Explanation Given: f ( x ) = tan x a t x = π 4 tan ( π 4 + h ) − 1 ≈ 2 h K = 6.2 Concept: Suppose that y = f ( x ) is a differentiable curve at x = a . Then the tangent line at x = a has equation y = f ( a ) + f ' ( a ) ( x − a ) We call the above equation the linear approximation or linearization of y = f ( x ) at the point ( a , f ( a ) ) and write f ( x ) ≈ L ( x ) = f ( a ) + f ' ( a ) ( x − a ) We sometimes write L a ( x ) to stress that the approximation is near a . Calculation: Description Step 1: Find f ' ( x ) Let f ( x ) = tan x Then f ( π 4 ) = 1 , f ' ( x ) = sec 2 x and f ' ( π 4 ) = 2 Step 2: By linear approximation, we will get… f ( π 4 + h ) − f ( π 4 ) = tan ( π 4 + h ) − 1 ≈ 2 h Moreover, f ' ' ( x ) = 2 sec 2 x tan x Because this is a decreasing function, it follows that for x ≥ 9 , K=max | f ' ' ( x ) | ≤ | f ' ' ( 9 ) | = 1 108 < 0

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 4.1, Problem 68E

To determine

Using a plot, the given statement regarding the second derivative of the given function pertaining to the linear approximation and also verify that the error numerically satisfies the given equation.

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Chapter 4.1, Problem 67E

Chapter 4.1, Problem 69E

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Chapter 4 Solutions

Calculus: Early Transcendentals

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Using a plot, the given statement regarding the second derivative of the given function pertaining to the linear approximation and also verify that the error numerically satisfies the given equation.

Solution Summary: The author analyzes the linear approximation of y = f(x) at the point (a,f)

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Using a plot, the given statement regarding the second derivative of the given function pertaining to the linear approximation and also verify that the error numerically satisfies the given equation.

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