Explanation For a graph of the function f , difference quotient computes the slope of a secant line through two points on the graph. The x -coordinates of these two points are x and x + h . Given: f ( x ) = x 5 / 2 Formula Used: The difference quotient is: f ( a + h ) − f ( a ) h f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h Calculation: The graph shows that the given function is an increasing function. It is evident from the graph that the slope of the secant line at x = 4 − h is less than the slope of the tangent line at x = 4 which is less than the slope of the secant line at x = 4 + h . (a) For smaller value of h = 0.0001 f ( 4 ) − f ( 3.999 ) 0.0001 = 0.019996 0.0001 = 19.996 f ' ( 4 ) = 20 f ( 4 + h ) − f ( 4 ) h = 0.00200004 0.0001 = 20

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 3.1, Problem 63E

To determine

We need to plot the graph of f(x)=x5/2 on [0,6]

(a) We need to justify the inequalities for h>0

f(4−h)−f(4)h≤f'(4)≤f(4+h)−f(4)h

(b) We need to compute f'(4)

(c) We need to plot y=f(x) and the tangent line at x=4

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Chapter 3.1, Problem 62E

Chapter 3.1, Problem 64E

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Chapter 3 Solutions

Calculus - Standalone book

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We need to plot the graph of f ( x ) = x 5 / 2 on [ 0 , 6 ] (a) We need to justify the inequalities for h > 0 f ( 4 − h ) − f ( 4 ) h ≤ f ' ( 4 ) ≤ f ( 4 + h ) − f ( 4 ) h (b) We need to compute f ' ( 4 ) (c) We need to plot y = f ( x ) and the tangent line at x = 4

Solution Summary: The author explains that the difference quotient computes the slope of a secant line through two points on the graph.

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We need to plot the graph of f(x)=x5/2 on [0,6]

(a) We need to justify the inequalities for h>0

f(4−h)−f(4)h≤f'(4)≤f(4+h)−f(4)h

(b) We need to compute f'(4)

(c) We need to plot y=f(x) and the tangent line at x=4

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Chapter 3 Solutions

We need to plot the graph of f ( x ) = x 5 / 2 on [ 0 , 6 ] (a) We need to justify the inequalities for h &gt; 0 f ( 4 − h ) − f ( 4 ) h ≤ f ' ( 4 ) ≤ f ( 4 + h ) − f ( 4 ) h (b) We need to compute f ' ( 4 ) (c) We need to plot y = f ( x ) and the tangent line at x = 4

Solution Summary: The author explains that the difference quotient computes the slope of a secant line through two points on the graph.

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We need to plot the graph of f(x)=x5/2 on [0,6] (a) We need to justify the inequalities for h>0 f(4−h)−f(4)h≤f'(4)≤f(4+h)−f(4)h (b) We need to compute f'(4) (c) We need to plot y=f(x) and the tangent line at x=4

Want to see the full answer?

Chapter 3 Solutions

We need to plot the graph of f ( x ) = x 5 / 2 on [ 0 , 6 ] (a) We need to justify the inequalities for h > 0 f ( 4 − h ) − f ( 4 ) h ≤ f ' ( 4 ) ≤ f ( 4 + h ) − f ( 4 ) h (b) We need to compute f ' ( 4 ) (c) We need to plot y = f ( x ) and the tangent line at x = 4

We need to plot the graph of f(x)=x5/2 on [0,6]

(a) We need to justify the inequalities for h>0

f(4−h)−f(4)h≤f'(4)≤f(4+h)−f(4)h

(b) We need to compute f'(4)

(c) We need to plot y=f(x) and the tangent line at x=4