati)esk once we get to the root cause$2ut certain interesting (atterns, lie ex(onential growth, ha)e an innite

number of com(onents8 Hou ha)e interest, which earns interest, which earnsinterest, which earns interest$$$ fore)er8 Hou can ne)er nd the single krootcausek of your ban account because an innite number of com(onents wentinto it .(retty tri((y/$

What happens if the raise goes negative?

&nteresting 1uestion$ 6s the raise goes negati)e, his salary will start lowering$2ut, as long as the salary is abo)e ero, the ban account will ee( rising86fter all, going from q00 to q?00 (er wee, while bad to you, still hel(s your

ban account$ )entually, a negati)e raise will o)er(ower the salary, maing itnegati)e, which means ;oe is now (aying his em(loyer$ 2ut u( until that (oint, ;oejs ban account would be growing$

How Euickly can we check for di;erences?

u((ose wejre measuring a stoc (ortfolio, not a ban account$ e might wanta second+by+second model of our salary and account balance$ 5he idea is tomeasure at inter)als short enough to get the detail we need ++ a large as(ect

of calculus is deciding what klimitk is enough to say kO, this is accurateenough for me8k$



5he calculus formulas you ty(ically see .integral of x 4 ?G T x/ are dierentfrom the kdiscretek formulas .sum of ? to n 4 ?G T n T .n S ?// because thediscrete case is using kchunyk inter)als$

Cey Ta$ea!ays

hy do & care about the analogy used3 5he traditional kdistance, )elocity,accelerationk doesnjt lead to the right 1uestions$ hatjs the next deri)ati)e of

acceleration3 .&tjs called k'erk, and itjs rarely used/$ uch a literal exam(le islie ha)ing ids thin multi(lication is only for nding area, and only wors ontwo numbers at a time$

%erejs the ey (oints:

• Calculus hel(s us nd related (atterns .ban account, to salary, to raises/• 5he kderi)ati)ek is going kdownk .nding wee+by+wee changes to get

your salary/• 5he kintegralk is going ku(k .adding u( your salary to get your ban

account/• e can gure out a formula for a (attern .gi)en my ban account, (redict

my salary/ or get a s(ecic )alue .whatjs my salary at wee <3/• Calculus is useful outside the hard sciences$ &f you ha)e a (attern or

formula .(roduction rate, sie of a (o(ulation, ^R7 of a country/ andwant to examine its beha)ior, calculus is the tool for you$

• 5extboo calculus in)ol)es memoriing the rules to deri)e and integrateformulas$ Learn the basics .xn, e, ln, sin, cos/ and lea)e the rest tomachines$ Our brain(ower is better s(ent learning how to translate our

thoughts into the language of math$&n my fantasy world, deri)ati)es and integrals are 'ust two e)eryday conce(ts$ 5heyjre kwhat you can dok to formulas, 'ust lie addition and subtraction arekwhat you can dok to numbers$

k%ey ids, we nd the total mass using addition .-ass? S -ass 4 -ass</$6nd to nd out how our (osition changes, we use the deri)ati)ek$

kRuh ++ addition is how you combine stu$ 6nd yeah, you tae the deri)ati)e tosee how your (osition is changing$ hat else would you do3k

One can always dream$ %a((y math$

1. !ohann sas!

%i alid,

5he thing about (hysics is that it’s more a((ro(riate fordescribing continuous)ariations$ -oney is a discrete (rocess and thusa case of non+continuous )ariations!2ut & agree it is a nice )iew to ex(lain it, and maybe lin it to the boringuse made in (hysics .(robably because it’s always (resented in the sameway!/$

6gain, great reading 8 ee( u( the good wor

http://naturelovesmath-en.blogspot.com/





2ests, ;ohann

". Prudh*i sas!

ingenious8it’s true that a lot of (eo(le get bored by the direct (hysics a((lication ofcalculus if its introduced too soon

money seems lie a natural way to understand itof course! it might not be able to so easily gi)e an intuiti)eunderstanding of the more grainy as(ects of calculus, but whate)er,thans8

#. Kalid sas!

;ohann: 5hans for dro((ing by, and great (oint about continuous )s$discrete$ 5he funny thing is that many (hysicists treat the formulas as“discrete" .i$e$ using innitesimal dx, dy, d 1uantities to mae a <d

cube, for exam(le/ and then “let it disa((ear" to mae it continuousagain$ 5he neat thing is that using discrete 1uantities really shows howthe error margin is there .the dierence between the actual sum ofs1uares and ?G T x/ and how limits G @iemann sum hel( us shrin this$

& agree though, that (hysics would be cool if it were shown to be anexam(le of these general (rinci(les .and not the “denition" as is oftenseen/$

7rudh)i: He(, there’s always details that you can’t get to when you

mae analogies$ 2ut you ha)e to start somewhere :/$

$. M! sas!

