Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5, Problem 39RE

To determine

To find the linearization of f at x=0

Expert Solution

Answer to Problem 39RE

The Linearization of function is L(x)=−1

Explanation of Solution

Given:

Let f be a function with f′(x)=sinx2 and f(0)=−1

Calculation:

If f is a differentiable at x=a , then the equation of the tangent line,

L(x)=f(a)+f′(a)(x−a)

Defines the linearization of f at a .

Since, f(0)=−1 , the point of tangency is (0,−1) .also, f′(x)=sinx2 , the slope of the tangent line is f′(0)=sin(0)2=0

So,

L(x)=f(a)+f′(a)(x−a)L(x)=f(0)+f′(0)(x−0)L(x)=−1+(0)(x−0)L(x)=−1

Therefore, the Linearization of function is L(x)=−1 .

To determine

To find the approximate value of f at x=0.1

Expert Solution

Answer to Problem 39RE

The approximate value of f at x=0.1 is −1 .

Explanation of Solution

Given:

Let f be a function with f′(x)=sinx2 and f(0)=−1

Calculation:

If f is a differentiable at x=a , then the equation of the tangent line,

L(x)=f(a)+f′(a)(x−a)

Defines the linearization of f at a . The approximation f(x)≈L(x) is the standard linear approximation of f at a .

Since , the Linearization of f is L(x)=−1 it means that for all values of x , f is approximately −1 .

Therefore, the approximate value of f at x=0.1 is −1 .

To determine

To find whether the actual value of f at x=0.1 is greater than or less than the approximation in (b).

Expert Solution

Answer to Problem 39RE

The approximate value is less than the actual value.

Explanation of Solution

Given:

Let f be a function with f′(x)=sinx2 and f(0)=−1

Calculation:

Since, the linearization of f is L(x)=−1 which was centered at x=0

Now f″(x)=2xcosx2 and for x>0 , f″(x)>0 means the curve is concave up.

Since, linearization provides tangent which is a line , so approximate value is less than the actual value.

Chapter 5, Problem 38RE

Chapter 5, Problem 40RE

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

Ch. 5.1 - Prob. 11QR Ch. 5.1 - Prob. 12QR Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - 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