essential calculus ch02 derivatives
Download
Skip this Video
Download Presentation
ESSENTIAL CALCULUS CH02 Derivatives

Loading in 2 Seconds...

play fullscreen
1 / 88

ESSENTIAL CALCULUS CH02 Derivatives - PowerPoint PPT Presentation


  • 191 Views
  • Uploaded on

ESSENTIAL CALCULUS CH02 Derivatives. In this Chapter:. 2.1 Derivatives and Rates of Change 2.2 The Derivative as a Function 2.3 Basic Differentiation Formulas 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Implicit Differentiation 2.7 Related Rates

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'ESSENTIAL CALCULUS CH02 Derivatives' - deo


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide2
In this Chapter:
  • 2.1 Derivatives and Rates of Change
  • 2.2 The Derivative as a Function
  • 2.3 Basic Differentiation Formulas
  • 2.4 The Product and Quotient Rules
  • 2.5 The Chain Rule
  • 2.6 Implicit Differentiation
  • 2.7 Related Rates
  • 2.8 Linear Approximations and Differentials

Review

slide14
1 DEFINITION The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope

m=line

Provided that this limit exists.

X→ a

Chapter 2, 2.1, P75

slide17
4 DEFINITION The derivative of a function f at a number a, denoted by f’(a), is

f’(a)=lim

if this limit exists.

h→ 0

Chapter 2, 2.1, P77

slide18
f’(a) =lim

x→ a

Chapter 2, 2.1, P78

slide19
The tangent line to y=f(X) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.

Chapter 2, 2.1, P78

slide23
6. Instantaneous rate of change=lim

∆X→0

X2→x1

Chapter 2, 2.1, P79

slide24
The derivative f’(a) is the instantaneous rate of change of y=f(X) with respect to x when x=a.

Chapter 2, 2.1, P79

slide25
9. The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions
  • What was the initial velocity of the car?
  • Was the car going faster at B or at C?
  • Was the car slowing down or speeding up at A, B, and C?
  • What happened between D and E?

Chapter 2, 2.1, P81

slide26
10. Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie.

(a) Describe and compare how the runners the race.

(b) At what time is the distance between the runners the greatest?

(c) At what time do they have the same velocity?

Chapter 2, 2.1, P81

slide27
15. For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning.

0 g’(-2) g’(0) g’(2) g’(4)

Chapter 2, 2.1, P81

slide28
the derivative of a function f at a fixed number a:

f’(a)=lim

h→ 0

Chapter 2, 2.2, P83

slide29
f’(x)=lim

h→ 0

Chapter 2, 2.2, P83

slide33
3 DEFINITION A function f is differentiable a if f’(a) exists. It is differentiable on an open interval (a,b) [ or (a,∞) or (-∞ ,a) or (- ∞, ∞)] if it is differentiable at every number in the interval.

Chapter 2, 2.2, P87

slide41
(a) f’(-3) (b) f’(-2) (c) f’(-1)
  • (d) f’(0) (e) f’(1) (f) f’(2)
  • (g) f’(3)

Chapter 2, 2.2, P91

slide42
2. (a) f’(0) (b) f’(1)

(c) f’’(2) (d) f’(3)

(e) f’(4) (f) f’(5)

Chapter 2, 2.2, P91

slide47
33. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.

Chapter 2, 2.2, P93

slide48
34. The figure shows graphs of f, f’, f”, and f”’. Identify each curve, and explain your choices.

Chapter 2, 2.2, P93

slide51
35. The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

Chapter 2, 2.2, P94

slide52
FIGURE 1

The graph of f(X)=c is the line y=c, so f’(X)=0.

Chapter 2, 2.3, P93

slide53
FIGURE 2

The graph of f(x)=x is the line y=x, so f’(X)=1.

Chapter 2, 2.3, P95

slide58
█GEOMETRIC INTERPRETATION OF THE CONSTANT MULTIPLE RULE

Multiplying by c=2 stretches the graph vertically by a factor of 2. All the rises have been doubled but the runs stay the same. So the slopes are doubled, too.

