3.1 – Derivative of a Function - PowerPoint Presentation

Slide1

3.1 –

Derivative of a FunctionSlide2

Slope of the Tangent Line

If

f

is defined on an open interval containing c and the limit exists, then and the line through (c, f (c)) with slope m is the line tangent to the graph of f at the point (c, f (c)).Slide3

The Slope of the Graph of a Linear Function

Find the slope of the graph of

at the point (2, 1).Slide4

The Slope of the Graph of a Nonlinear Function

Find the slope of the graph of

at the point (0, 1) and (-1, 2).Find the equation of the tangent line at each point.Slide5

Definition

Derivative – The

derivative

of f at x is given by provided the limit exists. For all x for which this limit exists, is a function of x.Slide6

Find the Derivative of the

Function Slide7

Alternate Definition

The

derivative

of the function f at the point x = a is the limit provided the limit exists.Slide8

Find the Derivative of the Functions Slide9

Homework

p.

105

~ 1-9 (O), 13-16, 17, 19Slide10

Reflection

p.

105

~ 1-9 (O), 13-16, 17, 19Slide11

Differentiation

Rules

3.3.1 (

also 3.2)Slide12

When Derivatives Do Not ExistSlide13

When Derivatives Do Not ExistSlide14

When Derivatives Do Not ExistSlide15

When Derivatives Do Not ExistSlide16

The Constant Rule

The derivative of a constant function is 0. That is, if

c

is a real number, then .Slide17

The Power Rule

If

n

is a rational number, then the function f (x) = xn is differentiable and For f to be differentiable at x = 0, n

must be a number such that xn–

1

is defined on an interval containing 0.Slide18

Find the Derivative of the FunctionSlide19

The Constant Multiple Rule

If

f

is a differentiable function and c is a real number, then cf is also differentiable and .Slide20

Find the Derivative of the FunctionSlide21

The Sum and Difference Rules

The sum (or difference) of two differentiable functions

f

and g is itself differentiable. Moreover, the derivative of f + g (or f – g) is the sum (or difference) of the derivatives of f and g

.Slide22

Find the Derivative of the FunctionSlide23

The Slope of a Graph

Find the slope of the graph of

when

, , and .Slide24

The Tangent Line

Find an equation of the tangent line to the graph

of when

.Slide25

The Product Rule

The product of two differentiable functions

f

and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function time the derivative of the first.Slide26

Find the DerivativeSlide27

Find the DerivativeSlide28

The Quotient Rule

The quotient of two differentiable functions

f

and g is itself differentiable for all values of x for which g(x) ≠ 0. Moreover, the derivative of f / g is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.Slide29

Find the DerivativeSlide30

Find the DerivativeSlide31

Homework

p

. 114

~ 1-10p. 124 ~ 1-5, 7-29 (O), 30, 32Slide32

Higher Order Derivatives

3.3.2Slide33

Which Rule Do I Use?

Find the DerivativeSlide34

Higher-order Derivative Notation

First Derivative:

Second Derivative:

Third Derivative: Fourth Derivative: nth Derivative: Slide35

Find the Second DerivativeSlide36

Instantaneous Rate of Change

A population of 500 bacteria is introduced into a culture and grows in number according to the equation

where

t is measured in hours. Find the rate at which the population is growing when t = 2Slide37

Velocity and Other Rates of Change

3.4Slide38

Position and Velocity

If a billiard ball is dropped from a height of 100 feet, its height

s

at time t is given by the position function where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals.[1, 2] [1, 1.5] [1, 1.1]Slide39

Position FunctionSlide40

Instantaneous Velocity

At time

t

= 0, a diver jumps from a platform diving board that is 32 feet above the water. the position of the diver is given by where s is measured in feet and t is measured in seconds.When does the diver hit the water?What is the diver’s velocity at impact?Slide41

Higher-order Derivatives

Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by

where s(t) is the height in meters and t is the time in seconds. What is the ratio of Earth’s gravitational force to the moon’s?Slide42

Free Fall

A silver dollar is dropped from the top of a building that is 1362 feet tall.

Determine the position, velocity, and acceleration functions for the coin.

Determine the average velocity on the interval [1, 2].Find the instantaneous velocities when t = 1 and t = 2.Find the time required for the coin to reach ground level.Find the velocity of the coin at impact.Slide43

Free-Fall

A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds?

What is the maximum height of the projectile?Slide44

Homework

p.135/1, 3, 7, 9-

15 (O),

19, 21Slide45

Derivatives of Trigonometric Functions

3.5Slide46

The Derivative of SineSlide47

Derivatives of Sine and Cosine FunctionsSlide48

Find the Derivative of the FunctionSlide49

Find the DerivativeSlide50

Simple Harmonic Motion

A weight hanging from a spring is stretched 5 units beyond its rest position (

x

= 0) and released at time t = 0 to bob up and down. Its position at any later time t is What are its velocity and acceleration at time t? Describe its motion.Slide51

Derivative of TangentSlide52

Derivatives of Trigonometric FunctionsSlide53

Find the DerivativeSlide54

Find the Second DerivativeSlide55

Homework

p

. 146/

1-9 (O), 11-23 (O), 27-39 (O), 43Slide56

The Chain Rule

4.1Slide57

The Chain Rule

If

y

= f (u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f (g(x)) is a differentiable function of

x and

or, equivalently,Slide58

Identify the inner and outer functions

Composite

y

= f (g(x)) Inner u = g(x) Outer y = f (u)

1.