&f someone had outright told me at any (oint in Calc & or Calc && that the]S C’ can be thought of as an initial condition, & might ha)e actuallyremembered to tac it to the end of integrals, instead of considering it anarbitrary annoyance that has little context$

5hat maes so much sense .and yet is so, so, so, (ainfully ob)ious/, that

it’s not e)en funny$

%. Kalid sas!

-;: 5hans for the comment P yeah, it’s TwayT too easy to thin of theSC as some mathematical details to ee( trac of, instead of something needed to gure out how to mae your model wor$



( @entle Introdu#tion To Learnin' 0al#ulus

by Kalid Azad · 31" comments

& ha)e a lo)eGhate relationshi( with calculus: it demonstrates the beauty ofmath and the agony of math education$

Calculus relates to(ics in an elegant, brain+bending manner$ -y closestanalogy is Rarwin’s 5heory of )olution: once understood, you start seeing

Bature in terms of sur)i)al$ Hou understand why drugs lead to resistant germs.sur)i)al of the ttest/$ Hou now why sugar and fat taste sweet .encourageconsum(tion of high+calorie foods in times of scarcity/$ &t all ts together$

Calculus is similarly enlightening$ Ron’t these formulas seem related in someway3

5hey are$ 2ut most of us learn these formulas inde(endently$ Calculus lets usstart with “circumference 4 T (i T r" and gure out the others \ the ^reeswould ha)e a((reciated this$

Fnfortunately+ calculus can epitomize what’s wrong with math

education$ -ost lessons feature contri)ed exam(les, arcane (roofs, andmemoriation that body slam our intuition ` enthusiasm$

&t really shouldn’t be this way$

3ath& art& and ideas

&’)e learned something from school: 'ath isn’t the hard part of mathG

motivation is.(ecically, staying encouraged des(ite

• 5eachers focused more on (ublishingG(erishing than teaching• elf+fullling (ro(hecies that math is diIcult, boring, un(o(ular or “not

your sub'ect"

• 5extboos and curriculums more concerned with (rots and test resultsthan insight

]6 -athematician’s Lament’ U(dfV is an excellent essay on this issuethat resonated withmany (eo(le:

https://web.archive.org/web/20081216021713/http:/www.redshift.com/~jmichael/html/feynman.html

http://www.maa.org/external_archive/devlin/LockhartsLament.pdf

http://reddit.com/info/6baz9/comments/

http://www.metafilter.com/70699/A-Mathematicians-Lament

http://www.math.princeton.edu/~bbukh/natural.html

https://web.archive.org/web/20081216021713/http:/www.redshift.com/~jmichael/html/feynman.html

http://www.maa.org/external_archive/devlin/LockhartsLament.pdf

http://reddit.com/info/6baz9/comments/

http://www.metafilter.com/70699/A-Mathematicians-Lament

http://www.math.princeton.edu/~bbukh/natural.html



“!if & had to design a mechanism for the ex(ress (ur(ose of destroying achild’s natural curiosity and lo)e of (attern+maing, & couldn’t (ossibly do asgood a 'ob as is currently being done \ & sim(ly wouldn’t ha)e the imaginationto come u( with the ind of senseless, soul+crushing ideas that constitutecontem(orary mathematics education$"

&magine teaching art lie this: *ids+ no "ngerpainting in

kindergarten. &nstead, let’s study (aint chemistry, the (hysics of light, andthe anatomy of the eye$ 6fter ? years of this, if the ids .now teenagers/ don’thate art already, they may begin to start coloring on their own$ 6fter all, theyha)e the “rigorous, testable" fundamentals to start a((reciating art$ @ight3

7oetry is similar$ &magine studying this 1uote .formula/:

“5his abo)e all else: to thine own self be true, and it must follow, as nightfollows day, thou canst not then be false to any man$" \illiam haes(eare,%amlet

&t’s an elegant way of saying “be yourself" .and if that means writingirre)erently about math, so be it/$ 2ut if this were math class, we’d be countingthe syllables, analying the iambic (entameter, and ma((ing out the sub'ect,)erb and ob'ect$

'ath and poetry are "ngers pointing at the moon. on’t confuse the

"nger for the moon. Formulas are a means to an end, a way to ex(ress amathematical truth$

e’)e forgotten that math is about ideas, not robotically mani(ulating theformulas that ex(ress them$

$ 6u6& !hat%s your 'reat idea?

