Chapter 6: Q. 2 (page 569)

In each problem that follows we are given values for $f (c)$ and $f' (c)$ for some $c$ Use this information to approximate $f (c + h)$ for the given value of $h$ .
a) $f (0) = 2, f' (0) = 1, h = 1$
b) $f (1) = - 1, f' (1) = - 3, h = 0.5$
c) $f (- 2) = - 3, f' (- 2) = 2, h = 0.25$
d) $f (1.5) = 6.2, f' (1.5) = 2.2, h = 0.1$

Short Answer

Expert verified

The linear approximation are:

a) $f (1) = 3$

b) $f (1.5) = - 2.5$

c) $f (- 1.75) = - 2.5$

d) $f (1.6) = 6.42$

Step by step solution

Step 1. Given information

a) $f (0) = 2, f' (0) = 1, h = 1$

b) $f (1) = - 1, f' (1) = - 3, h = 0.5$

c) $f (- 2) = - 3, f' (- 2) = 2, h = 0.25$

d) $f (1.5) = 6.2, f' (1.5) = 2.2, h = 0.1$

Part(a) step 1. Evaluate linear approximation.

The linear approximation is given by a formula: $f (c + h) = f (c) + h f' (c)$

Given $f (0 + 1) = f (0) + 1 . f' (0) f (1) = 2 + 1 (1) f (1) = 3$

Part (b) step 1. Evaluate linear approximation.

$f (c + h) = f (c) + h f' (c) f (1 + 0.5) = f (1) + (0.5) f' (1) f (1.5) = - 1 + (0.5) (- 3) f (1.5) = - 1 - 1.5 f (1.5) = - 2.5$

Part(c) step 1. Evaluate linear approximation.

$f (c + h) = f (c) + h f' (c) f (- 2 + 0.25) = f (- 2) + (0.25) f' (- 2) f (- 1.75) = - 3 + (0.25) (2) f (- 1.75) = - 3 + 0.5 f (- 1.75) = - 2.5$

Part(d) step 1. Evaluate linear approximation.

$f (c + h) = f (c) + h f' (c) f (1.5 + 0.1) = f (1.5) + (0.1) f' (1.5) f (1.6) = 6.2 + (0.1) (2.2) f (1.6) = 6.2 + 0.22 f (1.6) = 6.42$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given information

Part(a) step 1. Evaluate linear approximation.

Part (b) step 1. Evaluate linear approximation.

Part(c) step 1. Evaluate linear approximation.

Part(d) step 1. Evaluate linear approximation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In each problem that follows we are given values for f(c)and f'(c)for some cUse this information to approximate f(c+h)for the given value of h.a)f(0)=2,f'(0)=1,h=1b)f(1)=-1,f'(1)=-3,h=0.5c)f(-2)=-3,f'(-2)=2,h=0.25d)f(1.5)=6.2,f'(1.5)=2.2,h=0.1

Short Answer

Step by step solution

Step 1. Given information

Part(a) step 1. Evaluate linear approximation.

Part (b) step 1. Evaluate linear approximation.

Part(c) step 1. Evaluate linear approximation.

Part(d) step 1. Evaluate linear approximation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.