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 on Educational Website
**Problem Statement:**
Suppose that \( F'(x) = \frac{x^2}{x-2} \). Then \( F(7) - F(5) = \ldots \)
**Multiple Choice Options:**
- \( \cdot \cdot \cdot = 8 + 4\ln(3/5) \)
- \( \cdot \cdot \cdot = 16 + 4\ln(5/3) \)
- \( \cdot \cdot \cdot = 8 + 2\ln(3/5) \)
- \( \cdot \cdot \cdot = 16\ln(5/3) \)
**Explanation:**
This problem tests your understanding of the Fundamental Theorem of Calculus and how to apply integration to find the difference in function values. You will need to integrate \( F'(x) \) over the interval [5, 7] to find \( F(7) - F(5) \).
1. **Set Up the Integral:**
\[
F(7) - F(5) = \int_{5}^{7} \frac{x^2}{x-2} \, dx
\]
2. **Simplify the Integral Expression:**
The integral can be simplified using partial fraction decomposition or other integration techniques as appropriate.
3. **Evaluate the Integral:**
Calculate the definite integral to find the exact value of \( F(7) - F(5) \).
4. **Choose the Correct Option:**
Compare your result with the provided multiple choice options to select the correct answer.
Each step involves important calculus concepts, and understanding how to apply them is crucial for solving this problem correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03bb8038-6a00-49cd-8aea-6808bae4a834%2F30f6f10a-3437-4595-876c-34a1b000083d%2F7ihwbkk_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Question on Educational Website
**Problem Statement:**
Suppose that \( F'(x) = \frac{x^2}{x-2} \). Then \( F(7) - F(5) = \ldots \)
**Multiple Choice Options:**
- \( \cdot \cdot \cdot = 8 + 4\ln(3/5) \)
- \( \cdot \cdot \cdot = 16 + 4\ln(5/3) \)
- \( \cdot \cdot \cdot = 8 + 2\ln(3/5) \)
- \( \cdot \cdot \cdot = 16\ln(5/3) \)
**Explanation:**
This problem tests your understanding of the Fundamental Theorem of Calculus and how to apply integration to find the difference in function values. You will need to integrate \( F'(x) \) over the interval [5, 7] to find \( F(7) - F(5) \).
1. **Set Up the Integral:**
\[
F(7) - F(5) = \int_{5}^{7} \frac{x^2}{x-2} \, dx
\]
2. **Simplify the Integral Expression:**
The integral can be simplified using partial fraction decomposition or other integration techniques as appropriate.
3. **Evaluate the Integral:**
Calculate the definite integral to find the exact value of \( F(7) - F(5) \).
4. **Choose the Correct Option:**
Compare your result with the provided multiple choice options to select the correct answer.
Each step involves important calculus concepts, and understanding how to apply them is crucial for solving this problem correctly.
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