hyperbolic functions 5.8 hyperbolic functions · finding relative extrema find the relative extrema...

5.8 Hyperbolic Functions 383

5.8 Hyperbolic Functions

Develop properties of hyperbolic functions.Differentiate and integrate hyperbolic functions.Develop properties of inverse hyperbolic functions.Differentiate and integrate functions involving inverse hyperbolic functions.

Hyperbolic FunctionsIn this section, you will look briefly at a special class of exponential functions calledhyperbolic functions. The name hyperbolic function arose from comparison of the areaof a semicircular region, as shown in Figure 5.29, with the area of a region under ahyperbola, as shown in Figure 5.30.

Circle: Hyperbola:Figure 5.29 Figure 5.30

The integral for the semicircular region involves an inverse trigonometric (circular)function:

The integral for the hyperbolic region involves an inverse hyperbolic function:

This is only one of many ways in which the hyperbolic functions are similar to thetrigonometric functions.

�1

�1

�1 � x2 dx �12�x�1 � x2 � sinh�1

x �1

�1� 2.296.

�1

�1

�1 � x2 dx �12�x�1 � x2 � arcsin x �

1

�1�

�

2� 1.571.

�x2 � y2 � 1x2 � y2 � 1

x−1 1

2

y

y = 1 + x2

x−1 1

2

y = 1 − x2

y

Definitions of the Hyperbolic Functions

x � 0coth x �1

tanh x,tanh x �

sinh xcosh x

sech x �1

cosh xcosh x �

ex � e�x

2

x � 0csch x �1

sinh x,sinh x �

ex � e�x

2

FOR FURTHER INFORMATION For more information on the development of hyperbolic functions, see the article “An Introduction to Hyperbolic Functions in ElementaryCalculus” by Jerome Rosenthal in Mathematics Teacher. To view this article, go to MathArticles.com.

REMARK The notation is read as “the hyperbolic

sine of ” cosh as “the hyperbolic cosine of ” and so on.

x,xx,

sinh x

JOHANN HEINRICH LAMBERT(1728–1777)

The first person to publish a comprehensive study on hyperbolicfunctions was Johann HeinrichLambert, a Swiss-German mathe-matician and colleague of Euler.See LarsonCalculus.com to readmore of this biography.

American Institute of Physics (AIP) (use Emilio Serge Visual Archive)

9781285057095_0508.qxp 9/17/12 11:44 AM Page 383

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

The graphs of the six hyperbolic functions and their domains and ranges are shownin Figure 5.31. Note that the graph of can be obtained by adding the corresponding-coordinates of the exponential functions and Likewise, the

graph of can be obtained by adding the corresponding -coordinates of the exponential functions and

Many of the trigonometric identities have corresponding hyperbolic identities. Forinstance,

� 1.

�44

�e2x � 2 � e�2x

4�

e2x � 2 � e�2x

4

cosh2 x � sinh2 x � �ex � e�x

2 �2

� �ex � e�x

2 �2

hx �12e�x.f x �

12ex

ycosh xgx � �

12e�x.fx �

12exy

sinh x

384 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

HYPERBOLIC IDENTITIES

cosh 2x � cosh2 x � sinh2 xsinh 2x � 2 sinh x cosh x

cosh2 x �1 � cosh 2x

2sinh2 x �

�1 � cosh 2x2

coshx � y � cosh x cosh y � sinh x sinh y

coshx � y � cosh x cosh y � sinh x sinh ycoth2 x � csch2 x � 1

sinhx � y � sinh x cosh y � cosh x sinh ytanh2 x � sech2 x � 1

sinhx � y � sinh x cosh y � cosh x sinh ycosh2 x � sinh2 x � 1FOR FURTHER INFORMATIONTo understand geometrically therelationship between the hyperbolicand exponential functions, see thearticle “A Short Proof Linking the Hyperbolic and ExponentialFunctions” by Michael J. Seery in The AMATYC Review.

