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  • Calculus: Early Transcendentals
  • Applications of Differentiation

Calculus: Early Transcendentals

James Stewart

Chapter 4

Applications of Differentiation - all with Video Answers

Educators

+ 11 more educators

Section 6

Graphing with Calculus and Calculators

09:48

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 Barnes
Bobby Barnes
University of North Texas
09:17

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 Barnes
Bobby Barnes
University of North Texas
08:58

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 Barnes
Bobby Barnes
University of North Texas
11:23

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 Barnes
Bobby Barnes
University of North Texas
09:01

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 Barnes
Bobby Barnes
University of North Texas
07:48

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 Barnes
Bobby Barnes
University of North Texas
08:09

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 Barnes
Bobby Barnes
University of North Texas
07:20

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 Barnes
Bobby Barnes
University of North Texas
10:46

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 Barnes
Bobby Barnes
University of North Texas
07:38

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 Barnes
Bobby Barnes
University of North Texas
09:38

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 Barnes
Bobby Barnes
University of North Texas
12:06

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 Barnes
Bobby Barnes
University of North Texas
09:35

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 Barnes
Bobby Barnes
University of North Texas
08:08

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 Barnes
Bobby Barnes
University of North Texas
08:01

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 Barnes
Bobby Barnes
University of North Texas
06:12

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 Barnes
Bobby Barnes
University of North Texas
08:57

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 Barnes
Bobby Barnes
University of North Texas
08:01

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 Barnes
Bobby Barnes
University of North Texas
10:09

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 Barnes
Bobby Barnes
University of North Texas
05:34

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 Barnes
Bobby Barnes
University of North Texas
05:12

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 Barnes
Bobby Barnes
University of North Texas
12:51

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 Barnes
Bobby Barnes
University of North Texas
02:45

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 Barnes
Bobby Barnes
University of North Texas
04:26

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 Barnes
Bobby Barnes
University of North Texas
13:30

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 Barnes
Bobby Barnes
University of North Texas
17:55

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 Barnes
Bobby Barnes
University of North Texas
07:40

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 Barnes
Bobby Barnes
University of North Texas
05:43

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 Barnes
Bobby Barnes
University of North Texas
08:15

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 Barnes
Bobby Barnes
University of North Texas
10:04

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 Barnes
Bobby Barnes
University of North Texas
07:24

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 Barnes
Bobby Barnes
University of North Texas
09:49

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 Barnes
Bobby Barnes
University of North Texas
03:06

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 Conley
Ryan Conley
Numerade Educator
04:26

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 Conley
Ryan Conley
Numerade Educator
03:45

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 Conley
Ryan Conley
Numerade Educator
06:51

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 Conley
Ryan Conley
Numerade Educator
04:18

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 Conley
Ryan Conley
Numerade Educator
03:48

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 Conley
Ryan Conley
Numerade Educator
07:44

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 Barnes
Bobby Barnes
University of North Texas
06:57

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 Barnes
Bobby Barnes
University of North Texas

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