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Chapter 4

Applications of Differentiation

Educators


Problem 1

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = x^5 - 5x^4 - x^3 + 28x^2 - 2x $

Bobby B.
University of North Texas

Problem 2

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = -2x^6 + 5x^5 + 140x^3 - 110x^2 $

Bobby B.
University of North Texas

Problem 3

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = x^6 - 5x^5 + 25x^3 - 6x^2 - 48x $

Bobby B.
University of North Texas

Problem 4

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = \dfrac{x^4 - x^3 - 8}{x^2 - x - 6} $

Bobby B.
University of North Texas

Problem 5

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = \dfrac{x}{x^3 + x^2 + 1} $

Bobby B.
University of North Texas

Problem 6

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = 6\sin x - x^2 $, $ - 5 \leqslant x \leqslant 3 $

Bobby B.
University of North Texas

Problem 7

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = 6\sin x + \cot x $, $ - \pi \leqslant x \leqslant \pi $

Bobby B.
University of North Texas

Problem 8

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = e^x - 0.186x^4 $

Bobby B.
University of North Texas

Problem 9

Produce graphs of $ f $ that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.

$ f(x) = 1 + \dfrac{1}{x} + \dfrac{8}{x^2} + \dfrac{1}{x^3} $

Bobby B.
University of North Texas

Problem 10

Produce graphs of $ f $ that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.

$ f(x) = \dfrac{1}{x^8} - \dfrac{2 X 10^8}{x^4} $

Bobby B.
University of North Texas

Problem 11

(a) Graph the function
(b) Use l'Hospital's Rule to explain the behavior as $ x \to 0 $.
(c) Estimate the minimum value and intervals of concavity.
Then use calculus to find the exact values.

$ f(x) = x^2 \ln x $

Bobby B.
University of North Texas

Problem 12

(a) Graph the function
(b) Use l'Hospital's Rule to explain the behavior as $ x \to 0 $.
(c) Estimate the minimum value and intervals of concavity.
Then use calculus to find the exact values.

$ f(x) = xe^{1/x} $

Bobby B.
University of North Texas

Problem 13

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.

$ f(x) = \dfrac{(x + 4)(x - 3)^2}{x^4(x - 1)} $

Bobby B.
University of North Texas

Problem 14

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.

$ f(x) = \dfrac{(2x + 3)^2(x - 2)^5}{x^3(x - 5)^2} $

Bobby B.
University of North Texas

Problem 15

If $ f $ is the function considered in Example 3, use a computer algebra system to calculate $ f' $ and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate $ f" $ and use it to estimate the intervals of concavity and inflection points.

Bobby B.
University of North Texas

Problem 16

If $ f $ is the function of Exercise 14, find $ f' $ and $ f" $ and use their graphs to estimate the intervals of increase and decrease and concavity of $ f $.

Bobby B.
University of North Texas

Problem 17

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{x^3 + 5x^2 + 1}{x^4 + x^3 - x^2 + 2} $

Bobby B.
University of North Texas

Problem 18

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{x^{2/3}}{1 + x + x^4} $

Bobby B.
University of North Texas

Problem 19

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \sqrt{x + 5\sin x} $, $ x \leqslant 20 $

Bobby B.
University of North Texas

Problem 20

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = x - \tan^{-1} (x^2) $

Bobby B.
University of North Texas

Problem 21

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{1 - e^{1/x}}{1 + e^{1/x}} $

Bobby B.
University of North Texas

Problem 22

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{3}{3 + 2\sin x} $

Bobby B.
University of North Texas

Problem 23

Graph the function using as many viewing rectangles as you need to depict the true nature of the function.

$ f(x) = \dfrac{1 - \cos(x^4)}{x^8} $

Bobby B.
University of North Texas

Problem 24

Graph the function using as many viewing rectangles as you need to depict the true nature of the function.

$ f(x) = e^x + \ln |x - 4| $

Bobby B.
University of North Texas

Problem 25

(a) Graph the function.
(b) Explain the shape of the graph by computing the limit as $ x \to 0^+ $ or $ x \to \infty $.
(c) Estimate the maximum and minimum values and then use calculus to find the exact values.
(d) Use a graph of $ f" $ to estimate the $ x $-coordinates of the inflection points.

$ f(x) = x^{1/x} $

Bobby B.
University of North Texas

Problem 26

(a) Graph the function.
(b) Explain the shape of the graph by computing the limit as $ x \to 0^+ $ or $ x \to \infty $.
(c) Estimate the maximum and minimum values and then use calculus to find the exact values.
(d) Use a graph of $ f" $ to estimate the $ x $-coordinates of the inflection points.

$ f(x) = (\sin x)^{\sin x} $

Bobby B.
University of North Texas

Problem 27

In Example 4 we considered a member of the family of functions $ f(x) = \sin(x + \sin cx) $ that occur in FM synthesis. Here we investigate the function with $ c = 3 $. Start by graphing $ f $ in the viewing rectangle $ [0, \pi] $ by $ [-1.2, 1.2] $. How many local maximum points do you see? The graph has more than are visible to the naked eye. To discover the hidden maximum and minimum points you will need to examine the graph of $ f' $ very carefully. In fact, it helps to look at the graph of $ f" $ at the same time. Find all the maximum and minimum values and inflection points. Then graph $ f $ in the viewing rectangle $ [-2\pi, 2\pi] $ by $ [-1.2, 1.2] $ and comment on symmetry.

Bobby B.
University of North Texas

Problem 28

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = x^2 + cx $

Bobby B.
University of North Texas

Problem 29

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = x^2 + 6x + c/x $ (Trident of Newton)

Bobby B.
University of North Texas

Problem 30

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = x\sqrt{c^2 - x^2} $

Bobby B.
University of North Texas

Problem 31

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = e^x + ce^{-x} $

Bobby B.
University of North Texas

Problem 32

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = \ln(x^2 + c) $

Bobby B.
University of North Texas

Problem 33

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = \dfrac{cx}{1 + c^2x^2} $

Ryan C.
Numerade Educator

Problem 34

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = \dfrac{\sin x}{c + \cos x} $

Ryan C.
Numerade Educator

Problem 35

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = cx + \sin x $

Ryan C.
Numerade Educator

Problem 36

The family of functions $ f(t) = C(e^{-at} - e^{-bt}) $, where $ a $, $ b $, and $ C $ are positive numbers and $ b > a $, has been used to model the concentration of a drug injected into the bloodstream at time $ t = 0 $. Graph several members of this family. What do they have in common? For fixed values of $ C $ and $ a $, discover graphically what happens as $ b $ decreases. Then use calculus to prove what you have discovered.

Ryan C.
Numerade Educator

Problem 37

Investigate the family of curves given by $ f(x) = xe^{-cx} $, where $ c $ is a real number. Start by computing the limits as $ x \to \pm \infty $. Identify any transitional values of $ c $ where the basic shape changes. What happens to the maximum or minimum points and inflection points as $ c $ changes? Illustrate by graphing several members of the family.

Ryan C.
Numerade Educator

Problem 38

Investigate the family of curves given by the equation $ f(x) = x^4 + cx^2 + x $. Start by determining the transitional value of $ c $ at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of $ c $ at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.

Ryan C.
Numerade Educator

Problem 39

(a) Investigate the family of polynomials given by the equation $ f(x) = cx^4 - 2x^2 + 1 $. For what values of $ c $ does the curve have minimum points?
(b) Show that the minimum and maximum points of every curve in the family lie in the parabola $ y = 1 - x^2 $. Illustrate by graphing this parabola and several members of the family.

Bobby B.
University of North Texas

Problem 40

(a) Investigate the family of polynomials given by the equation $ f(x) = 2x^3 + cx^2 + 2x $. For what values of $ c $ does the curve have maximum and minimum points?
(b) Show that the minimum and maximum points of every curve in the family lie on the curve $ y = x - x^3 $. Illustrate by graphing this curve and several members of the family.

Bobby B.
University of North Texas