2

2−1−2

−2

−1

1

1

x

y

y = tanh x

Domain:Range: �1, 1

��, �

f(x) = ex

2h(x) = e−x

2

2

2−2

−2

−1

−1 1x

y y = cosh x

Domain:Range: �1, �

��, �

f(x) = ex

2

g(x) = −e−x

2

2

2

1

−1−2

−2

−1

1x

y

y = sinh x

Domain:Range: ��, �

��, �

2−1−2

−1

1

1

x

y

y = coth x = 1tanh x

Domain:Range: ��, �1 � 1, �

��, 0 � 0, �

2

2−1

−1

−2

−2 1x

y = sech x = 1cosh x

y

Domain:Range: 0, 1�

��, �

y = csch x = 1sinh x2

2

1

−1

−1 1x

y

Domain:Range:Figure 5.31

��, 0 � 0, ��, 0 � 0, �

9781285057095_0508.qxp 9/17/12 11:44 AM Page 384



Differentiation and Integration of Hyperbolic FunctionsBecause the hyperbolic functions are written in terms of and you can easilyderive rules for their derivatives. The next theorem lists these derivatives with the corresponding integration rules.

Proof Here is a proof of two of the differentiation rules. (You are asked to prove someof the other differentiation rules in Exercises 103–105.)

See LarsonCalculus.com for Bruce Edwards’s video of this proof.

Differentiation of Hyperbolic Functions

a.

b.

c.

d.ddx

�x � 1 cosh x � sin x� � x � 1 sinh x � cosh x � cosh x � x � 1 sinh x

ddx

�x sinh x � cosh x� � x cosh x � sinh x � sinh x � x cosh x

ddx

�lncosh x� �sinh xcosh x

� tanh x

ddx

�sinhx2 � 3� � 2x coshx2 � 3

� sech2 x

�1

cosh2 x

�cosh xcosh x � sinh x sinh x

cosh2 x

ddx

�tanh x� �ddx�

sinh xcosh x�

� cosh x

�ex � e�x

2

ddx

�sinh x� �ddx�

ex � e�x

2 �

e�x,ex


THEOREM 5.18 Derivatives and Integrals of Hyperbolic Functions

Let be a differentiable function of

�csch u coth u du � �csch u � Cddx

�csch u� � �csch u coth uu�

�sech u tanh u du � �sech u � Cddx

�sech u� � �sech u tanh uu�

�csch2 u du � �coth u � Cddx

�coth u� � �csch2 uu�

�sech2 u du � tanh u � Cddx

�tanh u� � sech2 uu�

�sinh u du � cosh u � Cddx

�cosh u� � sinh uu�

�cosh u du � sinh u � Cddx

�sinh u� � cosh uu�

x.u

9781285057095_0508.qxp 9/17/12 11:44 AM 85



Finding Relative Extrema

Find the relative extrema of

Solution Using the result of Example 1(d), set the first derivative of equal to 0.

So, the critical numbers are and Using the Second Derivative Test, youcan verify that the point yields a relative maximum and the point yields a relative minimum, as shown in Figure 5.32. Try using a graphing utility to confirm this result. If your graphing utility does not have hyperbolic functions, you canuse exponential functions, as shown.

When a uniform flexible cable, such as a telephone wire, is suspended from twopoints, it takes the shape of a catenary, as discussed in Example 3.

Hanging Power Cables

See LarsonCalculus.com for an interactive version of this type of example.

Power cables are suspended between two towers, forming the catenary shown in Figure 5.33. The equation for this catenary is

The distance between the two towers is Find the slope of the catenary at the pointwhere the cable meets the right-hand tower.

Solution Differentiating produces

At the point the slope (from the left) is

Integrating a Hyperbolic Function

Find

Solution

�sinh3 2x

6� C

�12�

sinh 2x3

3 � � C

u � sinh 2x�cosh 2x sinh2 2x dx �12�sinh 2x22 cosh 2x dx

�cosh 2x sinh2 2x dx.

m � sinh ba

.b, a coshb a,

y� � a�1a� sinh

xa

� sinh xa

.

2b.

y � a cosh xa

.

�12

xex � xe�x � 2ex

�12

xex � xe�x � ex � e�x � ex � e�x

fx � x � 1�12�ex � e�x �

12

ex � e�x

1, �sinh 10, �1x � 0.x � 1

x � 1 sinh x � 0

f

fx � x � 1 cosh x � sinh x.


