Essential calculus ch02 derivatives
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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

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ESSENTIAL CALCULUS CH02 Derivatives

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Essential calculus ch02 derivatives

ESSENTIAL CALCULUSCH02 Derivatives


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

  • 2.8 Linear Approximations and Differentials

    Review


Essential calculus ch02 derivatives

Chapter 2, 2.1, P73


Essential calculus ch02 derivatives

Chapter 2, 2.1, P73


Essential calculus ch02 derivatives

Chapter 2, 2.1, P73


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P74


Essential calculus ch02 derivatives

Chapter 2, 2.1, P75


Essential calculus ch02 derivatives

Chapter 2, 2.1, P75


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

Chapter 2, 2.1, P76


Essential calculus ch02 derivatives

Chapter 2, 2.1, P76


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

f’(a) =lim

x→ a

Chapter 2, 2.1, P78


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

Chapter 2, 2.1, P78


Essential calculus ch02 derivatives

Chapter 2, 2.1, P79


Essential calculus ch02 derivatives

Chapter 2, 2.1, P79


Essential calculus ch02 derivatives

6. Instantaneous rate of change=lim

∆X→0

X2→x1

Chapter 2, 2.1, P79


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

  • 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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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

f’(a)=lim

h→ 0

Chapter 2, 2.2, P83


Essential calculus ch02 derivatives

f’(x)=lim

h→ 0

Chapter 2, 2.2, P83


Essential calculus ch02 derivatives

Chapter 2, 2.2, P84


Essential calculus ch02 derivatives

Chapter 2, 2.2, P84


Essential calculus ch02 derivatives

Chapter 2, 2.2, P84


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

Chapter 2, 2.2, P88


Essential calculus ch02 derivatives

Chapter 2, 2.2, P88


Essential calculus ch02 derivatives

4 THEOREM If f is differentiable at a, then f is continuous at a .

Chapter 2, 2.2, P88


Essential calculus ch02 derivatives

Chapter 2, 2.2, P89


Essential calculus ch02 derivatives

Chapter 2, 2.2, P89


Essential calculus ch02 derivatives

Chapter 2, 2.2, P89


Essential calculus ch02 derivatives

Chapter 2, 2.2, P89


Essential calculus ch02 derivatives

  • (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


Essential calculus ch02 derivatives

2. (a) f’(0) (b) f’(1)

(c) f’’(2) (d) f’(3)

(e) f’(4) (f) f’(5)

Chapter 2, 2.2, P91


Essential calculus ch02 derivatives

Chapter 2, 2.2, P92


Essential calculus ch02 derivatives

Chapter 2, 2.2, P92


Essential calculus ch02 derivatives

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

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

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

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

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

Chapter 2, 2.2, P93


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

FIGURE 1

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

Chapter 2, 2.3, P93


Essential calculus ch02 derivatives

FIGURE 2

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

Chapter 2, 2.3, P95


Essential calculus ch02 derivatives

DERIVATIVE OF A CONSTANT FUNCTION

Chapter 2, 2.3, P95


Essential calculus ch02 derivatives

Chapter 2, 2.3, P95


Essential calculus ch02 derivatives

THE POWER RULE If n is a positive integer, then

Chapter 2, 2.3, P95


Essential calculus ch02 derivatives

THE POWER RULE(GENERAL VERSION) If n is any real number, then

Chapter 2, 2.3, P97


Essential calculus ch02 derivatives

█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


Essential calculus ch02 derivatives

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

(f+g)’=f’+g’

Chapter 2, 2.3, P97


Essential calculus ch02 derivatives

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

Chapter 2, 2.3, P97


Essential calculus ch02 derivatives

THE SUM RULE If f and g are both differentiable, then

Chapter 2, 2.3, P97


Essential calculus ch02 derivatives

THE DIFFERENCE RULE If f and g are both

differentiable, then

Chapter 2, 2.3, P98


Essential calculus ch02 derivatives

Chapter 2, 2.3, P100


Essential calculus ch02 derivatives

Chapter 2, 2.3, P100


Essential calculus ch02 derivatives

Chapter 2, 2.3, P101


Essential calculus ch02 derivatives

THE PRODUCT RULE If f and g are both

differentiable, then

Chapter 2, 2.4, P106


Essential calculus ch02 derivatives

THE QUOTIENT RULE If f and g are differentiable, then

Chapter 2, 2.4, P109


Essential calculus ch02 derivatives

Chapter 2, 2.4, P110


Essential calculus ch02 derivatives

DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

Chapter 2, 2.4, P111


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

  • 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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

█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


Essential calculus ch02 derivatives

  • 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


Essential calculus ch02 derivatives

Chapter 2, 2.8, P133


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

relative error

Chapter 2, 2.8, P136


Essential calculus ch02 derivatives

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

Chapter 2, Review, P139


Essential calculus ch02 derivatives

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

Chapter 2, Review, P139


Essential calculus ch02 derivatives

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


Essential calculus ch02 derivatives

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

Chapter 2, Review, P141


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