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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Question
![**Problem Statement:**
Find \( g'(8) \) given that \( f(8) = -6 \), \( f'(8) = 7 \), and \( g(x) = \frac{9x + 5}{f(x)} \). (Round your answer to four decimal places.)
**Solution:**
To solve for \( g'(8) \), we need to use the quotient rule for derivatives. The function \( g(x) = \frac{9x + 5}{f(x)} \) is in the form of \( \frac{u(x)}{v(x)} \) where \( u(x) = 9x + 5 \) and \( v(x) = f(x) \).
The quotient rule states:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
First, calculate the derivatives:
- \( u'(x) = 9 \) since the derivative of \( 9x + 5 \) is 9.
- \( v'(x) = f'(x) \).
Substitute the given values:
- \( u'(8) = 9 \)
- \( v(8) = f(8) = -6 \)
- \( v'(8) = f'(8) = 7 \)
- \( u(8) = 9 \times 8 + 5 = 77 \)
Using the quotient rule, calculate:
\[
g'(8) = \frac{9(-6) - 77(7)}{(-6)^2}
\]
Simplify:
\[
g'(8) = \frac{-54 - 539}{36} = \frac{-593}{36} \approx -16.4722
\]
Therefore, \( g'(8) \) is approximately \(-16.4722\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe88a8cea-da71-40ab-aaed-a0e0afa0772e%2F5ccd4f83-7935-4590-b10d-98fe0f0d4340%2Ftvr80q7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \( g'(8) \) given that \( f(8) = -6 \), \( f'(8) = 7 \), and \( g(x) = \frac{9x + 5}{f(x)} \). (Round your answer to four decimal places.)
**Solution:**
To solve for \( g'(8) \), we need to use the quotient rule for derivatives. The function \( g(x) = \frac{9x + 5}{f(x)} \) is in the form of \( \frac{u(x)}{v(x)} \) where \( u(x) = 9x + 5 \) and \( v(x) = f(x) \).
The quotient rule states:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
First, calculate the derivatives:
- \( u'(x) = 9 \) since the derivative of \( 9x + 5 \) is 9.
- \( v'(x) = f'(x) \).
Substitute the given values:
- \( u'(8) = 9 \)
- \( v(8) = f(8) = -6 \)
- \( v'(8) = f'(8) = 7 \)
- \( u(8) = 9 \times 8 + 5 = 77 \)
Using the quotient rule, calculate:
\[
g'(8) = \frac{9(-6) - 77(7)}{(-6)^2}
\]
Simplify:
\[
g'(8) = \frac{-54 - 539}{36} = \frac{-593}{36} \approx -16.4722
\]
Therefore, \( g'(8) \) is approximately \(-16.4722\).
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