1

31

−2

−1−2

−3

y

x

(0, −1)

(1, −sinh 1)

f (x) = (x − 1) cosh x − sinh x

so is a relative maximum. so is a relative minimum.Figure 5.32

1, �sinh 1f 1 > 0,0, �1f 0 < 0,

x

y

b−b

a

y = a cosh xa

CatenaryFigure 5.33

FOR FURTHER INFORMATIONIn Example 3, the cable is a catenary between two supports atthe same height. To learn about theshape of a cable hanging betweensupports of different heights,see the article “Reexamining the Catenary” by Paul Cella in The College Mathematics Journal. To view this article, go toMathArticles.com.

9781285057095_0508.qxp 9/17/12 11:44 AM Page 386



Inverse Hyperbolic FunctionsUnlike trigonometric functions, hyperbolic functions are not periodic. In fact, bylooking back at Figure 5.31, you can see that four of the six hyperbolic functions areactually one-to-one (the hyperbolic sine, tangent, cosecant, and cotangent). So, you canapply Theorem 5.7 to conclude that these four functions have inverse functions. Theother two (the hyperbolic cosine and secant) are one-to-one when their domains arerestricted to the positive real numbers, and for this restricted domain they alsohave inverse functions. Because the hyperbolic functions are defined in terms ofexponential functions, it is not surprising to find that the inverse hyperbolic functionscan be written in terms of logarithmic functions, as shown in Theorem 5.19.

Proof The proof of this theorem is a straightforward application of the properties ofthe exponential and logarithmic functions. For example, for

and

you can show that

and

which implies that is the inverse function of See LarsonCalculus.com for Bruce Edwards’s video of this proof.

f.g

g f x � x f gx � x

gx � lnx � �x2 � 1

fx � sinh x �ex � e�x

2


THEOREM 5.19 Inverse Hyperbolic Functions

Function Domain

��, 0 � 0, �csch�1 x � ln�1x

��1 � x2

�x� �0, 1�sech�1 x � ln

1 � �1 � x2

x

��, �1 � 1, �coth�1 x �12

ln x � 1x � 1

�1, 1tanh�1 x �12

ln 1 � x1 � x

�1, �cosh�1 x � lnx � �x2 � 1 ��, �sinh�1 x � lnx � �x2 � 1

TECHNOLOGY You can use a graphing utility to confirm graphically the resultsof Theorem 5.19. For instance, graph the following functions.

Hyperbolic tangent

Definition of hyperbolic tangent

Inverse hyperbolic tangent

Definition of inverse hyperbolic tangent

The resulting display is shown in Figure 5.34. As you watch the graphs being tracedout, notice that and Also notice that the graph of is the reflectionof the graph of in the line y � x.y3

y1y3 � y4.y1 � y2

y4 �12

ln 1 � x1 � x

y3 � tanh�1 x

y2 �ex � e�x

ex � e�x

y1 � tanh x3

−2

−3

2

y1 = y2

y3 = y4

Graphs of the hyperbolic tangent function and the inverse hyperbolictangent functionFigure 5.34

9781285057095_0508.qxp 9/17/12 11:44 AM Page 387



The graphs of the inverse hyperbolic functions are shown in Figure 5.35.

The inverse hyperbolic secant can be used to define a curve called a tractrix or pursuit curve, as discussed in Example 5.

A Tractrix

A person is holding a rope that is tied to a boat, as shown in Figure 5.36. As the person walks along the dock, the boat travels along a tractrix, given by the equation

where is the length of the rope. For feet, find the distance the person mustwalk to bring the boat to a position 5 feet from the dock.

Solution In Figure 5.36, notice that the distance the person has walked is

When this distance is

So, the person must walk about 41.27 feet to bring the boat to a position 5 feet from thedock.

� 20 ln 4 � �15 � 41.27 feet.y1 � 20 sech�1 520

� 20 ln 1 � �1 � 1 42

1 4

x � 5,

� 20 sech�1 x

20.

