X What is f'(x) given the integral function f(x) = f* cos² (u) du? f'(x) = cos²(x) o f'(x) = -cos²(x) o f'(x) = sin²(x) o f'(x) = cos(x) sin(x)

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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### 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 \(\boxed{ f'(x) = \cos^2(x) }\).
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 \(\boxed{ f'(x) = \cos^2(x) }\).
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