2.

3.

4.

Slide59

The General Power Rule

If

,

where u is a differentiable function of x and n is a rational number, then or, equivalently, Slide60

Find the DerivativeSlide61

Find the DerivativeSlide62

Homework

Chain Rule WorksheetSlide63

Factoring Out the Least Powers

Find the DerivativeSlide64

Factoring Out the Least Powers

Find the DerivativeSlide65

Factoring Out the Least Powers

Find the DerivativeSlide66

Find the DerivativeSlide67

Find the DerivativeSlide68

Trig Tangent Line

Find an equation of the tangent line to the graph of

at the point (π, 1). Then determine all values of x in the interval (0, 2π) at which the graph of f has a horizontal tangent.Slide69

Homework

p.153/ 1-11odd, 21-39odd, 59

Slide70

Implicit Differentiation

4.2Slide71

Find

dy

/dxSlide72

Guidelines for Implicit Differentiation

Differentiate both sides of the equation

with respect to x

.Collect all terms involving dy / dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy / dx out of the left side of the equation.Solve for dy / dx.Slide73

Find the derivativeSlide74

Homework

p.162/ 1-19odd, 49, 51Slide75

Example

Determine the slope of the tangent line to the graph of

at the point .Slide76

Example

Determine the slope of the tangent line to the graph of

at the point .Slide77

Finding the Second Derivative ImplicitlySlide78

Finding the Second Derivative ImplicitlySlide79

Example

Find the tangent and normal line to the graph given by

at the point .Slide80

Homework

p.162/ 21-25odd, 27-30, 31-43oddSlide81

Inverse Functions

4.3Slide82

Definition of Inverse Function

A function

g

is the inverse function of the function f if for each x in the domain of g. and for each x in the domain of f.Slide83

Verifying Inverse Functions

Show that the functions are inverse functions of each other.

and Slide84

The Existence of an Inverse Function

A function has an inverse function if and only if it is one-to-one.

If

f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.Slide85

Existence of an Inverse Function

Which of the functions has an inverse function?Slide86

Finding an Inverse

Find the inverse function of

.Slide87

The Derivative of an Inverse Function

Let

f

be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which . Moreover, Slide88

Example

Let .

What is the value of when

x = 3?What is the value of when x = 3?Slide89

Homework

p.

44/ 1-6, 7-23odd, 43

p. 170/ 28, 29bcSlide90

Inverse Trigonometric Functions

3.8Slide91

The Inverse Trigonometric Functions

Function

Slide92

Evaluating Inverse Trigonometric Functions

Evaluate each function.

Slide93

Solving an EquationSlide94

Using Right Triangles

a) Given

y

= arcsin x, where , find cos y.b) Given , find tan y.Slide95

Homework

3.8 Inverse Trig Review worksheetSlide96

Derivatives of Inverse Trigonometric Functions

FindSlide97

Derivatives of Inverse Trigonometric FunctionsSlide98

Differentiating Inverse Trigonometric Functions

Slide99

Differentiate and SimplifySlide100

Homework

p.

170/ 1-27odd, 31abSlide101

Derivatives of Exponential and Logarithmic Functions

4.4Slide102

Properties of Logarithms

If

a

and b are positive numbers and n is rational, then1. 2. 3. 4. Slide103

Expand the following logarithmsSlide104

Solving Equations

Solve 7 =

e

x + 1 Solve ln(2x  3) = 5Slide105

Derivative of the Natural Exponential FunctionSlide106

ExamplesSlide107

Derivatives for Bases Other than

e

Slide108

Find the derivative of each function.Slide109

Homework

p.44/ 33-38

p.178/ 1-13odd, 29, 30Slide110

Derivative of the Natural Logarithmic Function

Slide111

Example

Find the derivative of ln (2

x

).Slide112

Find the Derivative

1.

2. 3. Slide113

Derivatives for Bases Other than

e

Slide114

Example

Differentiate .Slide115

Example

Differentiate .Slide116

Example

Differentiate .Slide117

Find an equation of the tangent line to the graph at the given point.Slide118

Homework

p. 178/15-27odd, 37-41odd Slide119

Comparing Variable and ConstantsSlide120

Derivative Involving Absolute Value Slide121

Find the derivative Slide122

Use implicit differentiation to find

dy

/

dx.ln xy + 5x = 30Slide123

Logarithmic Differentiation

Differentiate .Slide124

Logarithmic DifferentiationSlide125

More Logarithmic DifferentiationSlide126

Homework

p. 179/ 31, 33-36, 43-55odd