� �20 sech�1 x

20� �202 � x2� � �202 � x2

y1 � y � �202 � x2

a � 20a

y � a sech�1 xa

� �a2 � x2


x

(0, y1)

(x, y)

10 20

x

20

202

− x2

Pers

on

y

y = 20 sech−1 − 202 − x2x20

A person must walk about 41.27 feet tobring the boat to a position 5 feet fromthe dock.Figure 5.36

1

1

2

2

3

3

−1−2

−2

−3

−3

y = tanh−1 x

x

y


�1, 1

1

1

2

2

3

3

−1−1

−2

−2

−3

−3

y = cosh−1 x

x

y


�1, �

1

1

2

2

3

3

−1−2

−2

−3

−3

y = sinh−1 x

x

y


��, �

1

1

2

2

3

3

−1

−2

−3

y = coth−1 x

x

y

Domain:Range: ��, 0 � 0, �

��, �1 � 1, �

1

1

2

2

3

3

−1−2 −1

−2

−3

−3

y = sech−1 x

x

y


0, 1�

1

1

2

2

3

3

−1

−3

y = csch−1 x

x

y

Domain:Range:Figure 5.35

��, 0 � 0, ��, 0 � 0, �

9781285057095_0508.qxp 9/17/12 11:44 AM Page 388



Inverse Hyperbolic Functions: Differentiation andIntegrationThe derivatives of the inverse hyperbolic functions, which resemble the derivatives of theinverse trigonometric functions, are listed in Theorem 5.20 with the corresponding inte-gration formulas (in logarithmic form). You can verify each of these formulas by applyingthe logarithmic definitions of the inverse hyperbolic functions. (See Exercises 106–108.)

Differentiation of Inverse Hyperbolic Functions

a.

b.

Integration Using Inverse Hyperbolic Functions

a.

b.

� 1

4�5 ln��5 � 2x

�5 � 2x� � C

12a

ln�a � ua � u� � C �

12� 1

2�5 ln��5 � 2x

�5 � 2x�� C

� dua2 � u2 � dx

5 � 4x2 �12� 2 dx

�5 2 � 2x2

�1a

ln a � �a2 � u2

�u� � C � �12

ln 2 � �4 � 9x2

�3x� � C

� du

u�a2 � u2� dx

x�4 � 9x2� � 3 dx

3x�4 � 9x2

�3x2

1 � x6

ddx

�tanh�1x3� �3x2

1 � x32

�2

�4x2 � 1

ddx

�sinh�12x� �2

�2x2 � 1


THEOREM 5.20 Differentiation and Integration Involving

Inverse Hyperbolic Functions

Let be a differentiable function of

� du

u�a2 ± u2� �

1a

ln a � �a2 ± u2

�u� � C

� dua2 � u2 �

12a

ln�a � ua � u� � C

� du�u2 ± a2

� lnu � �u2 ± a2 � C

ddx

�csch�1 u� ��u�

�u��1 � u2

ddx

�sech�1 u� ��u�

u�1 � u2

ddx

�coth�1 u� �u�

1 � u2

ddx

�tanh�1 u� �u�

1 � u2

ddx

�cosh�1 u� �u�

�u2 � 1

ddx

�sinh�1 u� �u�

�u2 � 1

x.u

REMARK Let andu � 3x.

a � 2

REMARK Let and u � 2x.

a � �5

9781285057095_0508.qxp 9/17/12 11:45 AM Page 389




Evaluating a Function In Exercises 1–6, evaluate the function. If the value is not a rational number, round youranswer to three decimal places.

1. (a) 2. (a)

(b) (b)

3. (a) 4. (a)

(b) (b)

5. (a) 6. (a)

(b) (b)

Verifying an Identity In Exercises 7–14, verify the identity.

7.

8.

9.

10.

11.

12.

13.

14.

Finding Values of Hyperbolic Functions In Exercises 15and 16, use the value of the given hyperbolic function to findthe values of the other hyperbolic functions at

15. 16.

Finding a Limit In Exercises 17–22, find the limit.

17. 18.

19. 20.

21. 22.

Finding a Derivative In Exercises 23–32, find the derivativeof the function.

23. 24.

25. 26.

27. 28.

29.

30.

31.

32.

Finding an Equation of a Tangent Line In Exercises33–36, find an equation of the tangent line to the graph of thefunction at the given point.

33.

34.

35.

36.

Finding Relative Extrema In Exercises 37–40, find any relative extrema of the function. Use a graphing utility to confirm your result.

37.

38.

39.

40.

Catenary In Exercises 41 and 42, a model for a power cablesuspended between two towers is given. (a) Graph the model,(b) find the heights of the cable at the towers and at themidpoint between the towers, and (c) find the slope of themodel at the point where the cable meets the right-hand tower.

41.

42.

