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 5.5, Problem 41E

To determine

To find the linearization f at x=0 .

Expert Solution

Answer to Problem 41E

The linearization L(x)=1+x

Explanation of Solution

Given information:

The given function is f with f(0)=1 and f′(x)=cos(x2)

Calculation: In order to find the linearization at x=0 , consider a=0 and then plug the values in the linearization equation.

a=0f(a)=f(0) =1f′(a)=f′(0) =cos(02) =1

Now substitute the values

L(x)=f(a)+f′(a)(x−a) =1+1(x−0) =1+x

Therefore,

The linearization L(x)=1+x

To determine

To estimate the value of f at x=0.1

Expert Solution

Answer to Problem 41E

Therefore the value of f at x=0.1 is 1.1

Explanation of Solution

Given information:

The given function is f with f(0)=1 and f′(x)=cos(x2)

Calculation:

Consider x=0.1 and substitute in the linearization equation.

L(x)=1+xL(0.1)=1+0.1 =1.1

Therefore the value of f at

x=0.1 is 1.1

To determine

To explain if the actual value of f at x=0.1 is greater or less than estimate in part (b).

Expert Solution

Answer to Problem 41E

The actual value at x=0.1 is less than the estimate.

Explanation of Solution

Given information:

The given function is f with f(0)=1 and f′(x)=cos(x2)

Here

f′(x)=cos(x2) Is positive for 0≤x2<π2or0≤x<π2≈1.2

f″(x)=−2xsin(x2)

f″ Is negative at x=0.1 , which means f(x) is concave down. The linearization is a line based on the function at f(0) , so if f is concave down it will be below the line at x=0.1 .

Therefore,

The actual value at x=0.1 is less than the estimate.

Chapter 5.5, Problem 40E

Chapter 5.5, Problem 42E

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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