Explanation Given: The function is f ( x ) = tan − 1 x and the value of a and b are 1 and 1.05 respectively. Concept used: Linear approximation: For a differentiable function at x = a the linearization is expressed as follows: f ( x ) ≈ f ( a ) + ( x − a ) f ′ ( a ) Error: The formula to calculate the error is as follows: Error = | Δ f − f ′ ( a ) Δ x | Calculation: The given function is expressed as follow: f ( x ) = ln ( x 2 + 1 ) ….(1) The expression Δ f represents the change is the value of f ( x ) as the value of value of x is changed from a to b . The value of Δ f is calculated as follows: Δ f = f ( b ) − f ( a ) = f ( 1.05 ) − f ( 1 ) Label the above equation as equation (2). Δ f = f ( 1.05 ) − f ( 1 ) …..(2) Substitute 1 for x in equation (1) as shown below: f ( 1 ) = ln ( 1 + 1 ) = ln ( 2 ) = 0.6931 Substitute 1.05 for x in equation (1) as shown below: f ( 1.05 ) = ln ( 1.05 + 1 ) = ln ( 2.05 ) = 0.7178 To obtain the value of Δ f substitute the value of f ( 1 ) and f ( 1.05 ) in equation (2) as shown below: Δ f = 0.7178 − 0.6931 = 0.0247 Differentiate equation (1) with respect to x as shown below: f ′ ( x ) = d ( ln ( x 2 + 1 ) ) d x = 2 x x 2 + 1 Substitute 1 for x in the above equation as shown below: f ′ ( 1 ) = 2 ⋅ 1 1 2 + 1 = 1 The value of Δ x is calculated as follows: Δ x = b − a = 1

4th Edition

ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 4.1, Problem 18E

To determine

To find:

The value of Δf, error and the value of f(b) for the function f(x)=tan−1x such that the value of a and b are 1 and 1.05 respectively.

Expert Solution & Answer

Chapter 4 Solutions

CALCULUS:EARLY TRANS.(LL)-W/ACCESS

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The value of Δ f , error and the value of f ( b ) for the function f ( x ) = tan − 1 x such that the value of a and b are 1 and 1.05 respectively.

Solution Summary: The author calculates the value of Delta f, error, and the linear approximation of the given function.

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To find:

The value of Δf, error and the value of f(b) for the function f(x)=tan−1x such that the value of a and b are 1 and 1.05 respectively.

Chapter 4 Solutions

The value of Δ f , error and the value of f ( b ) for the function f ( x ) = tan − 1 x such that the value of a and b are 1 and 1.05 respectively.

Solution Summary: The author calculates the value of Delta f, error, and the linear approximation of the given function.

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To find: The value of Δf, error and the value of f(b) for the function f(x)=tan−1x such that the value of a and b are 1 and 1.05 respectively.

Chapter 4 Solutions

To find:

The value of Δf, error and the value of f(b) for the function f(x)=tan−1x such that the value of a and b are 1 and 1.05 respectively.