Chapter 6: Exponential, Logarithmic, and Inverse Trigonometric Functions Summary: The main focus of this chapter is to introduce both logarithmic and exponential functions. In the process, the derivative and integration rules for both of these will be developed (making some use of implicit differentiation along the way). Then these derivative and integration rules will be used in the applications that were discussed in both Chapters 3 and 5. Later in the chapter, L’Hˆopital’s rule is introduced and discussed for evaluating limits that have an appropriate indeterminate form. The latter parts of the chapter look into describing the natural logarithmic function as a function that is defined in terms of the Fundamental Theorem of Calculus. Also, implicit differentiation is used again to discuss the derivatives and integrals that are associated with the inverse trigonometric functions. The chapter concludes by defining the hyperbolic trigonometric functions and discussing their properties and uses.

OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Define logarithmic and exponential functions (§6.1). 2. Take derivatives of functions involving logarithms (§6.2). 3. Use logarithmic differentiation to find derivatives of complicated functions (§6.2). 4. Evaluate integrals that involve logarithms (§6.2). 5. Relate the derivative of a function with the derivative of its inverse function (§6.3). 6. Take derivatives of exponential functions (§6.3). 7. Evaluate integrals that involve exponential functions (§6.3). 8. Solve various application problems that involve either logarithmic or exponential functions (§6.4). 119

120 9. Use L’Hˆopital’s rule to evaluate limits that have indeterminate forms such 0 ∞ as , , 0 · ∞, ∞ − ∞, 1∞ , 00 and ∞0 (§6.5). 0 ∞ 10. Define the logarithmic function using the Fundamental Theorem of Calculus (§6.6). 11. Define the inverse trigonometric functions (§6.7). 12. Differentiate and integrate inverse trigonometric functions (§6.7). 13. Learn the definitions and properties of the hyperbolic functions (§6.8).

6.1 Exponential and Logarithmic Functions PURPOSE: To introduce the functions ax and loga x. exponential: y = ax logarithmic: y = loga (x) Note: ax = b ⇔ loga (b) = x

Exponential and Logarithmic functions are introduced in this section. These are two types of functions that arise naturally as inverses of each other. One key idea to remember well is that a logarithm is simply a way of representing an exponent. For example, if y = ax then x = loga y is a way of expressing the exponent x in the first expression. Also because logarithms and exponential functions are inverses of each other, the cancelation equations for inverse functions result in these functions undoing each other. For example, aloga (x) = x and loga (ax ) = x There are some common properties of logarithms and exponentials that are easy to remember when paired together. To remember the properties of logarithms, it may be helpful to remember how exponents behave and then to think of logarithms as representing exponents. logb (x1 ) + logb (x2 ) = logb (x1 x2 ) logb (x1 ) − logb (x2 ) = logb (x1 /x2 ) logb b = 1 logb 1 = 0

⇔ ⇔ ⇔ ⇔

by1 +y2 = by1 by2 by1 −y2 = by1 /by2 b1 = b b0 = 1

IDEA: Any logarithmic function may be written in terms of the natural logarithm. Any exponential function may be written in terms of the natural exponential function.

The natural logarithm and the natural exponential functions are both important since they can be used to express other logarithms and other exponential functions. For example, logb (x) =

ln x ln b

ax = eln a


x

= ex ln a

These relationships will be useful in the next two sections where the derivatives and integrals of logarithmic and exponential functions are discussed.

121 Checklist of Key Ideas:  integer, rational and irrational powers  exponential functions  representing exponents with logarithms  logarithm with base b  bx and logb x are inverse functions  the number e and the natural exponential and logarithmic functions  change of base formulas for logarithms  exponential and logarithmic growth

6.2 Derivatives and Integrals Involving Logarithmic Functions PURPOSE: To give differentiate and integrate logarithmic functions and to discuss the process of logarithmic differentiation. This section develops the derivatives of logarithmic functions by using the limit description of e (see margin), the properties of logarithms and the definition of the derivative. The derivatives of logarithms give rise to a process called logarithmic differentiation. The process can be summarized by taking a logarithm of a given expression and then taking derivatives of the resulting expression. IDEA: The log property of log (ab) = log a + log b can remove the need for the product and/or quotient rules.

The idea behind logarithmic differentiation is to take advantage of the property of logarithms to turn multiplication inside a logarithm into a sum of terms. This also simplifies the process of taking derivatives of functions with powers and extends the power rule to cases with arbitrary real exponents. IDEA: Use the relationship logb x = ln x/ ln b to remember the derivative and integration rules for logb x.