Finding an Indefinite Integral In Exercises 43–54, findthe indefinite integral.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

Evaluating a Definite Integral In Exercises 55–60,evaluate the integral.

55. 56.

57. 58.

59. 60. �ln 2

0 2e�x cosh x dx��2 4

0

2�1 � 4x2

dx

�4

0

1�25 � x2

dx�4

0

125 � x2 dx

�1

0 cosh2 x dx�ln 2

0 tanh x dx

� cosh x�9 � sinh2 x

dx�csch1 x coth1 xx2 dx

�sech3 x tanh x dx�x csch2 x2

2 dx

�sech22x � 1 dx�cosh xsinh x

dx

� sinh x1 � sinh2 x

dx�cosh2x � 1 sinhx � 1 dx

�cosh �x�x

dx�sinh1 � 2x dx

�sech23x dx� cosh 2x dx

y � 18 � 25 cosh x

25, �25 x 25

y � 10 � 15 cosh x

15, �15 x 15

hx � 2 tanh x � x

gx � x sech x

f x � x sinhx � 1 � coshx � 1�4 x 4f x � sin x sinh x � cos x cosh x,

0, 1)y � esinh x,

0, 1)y � cosh x � sinh x2,

1, 1)y � xcosh x,

1, 0)y � sinh1 � x2,

gx � sech2 3x

f t � arctansinh ty � x cosh x � sinh x

hx �14

sinh 2x �x2

y � ln�tanh x2�f x � lnsinh x

f x � tanh4x2 � 3xy � sech5x2f x � cosh8x � 1f x � sinh 3x

limx→0�

coth xlimx→0

sinh x

x

limx→��

csch xlimx→�

sech x

limx→��

tanh xlimx→�

sinh x

tanh x �12

sinh x �32

x.

cosh x � cosh y � 2 cosh x � y

2 cosh

x � y2

sinh x � y � sinh x cosh y � cosh x sinh y

e2x � sinh 2x � cosh 2x

sinh 2x � 2 sinh x cosh x

sinh2 x ��1 � cosh 2x

2

cosh2 x �1 � cosh 2x

2

coth2 x � csch2 x � 1

tanh2 x � sech2 x � 1

coth�1 3sech�1 23

csch�1 2cosh�1 2

tanh�1 0cothln 5sinh�1 0cschln 2sech 1tanh�2cosh 0sinh 3

5.8 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

9781285057095_0508.qxp 9/17/12 11:45 AM Page 390





65.

66.

67. 68.

69. 70.

71.

72.

73.

74.

Finding an Indefinite Integral In Exercises 75–82, findthe indefinite integral using the formulas from Theorem 5.20.

75. 76.

77. 78.

79. 80.

81. 82.

Evaluating a Definite Integral In Exercises 83–86,evaluate the definite integral using the formulas from Theorem 5.20.

83. 84.

85. 86.

Differential Equation In Exercises 87–90, solve the differential equation.

87.

88.

89.

90.

Area In Exercises 91–94, find the area of the region.

91. 92.

93. 94.

95. Chemical Reactions Chemicals A and B combine in a 3-to-1 ratio to form a compound. The amount of compound being produced at any time is proportional to the unchangedamounts of A and B remaining in the solution. So, when 3 kilograms of A is mixed with 2 kilograms of B, you have

One kilogram of the compound is formed after 10 minutes.Find the amount formed after 20 minutes by solving the equation

�3k16

dt � � dxx2 � 12x � 32

.

dxdt

� k�3 �3x4 ��2 �

x4� �

3k16

x2 � 12x � 32.

