HOMEWORK HOMEWORK Section 10-1 Section 10-1 (page 801) (page 801) (evens) 2-18, 22-32, 34a (evens) 2-18, 22-32, 34a (16 problems) (16 problems)
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HOMEWORKHOMEWORK
Section 10-1 Section 10-1
(page 801)(page 801)
(evens) 2-18, 22-32, 34a(evens) 2-18, 22-32, 34a
(16 problems)(16 problems)
10-1Limits and MotionThe tangent problem.
What you’ll learn about• Average Velocity• Instantaneous Velocity• Limits Revisited• The Connection to Tangent Lines• The Derivative
… and whyThe derivative is fundamental to all of calculus. It allows us to
analyze rates of change for any function regardless of how complicated it is. The ability to find rates of change is essential in physics, economics, engineering,
and even history.
Vocabulary
66554433221100
feetfeet
0 5 10 15 20 25 30 350 5 10 15 20 25 30 35
secondsseconds
Average velocityAverage velocity:: the change in position the change in position divided by the change in timedivided by the change in time
6 feet6 feet
30 seconds30 secondsAve Vel.= 6/30 = 0.2 ft/secAve Vel.= 6/30 = 0.2 ft/sec
c
time
distVelAve
..
Average Rate of Change If ( ), then the of with respect to
( ) ( )on the interval [ , ] is .
Geometrically, this is the slope of the secant line through ( , ( )) and
( , ( )).
y f x y x
y f b f aa b
x b aa f a
b f b
average rate of change
aa bb
f(b)f(b)
f(a)f(a)
66554433221100
feetfeet
0 5 10 15 20 25 30 350 5 10 15 20 25 30 35secondsseconds
VelocityVelocity:: is it is it constantconstant over the entire interval? over the entire interval?
Ave Velocity= 6/30 = 0.2 ft/secAve Velocity= 6/30 = 0.2 ft/sec
Instantanious Velocity: the actual velocity atInstantanious Velocity: the actual velocity at an instant in timean instant in time. .
For many applications,For many applications, instantanious rate of changeinstantanious rate of change is more important than the is more important than the average rate of changeaverage rate of change..
sec0
0..
ft
time
distVelAve
We need infinitely small measurements that are still greater than zero.We need infinitely small measurements that are still greater than zero.
ExampleExample: The distance on object travels is given : The distance on object travels is given by the relation:by the relation: 3)(timedist For the interval: (2 sec --2.1 sec):For the interval: (2 sec --2.1 sec):
61.12
0.21.2
)0.2(1.2 33
time
distvel
For the interval: (2 sec --2.01 sec):For the interval: (2 sec --2.01 sec):
0601.12
0.201.2
)0.2(01.2 33
time
distvel
For the interval: (2 sec --2.001 sec):For the interval: (2 sec --2.001 sec):
006001.12
0.2001.2
)0.2(001.2 33
time
distvel
For the interval: (2 sec --2.0001 sec):For the interval: (2 sec --2.0001 sec):
00060001.12
0.20001.2
)0.2(0001.2 33
time
distvel
What is happeningWhat is happening to the velocity as to the velocity as our time interval our time interval approaches zero?approaches zero?
ExampleExample: The distance on object travels is given : The distance on object travels is given by the relation:by the relation: 3)(timedist
61.12
0.21.2
)0.2(1.2 33
time
distvel
53)( tts
006001.12
0.2001.2
)0.2(001.2 33
time
distvel
00060001.12
0.20001.2
)0.2(0001.2 33
time
distvel
The idea of the The idea of the limitlimit of a of a function is function is essentialessential to to calculating instantanious calculating instantanious rates of change.rates of change.
?0.2
)0.2(lim
33
2
t
tt
2
)42)(2(lim
2
2
t
tttt
42lim 22 ttt
42*222 12
Limits at a (Informal) When we write "lim ( ) ," we mean that ( ) gets arbitrarily
close to as gets arbitrarily close (but not equal) to .x af x L f x
L x a
I would prefer to replace I would prefer to replace arbitrarilyarbitrarily with the word with the word infinitelyinfinitely..
Finding the slopeFinding the slope of a curve at exactly one point. of a curve at exactly one point.
4433221100
0 1 2 30 1 2 3
xxx
xfxxf
x
xf
x
y
run
riseslope
)(
)()()(
xx
xfxf
)(
)()(lim 0
3xy
1
1lim
33
1
x
xt
1
)1)(1(lim
2
1
x
xxxt
1
)1)(1(lim
2
1
x
xxxt
3)1(lim 21 xxt
Finding the slope of a tangent Finding the slope of a tangent line and the instantaneous velocityline and the instantaneous velocity
are the exact same type of problem.are the exact same type of problem.
Your turn:
1. 1. Find the Find the instantaneous velocityinstantaneous velocity at time = 3 at time = 3 for the following time—distance relation: for the following time—distance relation:
63)( tts
tt
tsts
)(
)()(lim 0
3
)6)3(3()63(lim 3
t
tt
3
)3(3lim 3
t
tt
33lim 3 t
timeinchange
positioninchangetvel
....
....)3(
Derivative at a Point
The , denoted by '( ) and read
( ) ( )" prime of " is '( ) lim , provided the limit exists.
Geometrically, this is the slope of the tangent line through
x a
f a
f x f af a f a
x a
derivative of the function at = f x a
( , ( )).a f a
aa
f(a)f(a)
Not as good aNot as good a definition as thedefinition as the following slide.following slide.