Feisty, are we3 ell, here’s what & won’t do: recreate the existing textboos$ &fyou need answers right away for that big test, there’s (lenty of websites, class)ideos and 0+minute s(rints to hel( you out$

&nstead+ let’s share the core insights of calculus$ 1uations aren’t

enough \ & want the “aha8" moments that mae e)erything clic$

Formal mathematical language is one 'ust one way to communicate$ Riagrams,animations, and 'ust (lain talin’ can often (ro)ide more insight than a (agefull of (roofs$

5ut #al#ulus is hard

& thin anyone can a((reciate the core ideas of calculus$ e don’t need to bewriters to en'oy haes(eare$

&t’s within your reach if you now algebra and ha)e a general interest in math$Bot long ago, reading and writing were the wor of trained scribes$ Het todaythat can be handled by a ?0+year old$ hy3

http://www.sosmath.com/calculus/calculus.html

http://ocw.mit.edu/OcwWeb/Mathematics/index.htm


http://youtube.com/watch?v=EX_is9LzFSY

http://www.sosmath.com/calculus/calculus.html



http://youtube.com/watch?v=EX_is9LzFSY



2ecause we ex(ect it$ x(ectations (lay a huge (art in what’s (ossible$o expect that calculus is 'ust another sub'ect$ ome (eo(le get into the nitty+gritty .the writersGmathematicians/$ 2ut the rest of us can still admire what’sha((ening, and ex(and our brain along the way$

&t’s about how far you want to go$ &’d lo)e for e)eryone to understand the coreconce(ts of calculus and say “whoa"$

So !hat%s #al#ulus a6out?

ome dene calculus as “the branch of mathematics that deals with limits andthe dierentiation and integration of functions of one or more )ariables"$ &t’scorrect, but not hel(ful for beginners$

%ere’s my tae: Calculus does to algebra what algebra did to arithmetic$

• #rithmetic is about mani(ulating numbers .addition, multi(lication,

etc$/$

• #lgebra "nds patterns between numbers: a S b 4 c is a

famous relationshi(, describing the sides of a right triangle$ 6lgebra ndsentire sets of numbers \ if you now a and b, you can nd c$

• alculus "nds patterns between eEuations: you can see how one

e1uation .circumference 4 T (i T r/ relates to a similar one .area 4 (i Tr/$

#sing calculus, we can as all sorts of 1uestions:

• %ow does an e1uation grow and shrin3 6ccumulate o)er time3• hen does it reach its highestGlowest (oint3• %ow do we use )ariables that are constantly changing3 .%eat, motion,

(o(ulations, !/$• 6nd much, much more8

6lgebra ` calculus are a (roblem+sol)ing duo: calculus nds new e1uations,and algebra sol)es them$ $ike evolution+ calculus e!pands your

understanding of how /ature works.

(n )xam*le& Please

Let’s wal the wal$ u((ose we now the e1uation for circumference . T (i Tr/ and want to nd area$ hat to do3

:ealize that a "lled-in disc is like a set of :ussian dolls.

http://www.answers.com/calculus

http://www.answers.com/calculus



%ere are two ways to draw a disc:

• -ae a circle and ll it in• Rraw a bunch of rings with a thic marer

5he amount of “s(ace" .area/ should be the same in each case, right3 6nd howmuch s(ace does a ring use3

ell, the )ery largest ring has radius “r" and a circumference T (i T r$ 6s therings get smaller their circumference shrins, but it ee(s the (attern of T (iT current radius$ 5he nal ring is more lie a (in(oint, with no circumference atall$

Bow here’s where things get funy$ $et’s unroll those rings and line them

up. hat ha((ens3



• e get a bunch of lines, maing a 'agged triangle$ 2ut if we tae thinnerrings, that triangle becomes less 'agged .more on this in future articles/$

• One side has the smallest ring .0/ and the other side has the largest ring. T (i T r/

• e ha)e rings going from radius 0 to u( to “r"$ For each (ossible radius.0 to r/, we 'ust (lace the unrolled ring at that location$

• 5he total area of the “ring triangle" 4 ?G base T height 4 ?G T r T . T (iT r/ 4 (i T r, which is the formula for area8

Howa8 5he combined area of the rings 4 the area of the triangle 4 area ofcircle8

.&mage from ii(edia/

5his was a 1uic exam(le, but did you catch the ey idea3 e too a disc, s(litit u(, and (ut the segments together in a dierent way$ Calculus showed usthat a disc and ring are intimately related: a disc is really 'ust a bunch of rings$

