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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4.5 - Problem 7e
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4.5 - Problem 7e

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Linear approximation a. Write The equation of the line

ISBN: 9780321570567 2

Solution for problem 7E Chapter 4.5

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition

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Problem 7E

Linear approximation a. Write The equation of the line that represents the linear approximation to the following functions at the given point a. b. Graph the function and the linear approximation near x? = a ? . c. Use the linear approximation to estimate the given function value. d. Compute the percent error in your approximation:? 100 • l approx-exact | / |exact| f?? ?) = 12 ?? ? a = 2; ?? 2.1)

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SOLUTION STEP 1 Given f(x) = 12 x ;a = 2;f(2.1) f(x) = 2x (a). Equation of the line of linear approximation y = L(x) = f(a) + f(a)(x a) 2 Thus f(a) = 12 2 = 12 4 = 8 f(a) = 2 × 2 = 4 Therefore y = L(x) = f(a) + f(a)(x a) y = 8 + 4(x a) = 8 4x + 4a = 8 4x + 4 × 2 = 8 4x + 8 = 16 4x Thus we get y = 16 4x STEP 2 (b).Graph the function and the linear approximation near x = a. STEP 3 (c). Use the linear approximation to estimate the given function value. We have f(2.1) L(2.1) 16 4 × 2.1 16 8.4 = 7.6 Therefore f(2.1) 7.6 STEP 4 (d). |approximate exact| Compute the percent error in your approximation: 100 × |exact| 2 We get the exact value ,f(2.1)= 12 2.1 = 7.59 |7.67.59| Then percent error = 100 × |7.59| = 0.13%

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Linear approximation a. Write The equation of the line