   Chapter 4.9, Problem 40E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find f.f″(x) = 8x3 + 5, f(1) = 0, f′(1) = 8

To determine

To find: The general antiderivative for the function f(x)=8x3+5,f(1)=0,f(1)=8 .

Explanation

Given Data:

Write the given function as follows.

f(x)=8x3+5,f(1)=0,f(1)=8

Formula used:

The antiderivative function for the function xn is xn+1n+1+C .

Here, C is the constant.

Calculation of f(x) :

Rewrite the function f(x)=8x3+5 as follows.

f(x)=8x3+5x0 (1)

From the antiderivative function formula, the antiderivative for the function in equation (1) is written as follows.

f(x)=8(x3+13+1)+5(x0+10+1)+C=8(x44)+5x+C

f(x)=2x4+5x+C (2)

As f(1)=8 , substitute 1 for x in equation (2),

f(1)=2(1)4+5(1)+C=2+5+C=7+C

Rewrite the expression as follows.

C=f(1)7

Substitute 8 for f(1) ,

C=87=1

Substitute 1 for C in equation (2),

f(x)=2x4+5x+1

Calculation of f(x) :

Rewrite the function f(x)=2x4+5x+1 as follows.

f(x)=2x4+5x1+1x0 (3)

From the antiderivative function formula, the antiderivative for the function in equation (3) is written as follows

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