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: 9781319055905

Author: FRANZOSA

Publisher: VST

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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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Let y = tan10 (x5).Find ƒ, g, and h such that y = ƒ(u),where u = g(v) and v = h(x).

9n Let f(x) = 3/2 cos x and let a = Complete parts (a) through (d) below. 4 a. Find the linear approximation L for the function f at the point a. L(x) =| b. Graph f and L on the same set of axes. Choose the correct graph below. OA. OB. OD. c. Based on the graphs in part (b), state whether linear approximations to f near a are underestimates or overestimates. 9n Linear approximations to f(x) = 3/2 cos x near a = - because the graph of L lies the graph of f near a. are 4 d. Compute f''(a) to confirm the conclusion in part (c). 9n The value of f'"(x) at a = is f" which is V 0. This means that f is concave confirming the conclusion in part (c). = 4 4

Chapter 4 Solutions

EBK 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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