Explanation Given: Point P = ( 1 , 1 2 ) Graphs f ( x ) = 1 ( 1 + x 2 ) and L ( x ) = 1 2 + m ( x − 1 ) Formula Used: f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h Calculation: To determine if point P = ( 1 , 1 2 ) lies on the graph f ( x ) = 1 1 + x 2 ,we put it in the equation of the graph f ( 1 ) = 1 1 + 1 2 = 1 2 Similarly, we can determine that the point P = ( 1 , 1 2 ) lies on the graph L ( x ) = 1 2 + m ( x − 1 ) L ( 1 ) = 1 2 + m ( 1 − 1 ) = 1 2 As the point P satisfies the equations of the graphs for every value of m , hence it lies on both the graphs. The graphs of f ( x ) and L ( x ) are as shown:

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 3.1, Problem 64E

To determine

We need to determine if point P=(1,12) lies on the graphs f(x)=11+x2 and L(x)=12+m(x−1)

We need to plot the graphs y=f(x) and y=L(x) on the same axis for the several values of m and find the value of m such that y=L(x) is tangent to y=f(x)

We need to estimate f'(1)

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Chapter 3.1, Problem 63E

Chapter 3.1, Problem 65E

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Solution Summary: The author explains how to determine if the point P=(1,12) lies on the graphs.

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We need to determine if point P=(1,12) lies on the graphs f(x)=11+x2 and L(x)=12+m(x−1) We need to plot the graphs y=f(x) and y=L(x) on the same axis for the several values of m and find the value of m such that y=L(x) is tangent to y=f(x) We need to estimate f'(1)

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Chapter 3 Solutions

We need to determine if point P=(1,12) lies on the graphs f(x)=11+x2 and L(x)=12+m(x−1)

We need to plot the graphs y=f(x) and y=L(x) on the same axis for the several values of m and find the value of m such that y=L(x) is tangent to y=f(x)

We need to estimate f'(1)