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Calculus
1 If f(x) = compute f(2) and f'(2)

1 If f(x) = compute f(2) and f'(2)

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter4: Exponential And Logarithmic Functions
Section: Chapter Questions
Problem 3CC: If xis large, which function grows faster, f(x)=2x or g(x)=x2?
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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:** - \( f(2) = \frac{1}{8} \) - \( f'(2) = -\frac{3}{16} \)
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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