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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

ESSENTIAL CALCULUSCH02 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













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




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


f’(a) =lim

x→ a

Chapter 2, 2.1, P78


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


Chapter 2, 2.1, P78 (a, f(a)) whose slope is equal to f


Chapter 2, 2.1, P79 (a, f(a)) whose slope is equal to f


Chapter 2, 2.1, P79 (a, f(a)) whose slope is equal to f


6. Instantaneous rate of change=lim (a, f(a)) whose slope is equal to f

∆X→0

X2→x1

Chapter 2, 2.1, P79


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

Chapter 2, 2.1, P79


  • 9. (a, f(a)) whose slope is equal to fThe 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


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


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


the derivative of a function f at a fixed number a: following numbers in increasing order and explain your reasoning.

f’(a)=lim

h→ 0

Chapter 2, 2.2, P83


f following numbers in increasing order and explain your reasoning.’(x)=lim

h→ 0

Chapter 2, 2.2, P83


Chapter 2, 2.2, P84 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P84 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P84 following numbers in increasing order and explain your reasoning.


3 DEFINITION following numbers in increasing order and explain your reasoning. 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


Chapter 2, 2.2, P88 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P88 following numbers in increasing order and explain your reasoning.


4 THEOREM following numbers in increasing order and explain your reasoning. If f is differentiable at a, then f is continuous at a .

Chapter 2, 2.2, P88


Chapter 2, 2.2, P89 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P89 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P89 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P89 following numbers in increasing order and explain your reasoning.


  • (a) f following numbers in increasing order and explain your reasoning.’(-3) (b) f’(-2) (c) f’(-1)

  • (d) f’(0) (e) f’(1) (f) f’(2)

  • (g) f’(3)

Chapter 2, 2.2, P91


2. (a) f following numbers in increasing order and explain your reasoning.’(0) (b) f’(1)

(c) f’’(2) (d) f’(3)

(e) f’(4) (f) f’(5)

Chapter 2, 2.2, P91


Chapter 2, 2.2, P92 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P92 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P93 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P93 following numbers in increasing order and explain your reasoning.


33. The figure shows the graphs of f, f following numbers in increasing order and explain your reasoning.’, and f”. Identify each curve, and explain your choices.

Chapter 2, 2.2, P93


34. The figure shows graphs of f, f following numbers in increasing order and explain your reasoning.’, f”, and f”’. Identify each curve, and explain your choices.

Chapter 2, 2.2, P93


Chapter 2, 2.2, P93 following numbers in increasing order and explain your reasoning.


Chapter 2, 2.2, P93 following numbers in increasing order and explain your reasoning.


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


FIGURE 1 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.

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

Chapter 2, 2.3, P93


FIGURE 2 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.

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

Chapter 2, 2.3, P95


DERIVATIVE OF A CONSTANT FUNCTION 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, 2.3, P95


Chapter 2, 2.3, P95 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.


THE POWER RULE 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. If n is a positive integer, then

Chapter 2, 2.3, P95


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

Chapter 2, 2.3, P97


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.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


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

(f+g)’=f’+g’

Chapter 2, 2.3, P97


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

Chapter 2, 2.3, P97


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

Chapter 2, 2.3, P97


THE DIFFERENCE RULE 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.If f and g are both

differentiable, then

Chapter 2, 2.3, P98


Chapter 2, 2.3, P100 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, 2.3, P100 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, 2.3, P101 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.


THE PRODUCT RULE 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.If f and g are both

differentiable, then

Chapter 2, 2.4, P106


THE QUOTIENT RULE 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.If f and g are differentiable, then

Chapter 2, 2.4, P109


Chapter 2, 2.4, P110 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.


DERIVATIVE OF TRIGONOMETRIC FUNCTIONS 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, 2.4, P111


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


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


THE CHAIN RULE the functions whose graphs are shown. 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


F (g(x) = f the functions whose graphs are shown.’ (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


4. THE POWER RULE COMBINED WITH CHAIN RULE the functions whose graphs are shown. If n is any real number and u=g(x) is differentiable, then

Alternatively,

Chapter 2, 2.5, P116


49. A table of values for f, g, f the functions whose graphs are shown.’’, 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


Chapter 2, 2.5, P120


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


h(x)=f(f(x)) and g(x)=f(xWARNING 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


  • Steps in solving related rates problems: h(x)=f(f(x)) and g(x)=f(x

  • 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


Chapter 2, 2.8, P133 h(x)=f(f(x)) and g(x)=f(x


f(x) ~ f(a)+f h(x)=f(f(x)) and g(x)=f(x”(a)(x-a)

~

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

Chapter 2, 2.8, P133


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


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


relative error equation.

Chapter 2, 2.8, P136


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

Chapter 2, Review, P139


7. The figure shows the graphs of f, f following numbers in increasing order:’, and f”. Identify each curve, and explain your choices.

Chapter 2, Review, P139


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


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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