5his is a recurring theme in calculus: 6ig things are made from little

things. 6nd sometimes the little things are easier to wor with$

( note on exam*les

-any calculus exam(les are based on (hysics$ 5hat’s great, but it can be hardto relate: honestly, how often do you now the equation or velocity for anob'ect3 Less than once a wee, if that$

& (refer starting with (hysical, )isual exam(les because it’s how our mindswor$ 5hat ringGcircle thing we made3 Hou could build it out of se)eral (i(ecleaners, se(arate them, and straighten them into a crude triangle to see ifthe math really wors$ 5hat’s 'ust not ha((ening with your )elocity e1uation$

( note on ri'or +or the math 'ee$s,

& can feel the math (edants ring u( their eyboards$ ;ust a few words on“rigor"$

Rid you now we don’t learn calculus the way Bewton and Leibni disco)eredit3 5hey used intuiti)e ideas of “Yuxions" and “innitesimals" which werere(laced with limits because<)ure+ it works in practice. 6ut does it work

in theory?=$

http://upload.wikimedia.org/wikipedia/commons/0/07/TriangleFromCircle.gif

http://upload.wikimedia.org/wikipedia/commons/0/07/TriangleFromCircle.gif



e’)e created com(lex mechanical constructs to “rigorously" (ro)e calculus,but ha)e lost our intuition in the (rocess$

e’re looing at the sweetness of sugar from the le)el of brain+chemistry,instead of recogniing it as Bature’s way of saying “5his has lots of energy$ atit$"

& don’t want to .and can’t/ teach an analysis course or train researchers$ ould

it be so bad if e)eryone understood calculus to the “non+rigorous" le)el thatBewton did3 5hat it changed how they saw the world, as it did for him3

6 (remature focus on rigor dissuades students and maes math hard to learn$Case in (oint: e is technically dened by a limit, but the intuition of growth ishow it was disco)ered$ 5he natural log can be seen as an integral, or the timeneeded to grow$ hich ex(lanations hel( beginners more3

Let’s nger(aint a bit, and get into the chemistry along the way$ %a((y math$

.7: 6 ind reader has created an animated (ower(oint slideshow that hel(s(resent this idea more )isually .best )iewed in 7ower7oint, due to theanimations/$ 5hans8/

Andre says

^reat article8 & lo)e the insights$ &’m currently taing u( calculus class and &nd it hard to learn its essence 'ust by taing it u( in school$ & mean,de(ending u(on the style of one’s (rofessor, & thin math is a sub'ect one can

get by without much thining by 'ust nowing its (rocedure .exce(t integralcalculus, & thin/$ 2ut & nd myself being reluctant to score that way .and alsond integral calculus challenging/, so & surfed the &nternet to see for a websitethat would mae me understand what calculus really means, and your websiteturns out to be exactly what &’m looing for8& can really relate to you alid\& also feel that our math education systemtoday is being “head o)er the clouds" and must be more down+to+earth tobeginners$ Bot understanding the essence of mathematics maes the ma'orityof (eo(le not a((reciate it$ 5o gi)e an analogy, it’s lie they’re seeing music inwritten form and calling it music without e)en listening to it$ &n order to

understand what an abstract word really means, one must get a hold rst of itsmanifestations in the concrete world, and then how the abstract thereafterrelates to the concrete$ & thin whene)er (eo(le say they hate the beautifulsub'ect math, they 'ust don’t really understand what it means$&’)e read some of the others’ comments regarding e)olution$ & feel mo)ed toshare some facts, inferences and insights regarding its )alidity$Our scientic formulae are so (redicti)e only because each scientic formulare(resents a scientic generalisation that has been based on factualobser)ations$ &t’s because we ha)e obser)ed a set of (henomena to beconsistent that we classify them together and mae a scientic generalisation

out of them, taing ad)antage of their consistency to mae (redictions forfuture (ur(oses$ e ee( on obser)ing sets of (henomena in this way$%owe)er, that does not ex(lain how they can be consistent$ 5herefore one isleft with two general categories to ex(lain the consistency of each of them: .?/occurrences ensueA ./ otherwise, they’re being controlled$ hat do we call

