a. Answer The linearization of the function is L ( x ) = 10 x − 13 Explanation Given information: The given function is f ( x ) = x 3 − 2 x + 3 , x = 2 Since f ( 2 ) = 2 3 − 2 ⋅ 2 + 3 = 8 − 4 + 3 = 7 the point of tangency is ( 2 , 7 ) .the slope of the tangent line is f ′ ( 2 ) = 3 ( 2 2 ) − 2 = 10 thus, L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) L ( x ) = 7 + 10 ( x − 2 ) L ( x ) = 10 x − 13 b. To determine To check the accuracy of approximation L ( 2 + 0.1 ) ≈ f ( 2 + 0.1 ) Answer The approximation error is | 8.061 − 8 | ≈ 0.061 = 6.1 × 10 − 2 , the error is less than 10 − 1 Explanation Given information: The given value of a is 2 To approximate ( 2 + 0.1 ) 3 − 2 ( 2 + 0.1 ) + 3 , we use x = 2 + 0.1 : ( 2 + 0.1 ) 3 − 2 ( 2 + 0.1 ) + 3 = f ( 2 + 0.1 ) ≈ L ( 2 + 0.1 ) = 10 ( 2 + 0.1 ) − 13 = 8 The calculator gives ( 2 + 0.1 ) 3 − 2 ( 2 + 0.1 ) + 3 = 8.061 , so the approximation error is | 8.061 − 8 | ≈ 0.061 = 6.1 × 10 − 2 , the error is less than 10 − 1

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 1E

To determine

To find the linearization L(x)of f(x) at x=2

Expert Solution

Answer to Problem 1E

The linearization of the function is L(x)=10x−13

Explanation of Solution

Given information:

The given function is f(x)=x3−2x+3, x=2

Since f(2)=23−2⋅2+3=8−4+3=7 the point of tangency is (2,7) .the slope of the tangent line is f′(2)=3(22)−2=10 thus,

L(x)=f(a)+f′(a)(x−a)L(x)=7+10(x−2)L(x)=10x−13

To determine

To check the accuracy of approximation L(2+0.1)≈f(2+0.1)

Expert Solution

Answer to Problem 1E

The approximation error is |8.061−8|≈0.061=6.1×10−2 , the error is less than 10−1

Explanation of Solution

Given information:

The given value of a is 2

To approximate (2+0.1)3−2(2+0.1)+3 , we use x=2+0.1 :

(2+0.1)3−2(2+0.1)+3=f(2+0.1)≈L(2+0.1)=10(2+0.1)−13=8

The calculator gives (2+0.1)3−2(2+0.1)+3=8.061 , so the approximation error is

|8.061−8|≈0.061=6.1×10−2 , the error is less than 10−1

Chapter 5.5, Problem 10QR

Chapter 5.5, Problem 2E

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

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To find the linearization L(x)of f(x) at x=2

Answer to Problem 1E

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To check the accuracy of approximation L(2+0.1)≈f(2+0.1)

Answer to Problem 1E

Explanation of Solution

Chapter 5 Solutions

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