tx

x

y

−2−4 2 4−2

2

6

8

4x

y

−1−2−3−4 1 2 3 4

−4

1234

y �6

�x2 � 4y �

5x�x 4 � 1

−2 −1−3 1 2 3

−2

−3

2

1

3

x

y

−1−2−3−4 1 2 3 4

0.20.40.6

1.21.4

x

y

y � tanh 2xy � sech x2

dydx

�1 � 2x4x � x2

dydx

�x3 � 21x

5 � 4x � x2

dydx

�1

x � 1��4x2 � 8x � 1

dydx

�1

�80 � 8x � 16x2

�1

0

1�25x2 � 1

dx�1

�1

116 � 9x2 dx

�3

1

1

x�4 � x2 dx�7

3

1�x2 � 4

dx

� dxx � 2�x2 � 4x � 8

� �14x � x2 dx

� �x�1 � x3

dx� 1�x�1 � x

dx

� x9 � x 4 dx� 1

�1 � e2x dx

� 1

2x�1 � 4x2 dx� 1

3 � 9x2 dx

y � x tanh�1 x � ln�1 � x2

y � 2x sinh�12x � �1 � 4x2

0 < x < � 4y � sech�1cos 2x,y � csch�1 x2

y � tanh�1sin 2xy � sinh�1tan xf x � coth�1x2y � tanh�1�x

y � tanh�1 x2

y � cosh�13x

WRITING ABOUT CONCEPTS61. Comparing Functions Discuss several ways in which

the hyperbolic functions are similar to the trigonometricfunctions.

62. Hyperbolic Functions Which hyperbolic functionstake on only positive values? Which hyperbolic functionsare increasing on their domains?

63. Comparing Derivative Formulas Which hyperbolic derivative formulas differ from their trigonometric counterparts by a minus sign?

64. HOW DO YOU SEE IT? Use the graphs of and shown in the figures to answer the following.

(a) Identify the open interval(s) on which the graphs of and are increasing or decreasing.

(b) Identify the open interval(s) on which the graphs of and are concave upward or concave downward.g

f

gf

x

y

−1−2 1 2

−2

−1

2

1

g(x) = tanh x

x

y

−1−2 1 2−1

2

3

f (x) = cosh x

gf

9781285057095_0508.qxp 9/17/12 11:45 AM Page 391




96. Vertical Motion An object is dropped from a height of400 feet.

(a) Find the velocity of the object as a function of time (neglect air resistance on the object).

(b) Use the result in part (a) to find the position function.

(c) If the air resistance is proportional to the square of thevelocity, then where feet persecond per second is the acceleration due to gravity and is a constant. Show that the velocity as a function of time is by performing

and simplifying the result.

(d) Use the result of part (c) to find and give its interpretation.

(e) Integrate the velocity function in part (c) and find the position of the object as a function of Use a graphingutility to graph the position function when andthe position function in part (b) in the same viewing window. Estimate the additional time required for theobject to reach ground level when air resistance is notneglected.

(f) Give a written description of what you believe wouldhappen if were increased. Then test your assertion witha particular value of

97. Tractrix Consider the equation of the tractrix

,

(a) Find

(b) Let be the tangent line to the tractrix at the point When intersects the axis at the point show that thedistance between and is

98. Tractrix Show that the boat in Example 5 is always pointing toward the person.

99. Proof Prove that

100. Proof Prove that

101. Using a Right Triangle Show that

102. Integration Let and Show that

Proof In Exercises 103–105, prove the differentiation formula.

103.

104.

105.

Verifying a Differentiation Rule In Exercises 106–108,verify the differentiation formula.

106.

107.

108.ddx

�sech�1 x� ��1

x�1 � x2

ddx

�sinh�1 x� �1

�x2 � 1

ddx

�cosh�1 x� �1

�x2 � 1

ddx

�sech x� � �sech x tanh x

ddx

�coth x� � �csch2 x

ddx

�cosh x� � sinh x

�b

�b

ext dt �2 sinh bx

x.

b > 0.x > 0

arctansinh x � arcsintanh x.

sinh�1 t � lnt � �t2 � 1 .

�1 < x < 1.tanh�1 x �12

ln�1 � x1 � x�,

a.QPQ,y-L

P.L

dy dx.

a > 0.y � a sech�1�x/a� � �a2 � x2

k.k

k � 0.01t.s

limt→�

vt� dv 32 � kv2 � �� dt

vt � ��32 k tanh�32k tv

k�32dv dt � �32 � kv2,

PUTNAM EXAM CHALLENGE109. From the vertex of the catenary a

line is drawn perpendicular to the tangent to the catenaryat point Prove that the length of intercepted by theaxes is equal to the ordinate of the point

110. Prove or disprove: there is at least one straight line normal to the graph of at a point andalso normal to the graph of at a point

At a point on a graph, the normal line is the perpendicularto the tangent at that point. Also,and

These problems were composed by the Committee on the Putnam Prize Competition.© The Mathematical Association of America. All rights reserved.

sinh x � ex � e�x 2.�cosh x � ex � e�x 2

�c, sinh c.y � sinh x

a, cosh ay � cosh x

P.yLP.