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e

http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/


http://betterexplained.com/examples/files/CalcIntroAnimation.ppt

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e



http://betterexplained.com/examples/files/CalcIntroAnimation.ppt



these certainties in the uni)erse3 7hysical laws, which are certain, can’t be 'ustsome chance e)ents, which are random and uncertain$ Our lac of nowledge(ermits us to belie)e that some things 'ust ha((en by chance when we don’tnow what caused it, but that’s not the attitude of a scientistA scienceattem(ts to ex(lain causes or it won’t ha)e a cause$ &f the uni)erse wouldn’tfollow (hysical laws, we wouldn’t be able to classify anything .e$g$ atoms/, letalone obser)e any consistency$ hat intuition do you thin dro)e us to call(hysical laws laws3 Laws are commands$ Bothing comes from nothing$ 5he law

of conser)ation of energy signies this$ &f one wants to belie)e that somethingcan arise by itself, it shouldn’t be the uni)erse, because the uni)erse is underthe law of conser)ation of energy$ 5his therefore maes us conclude that theuni)erse has always existed from eternity (ast$ %owe)er, the uni)erse began$Our uni)erse is characterised by cosmic ex(ansion$ 5he second law ofthermodynamics indicates that the longer time has ela(sed, the greater theo)erall entro(y of the uni)erse shall be$ î)en that the uni)erse is currentlynot at a state of maximum entro(y, the rst and second laws ofthermodynamics indicate that the uni)erse must not ha)e always existed frometernity (ast$ -atter, energy, s(ace and time, which constitute the uni)erse,

ha)e not always existed$ 5herefore, because the uni)erse began to exist,either some 2eing or something must ha)e caused it$ 5his cause of theuni)erse must be immaterial, because the cause of the uni)erse cannot be theuni)erse itself, which is the totality of all material things, as nothing can causeitself that has not arisen from nothing$ &n other words, something causing itself is lie saying that it a((eared out of nowhere$ omething arising out of nothingcan only be true if that thing is not under the law of conser)ation of energy, or,if some 2eing xor some other thing caused it that, being able to create energy,is abo)e the law of conser)ation of energy$ 2ecause of laws such as the laws of thermodynamics, only the Creator can and will create the uni)erse fromnothing$ 2eing transcendent, the Creator of the uni)erse must (ossess auni1ue nature distinct from the uni)erse or from anything in it as much as theCreator of the uni)erse hasn’t caused the uni)erse or anything in it to bearresemblance to the Creator’s nature$ 5his nature then doesn’t necessarily ha)eto be tangible nor )isible to our eyes$ 5he theory of e)olution holds that millions and millions of years ago, shbegan e)ol)ing by means of little cumulati)e changes o)er long (eriods oftime$ O)er a((roximately ?D0000000 years, sh managed to e)ol)e toam(hibians$ O)er a((roximately W0000000 years, am(hibians e)ol)ed to

re(tiles$ ome of these re(tiles e)ol)ed to nonmoney mammals, still o)er along (eriod of time, which then e)ol)ed to moneys\sim(ly (ut, ourancestors$ Of course, sh came all the way from a common ancestor$ 5his iswhat Rarwin has (ro(osed$ 6fter the disco)ery of RB6, howe)er, the theory ofe)olution itself e)ol)ed to include nonli)ing chemicals that ha((ened to li)e bytime and incredible luc$ 5here is no substantial e)idence, howe)er, to su((ort this$ &t doesn’t followthat similarities in RB6 should indicate a common descent$ 5he assertion thatgenus e)ol)es to another genus o)er a )ery long (eriod of time is contrary toscience .genome is the total of all the genetic (ossibilities for a gi)en s(ecies,

and should not be confused for genoty(e/$ & understand that, in order toa((ear as though it was falsiable, and thus be con)incing, this assertionde(ends on natural selection$ 2ut it’s not the other way aroundA it is notre1uisite for this unobser)able assertion to be true in order for naturalselection to be true, or for natural things to ser)e some (ur(ose$ One (ur(ose



of natural selection is to eliminate the abnormal .mutations causeabnormalities/$ %owe)er, natural selection doesn’t cause ada(tationA all it doesare to eliminate the wea and the mutated and to s(are the sur)i)ors to li)e alonger re(roducti)e life$ 5oo much of this and extinction would occur$ Li)ingbeings ada(t to their surroundings because of the way they were designed Pnot because of natural selectionA without design in the rst (lace, naturalselection would be meaningless$ hat’s obser)able in nature are ada(tation,death and the fact that s(ecies can only (roduce s(ecies of their own ind$ Bo

