(a) Explanation Given: The value of the function is f ( 1 ) = 1 , f ′ ( 1 ) = 1 and f ′ ′ ( 1 ) = 1 . The value of the function is g ( 1 ) = 10 , g ′ ( 1 ) = 10 and g ′ ′ ( 1 ) = 10 . Definition used: Linear approximation “Suppose s is a differentiable on the interval I containing the point a . The linear approximation to s at a is a linear function L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) for x in I .” Calculation: Part 1 The value of the function is f ( 1 ) = 1 , f ′ ( 1 ) = 1 and f ′ ′ ( 1 ) = 1 with the point a = 1 . Write the function which satisfy the above function values is f ( x ) = 1 2 x 2 + 1 2 . Estimate the linear approximation to f near a = 1 through definition mentioned above with f ( 1 ) = 1 and f ′ ( 1 ) = 1 as follows, L ( x ) = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) = 1 + ( 1 ) ( x − 1 ) = 1 + x − 1 = x Calculate the value f at the point x = 1.1 as follows, f ( 1.1 ) = 1 2 ( 1.1 ) 2 + 1 2 = 1.21 2 + 1 2 = 0.605 + 0.5 = 1 (b) To determine To explain: The reason behind the error in the linear approximation to f near the point a is proportional to the magnitude of f ′ ′ ( a ) .

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 4.6, Problem 73E

(a)

To determine

The more accurate linear approximation near a given point for the two functions f and g as well as give the reason behind the accuracy of the linear approximation with graph.

(b)

To determine

To explain: The reason behind the error in the linear approximation to f near the point a is proportional to the magnitude of f′′(a) .

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Chapter 4.6, Problem 72E

Chapter 4.7, Problem 1QC

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Chapter 4 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Consider the function f(x) = x3. Where is the...Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Prob. 12E Ch. 4.1 - Absolute maximum/minimum values Use the following...

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The more accurate linear approximation near a given point for the two functions f and g as well as give the reason behind the accuracy of the linear approximation with graph.

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The more accurate linear approximation near a given point for the two functions f and g as well as give the reason behind the accuracy of the linear approximation with graph.

To explain: The reason behind the error in the linear approximation to f near the point a is proportional to the magnitude of f′′(a) .

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