Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 11.3, Problem 20E

To determine

To calculate: The derivative formula of the function, $f(x)=x3+1$ and hence find the derivative at the point $(−1,0)$ .

Expert Solution

Answer to Problem 20E

The derivative formula of the function, $f(x)=x3+1$ is $f'(x)=3x2$ and the derivative at the point $(−1,0)$ is 3.

Explanation of Solution

Given information: The function and point are $f(x)=x3+1$ and $(−1,0)$ respectively.

Formula used: The derivative of the function $f(x)$ is given by, $f'(x)=limh→0f(x+h)−f(x)h$ such that the limit exists.

Calculation: Find the derivative of the function, $f(x)=x3+1$ .

$f'(x)=limh→0[(x+h)3+1]−(x3+1)h=limh→0(x3+h3+3x2h+3xh2+1)−(x3+1)h=limh→0h(h2+3x2+3xh)h=limh→0h2+3x2+3xh$

Find the value of the limit when h approaches to 0.

$f'(x)=(0)2+3x2+3x(0)=3x2$

Thus, the derivative formula is $f'(x)=3x2$ .

Substitute $x=−1$ in the derivative formula.

$f'(−1)=3(−1)2=3(1)=3$

Thus, the derivative of the function is 3.

To determine

To calculate: The derivative of the function, $f(x)=x3+1$ at the point $(2,9)$ .

Expert Solution

Answer to Problem 20E

The derivative of the function, $f(x)=x3+1$ at the point $(2,9)$ is 12.

Explanation of Solution

Given: The function and point are $f(x)=x3+1$ and $(2,9)$ respectively.

Calculation: Substitute $x=2$ in the derivative formula, $f'(x)=3x2$ .

$f'(12)=3(2)2=3(4)=12$

Thus, the derivative of the function is 12.

Chapter 11.3, Problem 19E

Chapter 11.3, Problem 21E

Chapter 11 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

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The derivative formula of the function, f ( x ) = x 3 + 1 and hence find the derivative at the point ( − 1 , 0 ) .

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Answer to Problem 20E

Explanation of Solution

Answer to Problem 20E

Explanation of Solution

Chapter 11 Solutions