   Chapter 2, Problem 46RE

Chapter
Section
Textbook Problem

(a) Find the asymptotes of the graph of f ( x ) = 4 − x 3 + x and use them to sketch the graph.(h) Use your graph from part (a) to sketch the graph of f'.(c) Use the definition of a derivative to find f'(x).(d) Use a graphing device to graph f' and compare with your sketch in part (b).

(a)

To determine

To find: The asymptotes of the graph of f(x)=4x3+x.

Explanation

Calculation:

Recall that the line y=a is horizontal asymptote if limx±f(x)=a.

f(x)=4x3+x

Take limit on the above.

limxf(x)=limx4x3+x=limx4x13x+1=413+1=1

Thus, the horizontal asymptote is y=1.

Recall that the line x=b is vertical asymptote if limxbf(x)=.

f(x)=4x3+x approaches infinity if 3+x approaches which implies x approaches 3

(b)

To determine

To sketch: The graph of f using the graph of part a.

(c)

To determine

To find: The derivative of f using the definition.

(d)

To determine

To sketch: The graph of f and compare with the sketch in part b.

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