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...
Related questions
Question
![**Problem Statement:**
Given that
\[ f(x) = x^{11} h(x) \]
and
\[ h(-1) = 2 \]
\[ h'(-1) = 5 \]
Calculate \( f'(-1) \).
**Hint:** Use the product rule and the power rule.
---
**Solution Reference:**
To solve for \( f'(-1) \), apply the product rule for differentiation: if \( f(x) = u(x)v(x) \), then
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
For this problem:
- Let \( u(x) = x^{11} \) and \( v(x) = h(x) \).
- Therefore, \( u'(x) = 11x^{10} \) using the power rule.
Substitute into the product rule formula:
\[ f'(x) = 11x^{10}h(x) + x^{11}h'(x) \]
Evaluate \( f'(-1) \):
\[ f'(-1) = 11(-1)^{10}h(-1) + (-1)^{11}h'(-1) \]
Since:
\[ (-1)^{10} = 1 \]
and
\[ (-1)^{11} = -1 \]
Substitute the given values:
\[ f'(-1) = 11(1)(2) + (-1)(5) \]
\[ f'(-1) = 22 - 5 \]
\[ f'(-1) = 17 \]
So, \( f'(-1) = 17 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d09ffa2-b6c2-4e97-915d-74d176d987ab%2F127f0c58-56b1-4472-883c-370c2a546cd7%2F9lo5mbb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given that
\[ f(x) = x^{11} h(x) \]
and
\[ h(-1) = 2 \]
\[ h'(-1) = 5 \]
Calculate \( f'(-1) \).
**Hint:** Use the product rule and the power rule.
---
**Solution Reference:**
To solve for \( f'(-1) \), apply the product rule for differentiation: if \( f(x) = u(x)v(x) \), then
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
For this problem:
- Let \( u(x) = x^{11} \) and \( v(x) = h(x) \).
- Therefore, \( u'(x) = 11x^{10} \) using the power rule.
Substitute into the product rule formula:
\[ f'(x) = 11x^{10}h(x) + x^{11}h'(x) \]
Evaluate \( f'(-1) \):
\[ f'(-1) = 11(-1)^{10}h(-1) + (-1)^{11}h'(-1) \]
Since:
\[ (-1)^{10} = 1 \]
and
\[ (-1)^{11} = -1 \]
Substitute the given values:
\[ f'(-1) = 11(1)(2) + (-1)(5) \]
\[ f'(-1) = 22 - 5 \]
\[ f'(-1) = 17 \]
So, \( f'(-1) = 17 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning