Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 3E from Chapter 2.4 from Stewart's Calculus, 8th Edition.
Problem 3E
Chapter:
Problem:
Use the given graph ...to find a number such that
Step-by-Step Solution
Given information
We are given with following graph
https://imgur.com/XorDQgl
Step 1:
Expand the Inequality \[\begin{array}{l}\left| {x - 4} \right| < \delta \\ - \delta < x - 4 < \delta \\ - \delta + 4 < x < \delta + 4\end{array}\]For limits of x between $- \delta + 4$ and $\delta + 4$, the function value lies between 1.6 and 2.4
Step 2:
Consider the left side
\[\begin{array}{l}f\left( {4 - \delta } \right) = 1.6\\\sqrt {4 - \delta } = 1.6\\4 - \delta = {1.6^2}\\4 - \delta = 2.56\\\delta = 4 - 2.56\\\delta = 1.44\end{array}\]Therefore, \[{\delta _1} = 1.44\]
Step 3:
Consider the right side
\[\begin{array}{l}f\left( {4 + \delta } \right) = 2.4\\\sqrt {4 + \delta } = 2.4\\4 + \delta = {2.4^2}\\4 + \delta = 5.76\\\delta = 5.76 - 4\\\delta = 1.76\end{array}\]Therefore, \[{\delta _2} = 1.76\]
Step 4:
Thus, the required number $\delta $ is between 1.44 and 1.76. Thus minimum value of $\delta $ is 1.44
Therefore, \[{\delta } = 1.44\]
We are given with following graph
https://imgur.com/XorDQgl
Step 1:
Expand the Inequality \[\begin{array}{l}\left| {x - 4} \right| < \delta \\ - \delta < x - 4 < \delta \\ - \delta + 4 < x < \delta + 4\end{array}\]For limits of x between $- \delta + 4$ and $\delta + 4$, the function value lies between 1.6 and 2.4
Step 2:
Consider the left side
\[\begin{array}{l}f\left( {4 - \delta } \right) = 1.6\\\sqrt {4 - \delta } = 1.6\\4 - \delta = {1.6^2}\\4 - \delta = 2.56\\\delta = 4 - 2.56\\\delta = 1.44\end{array}\]Therefore, \[{\delta _1} = 1.44\]
Step 3:
Consider the right side
\[\begin{array}{l}f\left( {4 + \delta } \right) = 2.4\\\sqrt {4 + \delta } = 2.4\\4 + \delta = {2.4^2}\\4 + \delta = 5.76\\\delta = 5.76 - 4\\\delta = 1.76\end{array}\]Therefore, \[{\delta _2} = 1.76\]
Step 4:
Thus, the required number $\delta $ is between 1.44 and 1.76. Thus minimum value of $\delta $ is 1.44
Therefore, \[{\delta } = 1.44\]