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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![**Problem Statement:**
Given the function:
\[ f(x) = \frac{1}{x^3} \]
**Tasks:**
1. Compute \( f(2) \).
2. Compute the derivative \( f'(x) \) and evaluate it at \( x = 2 \), i.e., find \( f'(2) \).
**Solution:**
1. **Computing \( f(2) \)**:
\[ f(2) = \frac{1}{2^3} = \frac{1}{8} \]
2. **Finding the derivative \( f'(x) \)**:
\[ f(x) = x^{-3} \]
Using the power rule for differentiation:
\[ f'(x) = -3x^{-4} = -\frac{3}{x^4} \]
**Evaluate \( f'(2) \)**:
\[ f'(2) = -\frac{3}{2^4} = -\frac{3}{16} \]
**Answers:**
- \( f(2) = \frac{1}{8} \)
- \( f'(2) = -\frac{3}{16} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2da21713-669c-4fb3-a696-2121fe318bc8%2Fecbe153b-6ae1-462d-a0e5-0e61b9b9cffb%2F1953ei4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the function:
\[ f(x) = \frac{1}{x^3} \]
**Tasks:**
1. Compute \( f(2) \).
2. Compute the derivative \( f'(x) \) and evaluate it at \( x = 2 \), i.e., find \( f'(2) \).
**Solution:**
1. **Computing \( f(2) \)**:
\[ f(2) = \frac{1}{2^3} = \frac{1}{8} \]
2. **Finding the derivative \( f'(x) \)**:
\[ f(x) = x^{-3} \]
Using the power rule for differentiation:
\[ f'(x) = -3x^{-4} = -\frac{3}{x^4} \]
**Evaluate \( f'(2) \)**:
\[ f'(2) = -\frac{3}{2^4} = -\frac{3}{16} \]
**Answers:**
- \( f(2) = \frac{1}{8} \)
- \( f'(2) = -\frac{3}{16} \)
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