Fundamental of Calculus (Presentation Slide)A

7/21/2019 Fundamental of Calculus (Presentation Slide)A

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By :

Armita Pratiwi Ray

Heru Setiawan Girsang

Josepine Halcynon Sinaga

Mutiah Aima

Putri Alawiyah

Calculus



Calculus is a set of mathematical tools for solvingcertain prolem in the same way that geometry isthe stu!y of shape an! algera is the stu!y ofoperation an! theirs application to solving e"uation#

$t has to mayor ranches% !i&erential calculus' concerning rates of change an! slopes of curves (%an! integral calculus ' concerning accumulation of"uantities an! the areas un!er an! etweencurves(# )hese two ranches are relate! to each

other y fun!amental theorem of calculus# Bothranches ma*e use of the fun!amental nation ofconvergence of in+nite se"uences an! in+niteseries to a well,!e+ne! limit

-hat is Calculus .



/nglish physicist $saac 0ewton '1234,1545(an! German mathematician Gottfrie! -ilhelm6on 7eini8 '1232,1512( invente! calculus inthe 1299 s ecause it was nee!e! to solve thecutting,e!ge science an! math prolem oftheir time% inclu!ing how to calculate thelengths of curves% the areas oun!e! ycurves% an! the motion of oects that areaccelerating 'gaining spee!(

History of calculus



Calculus applie! to a wi!e range of prolem inme!icine% the social sciences% economics%iological growth an! !ecay% physics% an!engineering# $t is a ;e<ile language for

!escriing the physical wor!# =ects in motionchemical reaction% comple< surfaces an! volumes%heating an! cooling% the har!,to,imagineehaviors of space an! time% an! many other

events#Calculus is essential to the !esign of engines%

computers% an! another comple< machines % toeconomics% agriculture% an! physics#

-hy >o -e Stu!y Calculus



A ?function@ is a rule that relates one groupof numers to another# -hen f is a function of% it is common practice to write f as f '<(%which we rea! alou! as ?f of @# So thefunction f can also e written as f '<( ,

/<ample of unction

we can write a rule that each positive

numer% % to some negative numer% f ,# )his rule or function tells us that if e"uals D%f e"uals ,D

-hat $s unction



/<ample graph of )he unction

$t is often useful to ma*e a picture of a

function# )his is !one y pic*ing values fo< %applying the rule of the function% an! +n!ingout what values of f result# $n this way anynumer can e paire! with a numer f#

or /<ampleA graph of the function f '<(4 is shown inigure 1



igure 4 shows a line segment ust touching f'<( atthe point !irectly aove o 'the suscript ?o@ is ust

a lael to !istinguish o from other value of (# )hisline segment is sai! to e ?tangent@ to the curve#

)he Slop of a line tangent to a curve is calle! the!erivative of the curve at the point where the line

an! the curve touch#

-hat $s >erivative



A single tangent line shows the !erivative

only at one point% ut the !erivative can efoun! in the same way for every single pointalong the curve# )hese numers can e

graphe! as a curve in their own right#

)he !erivative of f'<( is often written :

Because it correspon!s to the slop or rate of

changes of f'<( at a single point <# thenumerator df stan!s for a very small verticalchange in f'<(% an! the !enominator dxstan!s for a very small hori8ontal change in <#

dx

df



A function f'<( might e !e+ne! either y aseries of measurements of some real wor!"uantity or y an e"uation#

)here is a set of stan!ar! rules that says

e<actly how to write !own that is startingwith an e"uation for f'<(#

)he !erivative of a function is ust another

function an! so you can ta*e its !erivativetoo# )his function is calle! the ?secon!!erivative of f'<(@ an! is written :

>erivatives

dxdf

2

2

dx

f d



or instance% the +rst !erivative of a functionthat !escries an! oectEs position !escriesoectEs spee!# )he secon! !erivative!escries the oectEs acceleration# Any!erivative eyon! the +rst is calle! a ?higher,or!er@ !erivative#

/<ample



)he area un!er the curve etween <9 an! <<o has een

sha!e! in#

)hese area is calle! the !e+nite integral of f'<( etween 9an! <o # )he !e+nite integral% li*e the !erivative at asingle

point% is simply a numer#

)he !e+nite integral can tell us the actual physical area of anoect with curving e!ges# $t can have other physical meaningas well#

)he $ntegral



or e<ample% the integral of an e"uation that!escries an oectEs velocity tells us how farthe oect has travelle!#

)he !e+nite integral of the velocity% v't(% fromtime 9 to time to is the area un!er the curve

from 9 to time to #



A series of !e+nite integrals is calle!

integrating f(x)% an! the resulting curve orfunction is calle! the indenite integral 'orsimply the integral( of f(x)#

)he in!e+nite integral of a function f'<( iswritten as follows:

∫ dx x f )(



/<ample of the integral:

)he oect moving at the constant 199 milesper hour% the integral of v(t) 199 is easy towrite !own as an e<act mathematical

e<pression:∫ += ct dt t v 100)(



An in!e+nite integral is more informative than a !e+niteintegral% ecause it can tell us the value of any !e+nite

integral#

$n calculus% !epen!ing on what real,wor! "uantity youEremeasuring an ?area@ maye the numer of miles !rivenor the pro+t ma!e y a usiness or the amount of oillea*e! from a eache! tan*er% or the proaility that aroc*et will e<plo!e efore reaching orit% or many

others things#

Fundamental of Calculus (Presentation Slide)A

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