one has e)er obser)ed actual e)olution ha((en naturally$ One only seessu((osed e)olution in some man+made boos with (ictures and in man+maderealistic <R animation mo)ies$ 6ll (ro(onents of the theory of e)olution canshow are some fossil remains with similarities, which ha)e already undergonedecom(osition$arnest 6$ %ooton, from %ar)ard #ni)ersity, states, “5o attem(t to restore thesoft (arts is an e)en more haardous undertaing$ 5he li(s, the eyes, the ears,and the nasal ti( lea)e no clues on the underlying bony (arts$ Hou can withe1ual facility model on a Beanderthaloid sull the features of a chim(anee orthe lineaments of a (hiloso(her$ 5hese alleged restorations of ancient ty(es of

man ha)e )ery little if any scientic )alue and are liely only to mislead the(ublic! o (ut not your trust in reconstructions$".#( From 5he 6(e ($ <</imilarities in RB6 do not indicate a common descentA similarities in RB6indicate a one language\the language of RB6 itself\and this we ha)ee)idence of$ 5he fact that one language was used to design, and to dictate allthe functions of, all li)ing beings on arth is 'ust undeniable$ 6fter all, all li)ingbeings on arth ha)e one thing in common\ life$ &f one has e)er used a(rogramming language before, one would understand the necessity of reusinga set of s(ecic codes to a number of dierent (rograms$ Com(uter(rogrammers though ha)e a way of con)erting lengthy codes to 'ust a shortone by sa)ing codes in header les because it would be tiring for humans torety(e lengthy codes o)er and o)er again$ &nformation is contained in our RB6,and our bodies were designed, and functions, as well, according to thes(ecications of this information$ hat ha((ens when a li)ing being isex(osed to harmful things such as radiation3 -utations are alterations thattae (lace in the RB6\damaging the information in it$ &t’s im(ossible for li)ingbeings to ac1uire new organs through mutations, because mutations do notadd new genetic information$ 5hings don’t 'ust ha((en by chance to an

omniscient beingA our lac of nowledge (ermits us to belie)e that somethings 'ust ha((en by chance when we don’t now what caused it, butintelligence identies with intelligence \ lie archaeologists do$ &nformationne)er originates by itself in matterA it always comes from an intelligent source$Ruring Rarwin’s time, this extremely com(lex chemical macromolecule calledthe RB6 was not yet disco)ered$ 5he outdated microsco(es of their time madethe )ery com(lex structure of the cell loo so sim(le$ %owe)er, if we wouldsubscribe to the current scientic disco)eries, as well as the technologies, ofour time, we would begin to a((rehend that the indications ne)er really(ointed to the theory of e)olution$ 6s science (rogresses, intelligent design

becomes more e)ident$ e shouldn’t limit oursel)es therefore in Rarwin’sworld)iew$Rarwin himself wrote, “&f it could be demonstrated that any com(lex organexisted, which could not (ossibly ha)e been formed by numerous, successi)e,slight modications, my theory would absolutely brea down$"



.5he Origin of (ecies ($ ?E>/Bote: %e didn’t write Umy theory would absolutely e)ol)eV$hat else is the meaning of e)idence3 )erywhere we loo, the more attenti)ewe are to the details, the more e)ident intelligent design becomes$-athematical formulae are symbolic re(resentations of mathematical ideas,and ideas can only be concei)ed by the mind$ e ex(erience this whene)erwe concei)e mathematical ideas$ 5he Fibonacci numbers is one such idea$Fibonacci numbers and golden section often occur in nature, e)en in our

bodies, and this re(etition goes against mere coincidences$ &t seems to methen that 'ust as we humans can mae something only out of that which hasalready been created, we can not concei)e mathematical ideas other than thatwhich has already been thought by an immaterial intelligent 2eing (rior to theuni)erse, as we humans rely u(on the uni)erse to deri)e our conclusions andmathematical ideas from$ 6ll mathematical ideas that we now of areembedded throughout the whole uni)erse$ 6s a matter of fact, mathematics isso (er)asi)e it e)en (ermeates science$ 5his does not contradict intelligence(rior to the uni)erse, but rather, (ro)es it$& understand that not all religions can be trusted to teach one what is true, but

lies exist not only in religion$ 7eo(le shouldn’t be “throwing the baby with thebathwater" and not lea)ing room for creation 'ust because some (eo(le whohold false beliefs ha((en to belie)e also in creation$ &t doesn’t follow thatcreation should be false due to that$ & thin (eo(le who dismiss intelligentdesign because of other (eo(le’s attitude against reason should be less biasedin their focus and consider also the scientists who belie)e in creation due tothe intelligently designed things that surround us$ One should learn u(on theinsights of the reasonable, rather than calling the untaught ignorant withoute)en educating them$& 'ust feel lie sharing these facts, inferences and insights of mine because &belie)e “iron shar(ens iron" and because & belie)e it’s im(ortant for you tonow the truth$ &’m always glad to hear others’ insights about the truth .whatis true/ in general$ & lo)e math because to me there is nothing more beautifulthan the truth, and math to me is also the realisation of the 1uantitati)eob'ecti)e as(ect of the truth .algebraic logic counts truth )alue P 0, ?/$



EB Learnin' The f#ial Terms

*e+ve een ale to -escrie our ste//ste rocess with analoies (/2as, 3ime/

lases, an- rins) an- -iarams!