Ly � c coshx c0, c

The Gateway Arch in St. Louis, Missouri, was constructed usingthe hyperbolic cosine function. The equation used for constructionwas

where and are measured in feet. Cross sections of the arch areequilateral triangles, and traces the path of the centers of massof the cross-sectional triangles. For each value of the area of thecross-sectional triangle is

(Source: Owner’s Manual for the Gateway Arch, Saint Louis, MO,by William Thayer)

(a) How high above the ground is thecenter of the highesttriangle? (At groundlevel, )

(b) What is the height of the arch? (Hint:For an equilateral triangle,where is one-half the base of the triangle, and the center ofmass of the triangle is located at two-thirds the height of the triangle.)

(c) How wide is the arch at ground level?

cA � �3c2,

y � 0.

A � 125.1406 cosh 0.0100333x.

x,x, y

yx

�299.2239 x 299.2239

y � 693.8597 � 68.7672 cosh 0.0100333x,

St. Louis Arch

Ken Nyborg/Shutterstock.com

9781285057095_0508.qxp 9/17/12 11:45 AM Page 392



Review Exercises 393

Review Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Sketching a Graph In Exercises 1 and 2, sketch the graphof the function and state its domain.

1.

2.

Expanding a Logarithmic Expression In Exercises 3 and4, use the properties of logarithms to expand the logarithmicexpression.

3.

4.

Condensing a Logarithmic Expression In Exercises 5and 6, write the expression as the logarithm of a single quantity.

5.

6.


7.

8.

9.

10.

11.

12.

Finding an Equation of a Tangent Line In Exercises 13and 14, find an equation of the tangent line to the graph of thefunction at the given point.

13.

14.


15. 16.

17. 18.

Evaluating a Definite Integral In Exercises 19–22,evaluate the definite integral.

19. 20.

21. 22.

Finding an Inverse Function In Exercises 23–28, (a) findthe inverse function of (b) graph and on the same set of coordinate axes, (c) verify that and

and (d) state the domains and ranges of and

23. 24.

25. 26.

27. 28.

Evaluating the Derivative of an Inverse Function InExercises 29–32, verify that has an inverse. Then use the function and the given real number to find (Hint:Use Theorem 5.9.)

29.

30.

31.

32.

Solving an Exponential or Logarithmic Equation InExercises 33–36, solve for x accurate to three decimal places.

33.

34.

35.

36.


37. 38.

39. 40.

41. 42.

Finding an Equation of a Tangent Line In Exercises 43and 44, find an equation of the tangent line to the graph of thefunction at the given point.

43. 44.

Implicit Differentiation In Exercises 45 and 46, use implicitdifferentiation to find

45. 46.


47. 48.

49. 50. � e2x � e�2x

e2x � e�2x dx� e4x � e2x � 1

ex dx

� x2e x 3�1 dx�xe1�x2 dx

cos x2 � xeyy ln x � y 2 � 0

dy/dx.

