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a) Consider f (x) = x³. Write the linear approximation of f for x near 2. Let's call this linear approximation L(x).

a) Consider f (x) = x³. Write the linear approximation of f for x near 2. Let's call this linear approximation L(x).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Question 5 In this problem we investigate the error involved in linear approximation. (a) Consider f(x) = x³. Write the linear approximation of f for x near 2. Let's call this linear approximation L(x). (b) Define the error of this linear approximation to be E(x) = f(x) – L(x). Show that E(x) lim x→2 x 0. (This means that not only is the error small for x 2 2, but that the error is small relative to (x – 2). It turns out that if f is any function that is differentiable at x = a, and E(x) is the error E(x) in the linear approximation, then lim our concrete example is one case of a far more x>a x – a general fact.) E(x) x→2 (x – 2)2 (c) Calculate lim What is the relationship between this limit and f"(2)? (It turns out that if f is twice differentiable at x = a, then E(x) f"(a) lim %3D x>a (x – a)2 2
Transcribed Image Text:Question 5 In this problem we investigate the error involved in linear approximation. (a) Consider f(x) = x³. Write the linear approximation of f for x near 2. Let's call this linear approximation L(x). (b) Define the error of this linear approximation to be E(x) = f(x) – L(x). Show that E(x) lim x→2 x 0. (This means that not only is the error small for x 2 2, but that the error is small relative to (x – 2). It turns out that if f is any function that is differentiable at x = a, and E(x) is the error E(x) in the linear approximation, then lim our concrete example is one case of a far more x>a x – a general fact.) E(x) x→2 (x – 2)2 (c) Calculate lim What is the relationship between this limit and f"(2)? (It turns out that if f is twice differentiable at x = a, then E(x) f"(a) lim %3D x>a (x – a)2 2
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