4owever, this is a ver elaorate wa to communicate. 4ere+s the 5fficial 6ath7

terms!

$ntuiti*e

-once+t0ormal Name Symbol

/2a (slit aart) 3a8e the -erivative (-erive) ddr

3ime/lase (lue

toether)3a8e the interal (interate) ∫

9rrow -irection:nterate or -erive ;with resect to;

a variale.

dr imlies movin

alon r

9rrow startsto =oun-s or rane of interation ∫endstart

lice:nteran- (shae ein lue-

toether, such as a rin)

?@uation, such

as2r

Let+s wal8 throuh the fanc names.



The Derivative

3he deri*ati*e is slittin a shae into sections as we move alon a ath (i.e., /

2ain it). Aow here+s the tric8! althouh the -erivative enerates the entire se@uence

of sections (the lac8 line), we can also extract a single one.

3hin8 aout a function li8e f (x)=x2. :t+s a curve that -escries a iant list of

ossiilities (1, $, , 1&, "%, etc.). *e can rah the entire curve, sure, or examine the

value of f(x) at a secific value, li8e x=3.

3he -erivative is similar. 5fficiall, it+s the entire attern of sections, ut we can Bero

into a secific one as8in for the -erivative at a certain value. (3he -erivative is a

function, Cust li8e f (x)=x2D if not otherwise secifie-, we+re -escriin the entirefunction.)

*hat -o we nee- to fin- the -erivativeE 3he shae to slit aart, an- the ath to follow

as we cut it u (the orane arrow). For examle!

• 3he -erivative of a circle with respect to the ra-ius creates rins

• 3he -erivative of a circle with respect to the erimeter creates slices

• 3he -erivative of a circle with respect to the x/axis creates oar-s

: aree that ;with resect to; soun-s formal! Honorable Grand Poombah radius, it is

with respect to you that we derive. 6ath is a entleman+s ame, : suose.

3a8in the -erivative is also calle- ;-ifferentiatin;, ecause we are fin-in the

-ifference etween successive ositions as a shae rows. (9s we row the ra-ius of a

circle, the -ifference etween the current -isc an- the next siBe u is that outer rin.)

The Inte'ral& (rro!s& and Sli#es

3he integral is luein toether (time/lasin) a rou of sections an- measurin the

final result. For examle, we lue- toether the rins (into a ;rin trianle;) an- saw it

accumulate- tor2, a8a the area of a circle.

4ere+s what we nee- to fin- the interal!



• 2hich direction are we gluing the ste+s together3 9lon the orane line

(the ra-ius, in this case)

• 2hen do we start and sto+3 9t the start an- en- of the arrow (we start at 0,

no ra-ius, an- move to r, the full ra-ius)

• How big is each ste+3 *ellG each item is a ;rin;. :sn+t that enouhE

AoeH *e nee- to e secific. *e+ve een sain we cut a circle into ;rins; or ;iBBa

slices; or ;oar-s;. =ut that+s not secific enouhD it+s li8e a ==I recie that sas ;Joo8

meat. Flavor to taste.;

6ae an exert 8nows what to -o, ut we nee- more secifics. 4ow lare, exactl, is

each ste (technicall calle- the ;interan-;)E

9h. 9 few notes aout the variales!

• :f we are movin alon the ra-ius r, then dr is the little chun8 of ra-ius in the

current ste

• 3he heiht of the rin is the circumference, or2r

3here+s several otchas to 8ee in min-.

First, dr is its own variale, an- not ;- times r;. :t reresents the tin section of the

ra-ius resent in the current ste. 3his smol (dr, dx, etc.) is often searate- from

the interan- Cust a sace, an- it+s assume- to e multilie- (written2r dr).

Aext, if r is the onl variale use- in the interal, then dr is assume- to e there. o if

ou see∫2r this still imlies we+re -oin the full∫2r dr. (9ain, if there are two



variales involve-, li8e ra-ius an- erimeter, ou nee- to clarif which ste we+re

usin! dr or d(E)

Last, rememer that r (the ra-ius) chanes as we time/lase, startin at 0 an-

eventuall reachin its final value. *hen we see r in the context of a step, it means ;the

siBe of the ra-ius at the current ste; an- not the final value it ma ultimatel have.

