Explanation Use the given equation (3) and obtain the plot of its fourth derivative to observe its maximum value in the specified range. Given: d d x tan − 1 x = 1 x 2 + 1 ; ∫ d x x 2 + 1 = tan − 1 x + C ----- (3) Calculation: From equation (3), d d x tan − 1 x = 1 x 2 + 1 = > ∫ 0 1 To determine (b) A value of N such that S N approximates the integral with an error of utmost 10 − 6 and calculate the corresponding approximation and confirm that you have computed π 4 to at least four places.

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.9, Problem 51E

To determine

(a)

To show that $\int_{0}^{1} \frac{d x}{1 + x^{2}} = \frac{π}{4}$ and use a computer algebra system to graph the function $f^{(4)}$ for $f (x) = {(1 + x^{2})}^{- 1}$ and find its maximum on $[0, 1]$

To determine

(b)

A value of $N$ such that $S_{N}$ approximates the integral with an error of utmost $10^{- 6}$ and calculate the corresponding approximation and confirm that you have computed $\frac{π}{4}$ to at least four places.

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Chapter 8.9, Problem 50E

Chapter 8.9, Problem 52E

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(a) To show that ∫ 0 1 d x 1 + x 2 = π 4 and use a computer algebra system to graph the function f ( 4 ) for f ( x ) = ( 1 + x 2 ) − 1 and find its maximum on [ 0 , 1 ]

Solution Summary: The author explains how to determine the value of N such that S N approximates the integral with an error of utmost 10 6 is 20.

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(a)

To show that $\int_{0}^{1} \frac{d x}{1 + x^{2}} = \frac{π}{4}$ and use a computer algebra system to graph the function $f^{(4)}$ for $f (x) = {(1 + x^{2})}^{- 1}$ and find its maximum on $[0, 1]$

(b)

A value of $N$ such that $S_{N}$ approximates the integral with an error of utmost $10^{- 6}$ and calculate the corresponding approximation and confirm that you have computed $\frac{π}{4}$ to at least four places.

Want to see the full answer?

Chapter 8 Solutions

(a) To show that ∫ 0 1 d x 1 + x 2 = π 4 and use a computer algebra system to graph the function f ( 4 ) for f ( x ) = ( 1 + x 2 ) − 1 and find its maximum on [ 0 , 1 ]

Solution Summary: The author explains how to determine the value of N such that S N approximates the integral with an error of utmost 10 6 is 20.

Concept explainers

Videos

(a) To show that ∫01dx1+x2=π4 and use a computer algebra system to graph the function f(4) for f(x)=(1+x2)−1 and find its maximum on [0,1]

(b) A value of Nsuch that SN approximates the integral with an error of utmost 10−6 and calculate the corresponding approximation and confirm that you have computed π4 to at least four places.

Want to see the full answer?

Chapter 8 Solutions

(a)

To show that $\int_{0}^{1} \frac{d x}{1 + x^{2}} = \frac{π}{4}$ and use a computer algebra system to graph the function $f^{(4)}$ for $f (x) = {(1 + x^{2})}^{- 1}$ and find its maximum on $[0, 1]$

(b)

A value of $N$ such that $S_{N}$ approximates the integral with an error of utmost $10^{- 6}$ and calculate the corresponding approximation and confirm that you have computed $\frac{π}{4}$ to at least four places.