Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Problem Statement:** If \( f(6) = 7 \) and \( f'(6) = 4 \), estimate \( f(5.8) \). **Solution:** To estimate \( f(5.8) \), we use the linear approximation formula: \[ f(x) \approx f(a) + f'(a)(x - a) \] Given: - \( a = 6 \) - \( f(6) = 7 \) - \( f'(6) = 4 \) - \( x = 5.8 \) Substitute the values into the formula: \[ f(5.8) \approx 7 + 4(5.8 - 6) \] Calculate the difference: \[ 5.8 - 6 = -0.2 \] Then calculate: \[ f(5.8) \approx 7 + 4(-0.2) = 7 - 0.8 = 6.2 \] **Estimate:** \( f(5.8) \approx 6.2 \)
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Transcribed Image Text:**Problem Statement:** If \( f(6) = 7 \) and \( f'(6) = 4 \), estimate \( f(5.8) \). **Solution:** To estimate \( f(5.8) \), we use the linear approximation formula: \[ f(x) \approx f(a) + f'(a)(x - a) \] Given: - \( a = 6 \) - \( f(6) = 7 \) - \( f'(6) = 4 \) - \( x = 5.8 \) Substitute the values into the formula: \[ f(5.8) \approx 7 + 4(5.8 - 6) \] Calculate the difference: \[ 5.8 - 6 = -0.2 \] Then calculate: \[ f(5.8) \approx 7 + 4(-0.2) = 7 - 0.8 = 6.2 \] **Estimate:** \( f(5.8) \approx 6.2 \)
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