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Chapter 2: Q. 32 (page 166)

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Short Answer

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The graph is:

Step by step solution

Step 1. Given Information

We are given the associated slope function f' for some unknown function f and we need to sketch a possible graph of f.

Step 2. Finding the graph

The function increases from $x < - 0.9$ and $x > 0.9$ and decreases from $- 0.9 < x < 0.9$ .

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Most popular questions from this chapter

Last night Phil went jogging along Main Street. His distance from the post office t minutes after $6 : 00$ p.m. is shown in the preceding graph at the right.

(a) Give a narrative (that matches the graph) of what Phil did on his jog.

(b) Sketch a graph that represents Phil’s instantaneous velocity t minutes after $6 : 00$ p.m. Make sure you label the tick marks on the vertical axis as accurately as you can.

(c) When was Phil jogging the fastest? The slowest? When was he the farthest away from the post office? The closest to the post office?

For each function f that follows find all the x-values in the domain of f for which $f' (x) = 0$ and all the values for which $f' (x)$ does not exist in later section we will call these values the critical points of f

localid="1648604345877" $a) f (x) = x^{3} - 2 x b) f (x) = \sqrt{x} - x c) f (x) = \frac{1}{1 + \sqrt{x}} d) f (x) = \frac{x^{2} (x - 1)}{{(x - 2)}^{2}}$

Taking the limit: We have seen that if f is a smooth function, then $f^{'} (c) \approx \frac{f (c + h) - f (c)}{h}$ This approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

$f^{'} (c) = \lim_{h \to 0} \frac{f (c + h) - f (c)}{h}$ .

Use the limit just defined to calculate the exact slope of the tangent line to $f (x) = x^{2}$ at $x = 4 .$

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = {(x^{- \frac{1}{2}} (x^{2} - 1))}^{3}$

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function $E (t) = 123 {(1.025)}^{t}$ , where expenditures are measured in billions of dollars and time is measured in years since 1990.

(a) Estimate the total yearly expenditures by these colleges and universities in 1995.

(b) Compute the average rate of change in yearly expenditures between 1990 and 2000.

(d) Estimate the rate at which yearly expenditures of public colleges and universities were increasing in 1995.

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Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the graph

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Most popular questions from this chapter

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