If ƒ(6) = 7 and ƒ’(6) = 4, estimate ƒ(5.8). f(5.8) ~

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:**

If \( f(6) = 7 \) and \( f'(6) = 4 \), estimate \( f(5.8) \).

**Solution:**

To estimate \( f(5.8) \), we use the linear approximation formula:

\[
f(x) \approx f(a) + f'(a)(x - a)
\]

Given:
- \( a = 6 \)
- \( f(6) = 7 \)
- \( f'(6) = 4 \)
- \( x = 5.8 \)

Substitute the values into the formula:

\[
f(5.8) \approx 7 + 4(5.8 - 6)
\]

Calculate the difference:

\[
5.8 - 6 = -0.2
\]

Then calculate:

\[
f(5.8) \approx 7 + 4(-0.2) = 7 - 0.8 = 6.2
\]

**Estimate:**

\( f(5.8) \approx 6.2 \)
Transcribed Image Text:**Problem Statement:** If \( f(6) = 7 \) and \( f'(6) = 4 \), estimate \( f(5.8) \). **Solution:** To estimate \( f(5.8) \), we use the linear approximation formula: \[ f(x) \approx f(a) + f'(a)(x - a) \] Given: - \( a = 6 \) - \( f(6) = 7 \) - \( f'(6) = 4 \) - \( x = 5.8 \) Substitute the values into the formula: \[ f(5.8) \approx 7 + 4(5.8 - 6) \] Calculate the difference: \[ 5.8 - 6 = -0.2 \] Then calculate: \[ f(5.8) \approx 7 + 4(-0.2) = 7 - 0.8 = 6.2 \] **Estimate:** \( f(5.8) \approx 6.2 \)
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