3hese issues are extremel confusin. :+- refer we use r-r to in-icate an interme-iate

;r at the current ste; instea- of a eneral/urose ;r; that+s easil confuse- with the

max value of the ra-ius. : can+t chane the smols at this oint, unfortunatel.

Pra#ti#in' The Lin'o

Let+s learn to tal8 li8e calculus natives. 4ere+s how we can -escrie our /2astrateies!

$ntuiti*e

/isualization0ormal descri+tion Symbol

derive the area of a circle with respect to

the radius

ddr6rea


the perimeter

dd(6r

ea


the x-axis

ddx6re

a



2ememer, the -erivative Cust slits the shae into (hoefull) eas/to/measure stes,

such as rins of siBe2r dr. *e ro8e aart our leo set an- have ieces scattere- on

the floor. *e still nee- an interal to lue the arts toether an- measure the new siBe.

3he two comman-s are a ta team!

• 3he -erivative sas! ;58, : slit the shae aart for ou. :t loo8s li8e a unch of

ieces2r tall an- dr wi-e.;

• 3he interal sas! ;5h, those ieces resemle a trianle // : can measure thatH

3he total area of that trianle is 12baseheight, which wor8s out to r2in this

case.;.

4ere+s how we+- write the interals to measure the stes we+ve ma-e!

0ormal descri+tion SymbolMeasures #otal

Size O,

integrate 2 * pi * r * dr

from r=0 to r=r ∫r02r dr

integrate [a pizza slice]

from [p = min perimeter] to

[p = max perimeter]

∫(=max(=min((ia slic

e)d(

integrate [a board] from [x =

min value] to [x = max value]

∫x=maxx=minboard dx

9 few notes!



• 5ften, we write an interan- as an unsecifie- ;iBBa slice; or ;oar-; (use a

formal/soun-in name li8e s(()or b(x)if ou li8e). First, we setu the

interal, an- then we worr aout the exact formula for a oar- or slice.

• =ecause each interal reresents slices from our oriinal circle, we 8now the

will e the same. Kluin an set of slices shoul- alwas return the total area,

rihtE

• 3he interal is often -escrie- as ;the area un-er the curve;. :t+s accurate, ut

shortsihte-. es, we are luin toether the rectanular slices un-er the curve.

=ut this comletel overloo8s the rece-in /2a an- 3ime/Lase thin8in.

*h are we -ealin with a set of slices vs. a curve in the first laceE 6ost li8el,

ecause those slices are easier than analBin the shae itself (how -o ou

;-irectl; measure a circleE).

Auestions

1) Jan ou thin8 of another activit which is ma-e simpler shortcuts an- notation,

vs. written ?nlishE

") :ntereste- in erformanceE Let+s -rive the calculus car, even if ou can+t uil- it et.

4uestion 5. 4ow woul- ou write the interals that cover half of a circleE

?ach shoul- woul- e similar to!

integrate [size of step] from [start] to [end] with respect to [path variable]

(9nswer for the first half an- the secon- half . 3his lin8s to *olfram 9lha, an online

calculator, an- we+ll learn to use it later on.)

http://www.wolframalpha.com/input/?i=integrate+2+*+pi+*+r+*+dr+from+r%3D0+to+r%3D0.5r


http://www.wolframalpha.com/input/?i=integrate+2+*+pi+*+r+*+dr+from+r%3D0.5r+to+r%3Dr






4uestion 6. Jan ou fin- the comlete wa to -escrie our ;iBBa/slice; aroachE

3he ;math comman-; shoul- e somethin li8e this!

integrate [size of step] from [start] to [end] with respect to [path variable]

2ememer that each slice is asicall a trianle (so what+s the areaE). 3he slices move

aroun- the erimeter (where -oes it start an- stoE). 4ave a uess for the comman-E

4ere it is, the slice//slice -escrition.

4uestion 7. Jan ou fiure out how to move from volume to surface areaE

http://www.wolframalpha.com/input/?i=integrate+1%2F2+*+r+*+dp+from+p%3D0+to+p%3D2*pi*r

http://www.wolframalpha.com/input/?i=integrate+1%2F2+*+r+*+dp+from+p%3D0+to+p%3D2*pi*r



9ssume we 8now the volume of a shere is 4! * pi * r"!. 3hin8 aout the instructions

to searate that volume into a se@uence of shells. *hich variale are we movin

throuhE

An Intuitive Introduction to Limits

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