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Chapter 2: Q. 71 (page 234)

In Exercises 66鈥�71, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process.

f'(x)=34-9x2

Short Answer

Expert verified

The function that has the derivative f'(x)=34-9x2 is f(x)=sin-1(3x2).

Step by step solution

01

Step 1. Given Information.

The derivative:

f'(x)=34-9x2

02

Step 2. Guess the formula for the given derivative.

We know that,

ddx(sin-1x)=11-x2

So we can conclude that the derivative will be of the form of sinefunction.

03

Step 3. Check guess-and-check process.

Consider

f(x)=sin-1(3x)f'(x)=11-(3x)2.3=31-9x2

which is not a given derivative.

04

Step 4. Check another function.

Consider the function,

f(x)=sin-1(3x2)f'(x)=11-(3x2)2.32=11-9x24.32=24-9x2.32=34-9x2

which is a given derivative.

So the function is f(x)=sin-1(3x2).

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Most popular questions from this chapter

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c)for each function f and value x=c in Exercises 23鈥�38.

23.f(x)=x2,x=-3

Use the limit you just found to calculate the exact slope of the tangent line to f(x)=x2at x=4. Obviously you should get the same final answer as you did earlier.

Each graph in Exercises 31鈥�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.

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

f(x)=x2-x3+5x9x-1

For each function f(x)and interval [a,b]in Exercises 81-86, use the Intermediate Value Theorem to argue that the function must have at least one real root on a,b. Then apply Newton鈥檚 method to approximate that root.

localid="1648369345806" f(x)=x4-2,a,b=1,2.
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