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
![### Calculus Question: Derivative of an Integral Function
#### Question
What is \( f'(x) \) given the integral function \( f(x) = \int_0^x \cos^2(u) \, du \)?
#### Answer Choices
1. \(\mathbf{ f'(x) = \cos^2(x)}\) *(correct answer)*
2. \( f'(x) = -\cos^2(x) \)
3. \( f'(x) = \sin^2(x) \)
4. \( f'(x) = \cos(x) \sin(x) \)
#### Explanation
To find \( f'(x) \) from the given integral function \( f(x) = \int_0^x \cos^2(u) \, du \), we use the Fundamental Theorem of Calculus, which states that if \( F(x) = \int_a^x f(t) \, dt \), then \( F'(x) = f(x) \).
In this problem:
- The function inside the integral is \( \cos^2(u) \).
- \( f(x) = \int_0^x \cos^2(u) \, du \).
Therefore, by the Fundamental Theorem of Calculus:
\[ f'(x) = \cos^2(x) \]
Thus, the correct answer is \(\boxed{ f'(x) = \cos^2(x) }\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8332b21a-7711-4793-b2df-7a1f1091dd0f%2F4e66bc79-c0e0-40b1-8a23-5bc1aedd8007%2Fuevjj9_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Question: Derivative of an Integral Function
#### Question
What is \( f'(x) \) given the integral function \( f(x) = \int_0^x \cos^2(u) \, du \)?
#### Answer Choices
1. \(\mathbf{ f'(x) = \cos^2(x)}\) *(correct answer)*
2. \( f'(x) = -\cos^2(x) \)
3. \( f'(x) = \sin^2(x) \)
4. \( f'(x) = \cos(x) \sin(x) \)
#### Explanation
To find \( f'(x) \) from the given integral function \( f(x) = \int_0^x \cos^2(u) \, du \), we use the Fundamental Theorem of Calculus, which states that if \( F(x) = \int_a^x f(t) \, dt \), then \( F'(x) = f(x) \).
In this problem:
- The function inside the integral is \( \cos^2(u) \).
- \( f(x) = \int_0^x \cos^2(u) \, du \).
Therefore, by the Fundamental Theorem of Calculus:
\[ f'(x) = \cos^2(x) \]
Thus, the correct answer is \(\boxed{ f'(x) = \cos^2(x) }\).
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images

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