Derivative at a Point (easier for computing)
0
The , denoted by '( ) and read
( ) ( )" prime of " is '( ) lim , provided the limit exists.
h
f a
f a h f af a f a
h
derivative of the function at = f x a
Remember the idea of Remember the idea of instantaneous rate of changeinstantaneous rate of change:: (as f(a + h) gets infinitely (as f(a + h) gets infinitely closeclose to f(a)). to f(a)).
NumeratorNumerator: “rise” (with the rise being infinitely small): “rise” (with the rise being infinitely small)
DenominatorDenominator: “run” (with the fun being infinitely small): “run” (with the fun being infinitely small)
DerivativeDerivative think think slopeslope (of the tangent line at a point on the curve)(of the tangent line at a point on the curve)
Finding a Derivative (slope) at a PointFind '(3) if ( ) 2 4.f f x x
h
xfhxfxf h
)()(lim)( 0
'
This is a This is a trivialtrivial problem. You problem. You know the slope of this line at know the slope of this line at any point on the line is 2.any point on the line is 2.
h
fhff h
)3()3(lim)3( 0
'
h
hf h
)43*2(]4)3(2[lim)3( 0
'
h
hf h
)46426lim)3( 0
'
General form of a derivativeGeneral form of a derivative
Replace ‘x’ in the general formReplace ‘x’ in the general form with the ‘x’ value of the pointwith the ‘x’ value of the point
simplifysimplify
22
lim)3( 0' h
hf h Simplify Simplify
we already knew it would be 2we already knew it would be 2
Rewrite f(x) using the Rewrite f(x) using the input values aboveinput values above
Derivative
0
If ( ), then
, is the function ' whose value at is
( ) ( )'( ) lim for all values of where the
limit exists.
h
y f x
f x
f x h f xf x x
h
derivative of the function with respect to
at =
f
x x a
h
xfhxfxf h
)()(lim)( 0
'
Finding the Derivative of a Function
2Find '( ) if ( ) 2 .f x f x x
(gives an equation that you can use to find the slope (gives an equation that you can use to find the slope at any at any point point in the original function)in the original function)
h
xfhxfxf h
)()(lim)( 0
'
General form of a derivativeGeneral form of a derivative
Input values are ‘x+h’ and ‘x’ intoInput values are ‘x+h’ and ‘x’ into the function.the function.h
xhxxf h
22
0' 2)(2
lim)(
h
xhxhxxf h
222
0' 2)2(2
lim)(
SimplifySimplify
h
hxhxf h
2
0' 24
lim)(
SimplifySimplify
hxxf h 24lim)( 0' SimplifySimplify xxf 4)('
Your turn:
h
xfhxfxf h
)()(lim)( 0
'
1. Find: f’(x) for 1. Find: f’(x) for 3)( xxf
Using: Using:
Finding the derviative of a point (alternative method)
h
xfhxfxf h
)()(lim)( 0
'
1. Find: f’(x) for the function.1. Find: f’(x) for the function.
xxf 6)('
2.2. Use ‘x’ value of the point as an input value to f’(x)Use ‘x’ value of the point as an input value to f’(x)
h
xhxxf h
22
0' 3)(3
lim)(
h
xhxhxxf h
222
0' 3)2(3
lim)(
hxxf h 36lim)( 0'
23)( xxf )2(': ffind
h
hxhxf h
2
0' 36
lim)(
12)2(6)2(' f
2.2. Find the instaneous velocity at time t = 4Find the instaneous velocity at time t = 4 for the following time—distance relationfor the following time—distance relation
using: using:
Your turn:
h
xfhxfxf h
)()(lim)( 0
' 53)( tts
Pretty silly really. You know the slope anywhere on Pretty silly really. You know the slope anywhere on the graph is 3 (y = mx + b)the graph is 3 (y = mx + b)
3)(' tf
h
thttf h
)53(]5)(3[lim)( 0
'
hxxf h 36lim)( 0'
3)4(' f
h
thttf h
53533lim)( 0
'
h
htf h
3lim)( 0
'
Newton vs. LiebnitzBoth independently discovered calculus.Both independently discovered calculus.
They came up with different notation to describe the derivative. They came up with different notation to describe the derivative.
Liebnitz’s notationLiebnitz’s notation is much more useful later on in calculus. is much more useful later on in calculus.
dx
dyxf )('
h
xfhxfxf h
)()(lim)( 0
'
h
xfhxf
dx
dyh
)()(lim 0
This is too hard!!Figuring out these problems using following formula:Figuring out these problems using following formula:
h
xfhxf
dx
dyh
)()(lim 0
is meant to help you figure out the is meant to help you figure out the relationshiprelationship between the between the limit and the slope of the tangent linelimit and the slope of the tangent line at a point. at a point.
(Infinitesimally small change in (Infinitesimally small change in rise over rise over an an infinitesimally small change in infinitesimally small change in runrun))
What you’ll learn in Caclulus.
There are some very There are some very simplesimple methods of finding the methods of finding the derviatives of various functions. Each method is different for derviatives of various functions. Each method is different for different classes of functions.different classes of functions.
Polynomials –way easyPolynomials –way easy
Trig Functions – pretty easy for simple onesTrig Functions – pretty easy for simple ones
Composition of functions – another “cool” methodComposition of functions – another “cool” method
Product of functions– really “slick” methodProduct of functions– really “slick” method
Quotient of functions – not too hardQuotient of functions – not too hard
Rational functions (ratio of functions)--harderRational functions (ratio of functions)--harder
I’ll teach you the method for polynomials – way easy
naxxf )(
1*)(' naxnxf
23)( xxf 123*2)(' xxf
xxf 6)('
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