�4, 1�f �x� � ex�4,�0, 1�f �x� � e6x,

y � 3e�3�tg�x� �x2

ex

h�z� � e�z2�2y � �e2x � e�2x

g�x� � ln ex

1 � exg�t� � t2et

ln x � ln�x � 3� � 0

ln �x � 1 � 2

�4 � 3e�2x � 6

e3x � 30

a � 00 � x � �,f �x� � cos x,

a ��33

��

4� x �

�

4,f �x� � tan x,

a � 4f �x� � x�x � 3,

a � �1f �x� � x3 � 2,

� f �1�� a�.aff

x ≥ 0f �x� � x2 � 5,f �x� � 3�x � 1

f �x� � x3 � 2f �x� � �x � 1

f �x� � 5x � 7f �x� �12x � 3

f �1.ff � f �1�x�� x,

f �1� f �x�� xf �1ff,

��

0tan

�

3 d��3

0sec � d�

�e

1

ln xx

dx�4

1 2x � 1

2x dx

� ln �x

x dx� sin x

1 � cos x dx

� x2

x3 � 1 dx� 1

7x � 2 dx

�1, 2�y � 2x2 � ln x2,

��1, 2�y � ln�2 � x� �2

2 � x,

y � ln 4xx � 6

y � ln�x2 � 4x2 � 4

f �x� � �ln�2x��3

f �x� � x�ln x

f �x� � ln�3x2 � 2x�g�x� � ln �2x

3�ln x � 2 ln�x2 � 1�� 2 ln 5

ln 3 �13 ln�4 � x2� � ln x

ln��x2 � 1��x � 1��

ln 5�4x2 � 14x2 � 1

f �x� � ln�x � 3�f �x� � ln x � 3

9781285057095_050R.qxp 9/17/12 11:45 AM Page 393




Evaluating a Definite Integral In Exercises 51–54,evaluate the definite integral.

51. 52.

53. 54.

55. Area Find the area of the region bounded by the graphs of

and

56. Depreciation The value of an item years after it is purchased is for

(a) Use a graphing utility to graph the function.

(b) Find the rates of change of with respect to when and

(c) Use a graphing utility to graph the tangent lines to thefunction when and

Sketching a Graph In Exercises 57 and 58, sketch thegraph of the function by hand.

57. 58.


59. 60.

61. 62.

63. 64.

Finding an Indefinite Integral In Exercises 65 and 66,find the indefinite integral.

65. 66.

67. Climb Rate The time (in minutes) for a small plane toclimb to an altitude of feet is

where 18,000 feet is the plane’s absolute ceiling.

(a) Determine the domain of the function appropriate for thecontext of the problem.

(b) Use a graphing utility to graph the time function andidentify any asymptotes.

(c) Find the time when the altitude is increasing at the greatest rate.

68. Compound Interest

(a) How large a deposit, at 5% interest compounded continuously, must be made to obtain a balance of $10,000in 15 years?

(b) A deposit earns interest at a rate of percent compoundedcontinuously and doubles in value in 10 years. Find

Evaluating an Expression In Exercises 69 and 70, evaluateeach expression without using a calculator. (Hint: Make asketch of a right triangle.)

69. (a) 70. (a)

(b) (b)


71.

72.

73.

74.

75.

76.


77. 78.

79. 80.

81. 82.

Area In Exercises 83 and 84, find the area of the region.

83. 84.


85. 86.

87. 88.

89. 90.


91. 92.

93. 94.

95. 96. � x

�x4 � 1 dx� 1

9 � 4x2 dx

�csch4�3x�coth�3x� dx�sech2 xtanh x

dx

�sinh 6x dx� x2 sech2 x3 dx

y � x tanh�1 2xy � sinh�1�4x�y � ln�cosh x�y � coth�8x2�y � 2x � cosh �xy � sech�4x � 1�

x

y

−1 1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

x

y

−1−2 1 2

1

2

3

4

y �6

16 � x2y �4 � x

�4 � x2

� arcsin 2x�1 � 4x2

dx� arctan�x�2�

4 � x2 dx

� 1

x�9x2 � 49 dx�

x�1 � x4

dx

� 1

3 � 25x2 dx� 1

e2x � e�2x dx

2 < x < 4y � �x2 � 4 � 2 arcsec x2

,

y � x�arcsin x�2 � 2x � 2�1 � x2 arcsin x

y �12 arctan e2x

y � x arcsec x

y � arctan�2x2 � 3�y � tan�arcsin x�

cos�arcsec �5 �cos�arcsin 12�tan�arccot 2�sin�arcsin 12�

r.r

t � 50 log10 18,000

18,000 � h

ht

�2�1�t

t2 dt��x � 1�5�x�1�2 dx

h�x� � log5 x

x � 1g�x� � log3 �1 � x

f �x� � x�4�3x�y � x2x�1

f �x� � 53xf �x� � 3x�1

y � 14

x

y � 3 x�2

t � 4.t � 1

t � 4.t � 1tV

0 � t � 5.V � 9000e�0.6t

tV

x � 2.x � 0,y � 0,y � 2e�x,

�2

0

e2x

e2x � 1 dx�3

1

ex

e x � 1 dx

�2

1�2 e1�x

x2 dx�1

0 xe�3x 2

dx

9781285057095_050R.qxp 9/17/12 11:45 AM Page 394



hyperbolic functions 5.8 hyperbolic functions · finding relative extrema find the relative extrema...

Documents