Chapter 2, 2.3, P97

slide59
█ Using prime notation, we can write the Sum Rule as

(f+g)’=f’+g’

Chapter 2, 2.3, P97

slide60
THE CONSTANT MULTIPLE RULE If c is a constant and f is a differentiable function, then

Chapter 2, 2.3, P97

slide62
THE DIFFERENCE RULE If f and g are both

differentiable, then

Chapter 2, 2.3, P98

slide66
THE PRODUCT RULE If f and g are both

differentiable, then

Chapter 2, 2.4, P106

slide70
43. If f and g are the functions whose graphs are shown, left u(x)=f(x)g(X) and v(x)=f(X)/g(x)

Chapter 2, 2.4, P112

slide71
44. Let P(x)=F(x)G(x)and Q(x)=F(x)/G(X), where F and G and the functions whose graphs are shown.

Chapter 2, 2.4, P112

slide72
THE CHAIN RULE If f and g are both differentiable and F =f。g is the composite function defined by F(x)=f(g(x)), then F is differentiable and F’ is given by the product

F’(x)=f’(g(x))‧g’(x)

In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then

Chapter 2, 2.5, P114

slide73
F (g(x) = f’ (g(x)) ‧ g’(x)

outer evaluated derivative evaluated derivative

function at inner of outer at inner of inner

function function function function

Chapter 2, 2.5, P115

slide74
4. THE POWER RULE COMBINED WITH CHAIN RULE If n is any real number and u=g(x) is differentiable, then

Alternatively,

Chapter 2, 2.5, P116

slide75
49. A table of values for f, g, f’’, and g’ is given
  • If h(x)=f(g(x)), find h’(1)
  • If H(x)=g(f(x)), find H’(1).

Chapter 2, 2.5, P120

slide76
51. IF f and g are the functions whose graphs are shown, let u(x)=f(g(x)), v(x)=g(f(X)), and w(x)=g(g(x)). Find each derivative, if it exists. If it dose not exist, explain why.
  • u’(1) (b) v’(1) (c)w’(1)

Chapter 2, 2.5, P120

slide77
52. If f is the function whose graphs is shown, let h(x)=f(f(x)) and g(x)=f(x2).Use the graph of f to estimate the value of each derivative.

(a) h’(2) (b)g’(2)

Chapter 2, 2.5, P120

slide78
█WARNING A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation.

Chapter 2, 2.7, P129

slide79
Steps in solving related rates problems:
  • Read the problem carefully.
  • Draw a diagram if possible.
  • Introduce notation. Assign symbols to all quantities that are functions of time.
  • Express the given information and the required rate in terms of derivatives.
  • Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3).
  • Use the Chain Rule to differentiate both sides of the equation with respect to t.
  • Substitute the given information into the resulting equation and solve for the unknown rate.

Chapter 2, 2.7, P129

slide81
f(x) ~ f(a)+f”(a)(x-a)

~

Is called the linear approximation or tangent line approximation of f at a.

Chapter 2, 2.8, P133

slide82
The linear function whose graph is this tangent line, that is ,

is called the linearization of f at a.

L(x)=f(a)+f’(a)(x-a)

Chapter 2, 2.8, P133

slide83
The differential dy is then defined in terms of dx by the equation.

So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f, then the numerical value of dy is determined.

dy=f’(x)dx

Chapter 2, 2.8, P135

slide84
relative error

Chapter 2, 2.8, P136

slide85
1. For the function f whose graph is shown, arrange the following numbers in increasing order:

Chapter 2, Review, P139

slide86
7. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.

Chapter 2, Review, P139

slide87
50. If f and g are the functions whose graphs are shown, let P(x)=f(x)g(x), Q(x)=f(x)/g(x), and C(x)=f(g(x)). Find (a) P’(2), (b) Q’(2), and (c)C’(2).

Chapter 2, Review, P140

slide88
61. The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.

Chapter 2, Review, P141

ad