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Chapter 4: Q. 1 (page 364)

Find a function f that has the given derivative f' and value f(c), if possible.

f'(x)=x8鈭�x2+1,f(0)=鈭�1

Short Answer

Expert verified

The function is thisx99-x33+x+c

and value of f(-1)=-209

Step by step solution

01

Step 1.Given

The given function isf'(x)=x8鈭�x2+1,f(0)=鈭�1

02

Step 2. Finding the function using integration

Integratingbothsideswithrespecttox,鈭�f'(x)=鈭�x8鈭�x2+1dx=x99-x33+x+cwherecisconstant

03

Step 3. Finding the value of f(c).

whenf(0)=-1f(x)=x99-x33+x+cf(0)=0-0+0+c-1=cf(c)=f(-1)=(-1)99-(-1)33+(-1)+-1=19-13-1-1=-29-2=-209

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Most popular questions from this chapter

If 鈭�-23f(x)dx=4,鈭�-26f(x)dx=9,鈭�-23g(x)dx=2and 鈭�36g(x)dx=3,then find the values of each definite integral in Exercises 29-40. If there is not enough information, explain why.

鈭�36fxdx.

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�613(1-2x)2+4xdx

What is the difference between an antiderivative of a function and the indefinite integral of a function?

For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

f(x)=x2,[a,b]=[0,3]left sum with

a) n = 3 b) n = 6

Suppose f is a function whose average value on

[-2,5]is 10and whose average rate of change on

the same interval is -3. Sketch a possible graph for f .

Illustrate the average value and the average rate of change

on your graph of f.
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