Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 9E from Chapter 2.1 from Stewart's Calculus, 8th Edition.
Problem 9E
Chapter:
Problem:
The point P(1, 0) lies on the curve y − sin(10 Pi/x)...
Step-by-Step Solution
Step 1
The slope of a secant line joining P($x_1,y_1$) and Q($x_2,y_2$ is given by:\[{\rm{Slope}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Step 2: (a)
Consider the Table of points along with calculated Slopes:
https://imgur.com/nFvNoUS
The slopes do not appear to approach a limit
Step 3: Graph of the Function
Consider the graph of the function given below. As we can see that slopes of the secant lines are not close to each other as it fluctuates largely even for small change in x
https://imgur.com/1BVUxlx
Step 4: c
To find the slope of tangent line, choose point Q very close to P. Let Q be at x=1.00001
\[y\left( {0.00001} \right) = - 0.000314153\]The point Q is: \[Q = \left( {1.00001,- 0.000314153} \right)\]So the slope at P is: \[\begin{array}{l}{S_P} = \dfrac{{0 - \left( { - 0.000314153} \right)}}{{0.00001}}\\ = 31.4153\end{array}\]https://imgur.com/tTLP2YRTherefore,
\[{S_P} = 31.4153\]
The slope of a secant line joining P($x_1,y_1$) and Q($x_2,y_2$ is given by:\[{\rm{Slope}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Step 2: (a)
Consider the Table of points along with calculated Slopes:
https://imgur.com/nFvNoUS
The slopes do not appear to approach a limit
Step 3: Graph of the Function
Consider the graph of the function given below. As we can see that slopes of the secant lines are not close to each other as it fluctuates largely even for small change in x
https://imgur.com/1BVUxlx
Step 4: c
To find the slope of tangent line, choose point Q very close to P. Let Q be at x=1.00001
\[y\left( {0.00001} \right) = - 0.000314153\]The point Q is: \[Q = \left( {1.00001,- 0.000314153} \right)\]So the slope at P is: \[\begin{array}{l}{S_P} = \dfrac{{0 - \left( { - 0.000314153} \right)}}{{0.00001}}\\ = 31.4153\end{array}\]https://imgur.com/tTLP2YRTherefore,
\[{S_P} = 31.4153\]