Checklist of Key Ideas:  derivatives of logarithmic functions  using properties of logarithmic functions before differentiation  logarithmic differentiation  extended power rule for general powers  integration rules for logarithmic functions

lim (1 + h)1/h = e

h→0

1. take logarithms of the expression 2. differentiate 3. solve for dy/dx logarithm property ln (ab) = ln a + ln b

122

6.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions PURPOSE: To relate the derivative of a function with the derivative of its inverse and to give rules for differentiating and integrating exponential functions.

differentiability of f −1

In this section, implicit differentiation and the properties of inverse functions are both used to find the derivatives of exponential functions. The first observation is that if a function is differentiable at a point (a, b) and its slope is not zero then the inverse of the function should be differentiable at the point (b, a). In particular, the slopes of the function and its inverse at these points should be the reciprocals of each other. IDEA: If f (a) = b then f goes through the point (a, b) and f −1 goes through the point (b, a). The slope of f at (a, b) is the reciprocal of the slope of f −1 at the point (b, a).

increasing and decreasing

derivatives of ax and ex

This result can be seen as a geometric description of inverses since f and f −1 are reflections of each other across the line y = x. The concepts of increasing and decreasing are briefly mentioned in order to discuss the invertibility of a given function (see Chapter 4 for more on increasing and decreasing). This is important because when a function is strictly decreasing or increasing then it will be one-toone (or passes the horizontal line test) and so will have an inverse function. This is the case with both ex and ln x which are both strictly increasing functions. Using implicit differentiation and the relationship between inverses, the derivatives of exponential functions may now be found. For example, the function y = ex may also be written as x = ln y. Then dy/dx may be found using implicit differentiation together with the derivatives of logarithms from the previous section. IDEA: Use the relationship ax = ex ln a to remember the derivative and integration rules for ax .

Checklist of Key Ideas:  differentiability of inverse functions  finding derivatives of inverse functions  increasing and decreasing functions and f ′ (x)  derivatives of exponential functions  derivatives of functions of the form y = f (x)g(x)  integration rules of exponential functions

123

6.4 Graphs and Applications Involving Logarithmic and Exponential Functions PURPOSE: To examine graphs of logarithmic and exponential functions and to show some applications of logarithmic and exponential functions. The main purpose of this section is to use the derivative and integral rules that have now been developed for logarithmic and exponential functions in various applications. There are many applications that have been discussed so far (see Chapters 3 and 5). These applications may be separated into two broad groups: those that involve differentiation and those that involve integration. Applications involving differentiation are discussed in more detail in Chapter 3. These applications are now applied to logarithmic and exponential functions. For example, sketching exponential and logarithmic functions may be done in the usual fashion by taking the first and second derivatives and looking for relevant information such as intervals of increase and decrease, concavity, relative extrema and points of inflection. On a closed interval, if the given function is continuous, then an absolute maximum or minimum may be found by the usual methods as well.

applications involving differentiation → Chapter 3

Applications involving integration are discussed in Chapter 5. These applications include finding the area under a curve, finding the volume that results when a given curve is revolved around an axis, finding the length of a given curve between two points, finding the surface area of a solid of revolution, finding the average value of a function and others. All of these may now be discussed using the integration and derivative rules that have been developed for both logarithmic and exponential functions.

applications involving integration → Chapter 5

Checklist of Key Ideas:  graphing techniques for functions  relative and absolute extrema  critical points and inflection points  intervals of increase and decrease  concavity of a function  area between two curves  calculating volume of a solid of revolution by disks  calculating volume of a solid of revolution by shells  arc length of a curve defined either explicitly or parametrically  surface area of solid of revolution

124  rectilinear motion  average value of a function on an interval  work of a variable force

6.5 L’Hˆopital’s Rule; Indeterminate Forms PURPOSE: To introduce a method for finding limiting values when a limit evaluates as an indeterminate form.

Note: Multiple applications of L’Hˆopital’s rule may be needed

L’Hˆopital’s rule cannot be applied directly to indetermine products, quotients or powers

0∞ not indeterminate

The purpose of this section is to develop a method for finding limiting values when a limit evaluates as an indeterminate form. The method in this section is called L’Hˆopital’s rule. The primary usefulness of L’Hˆopital’s rule is when a limit has the indeterminate form of 0/0 or ∞/∞. In either of these cases, the rule says that f (x) , will be equivalent to the limit the limit of the ratio of the functions, i.e., lim x→a g(x) f ′ (x) of the ratio of their derivatives, i.e., lim ′ . x→a g (x) The power of L’Hˆopital’s rule is often abused. If a limit does not have one of the indeterminate forms 0/0 or ∞/∞ then the rule cannot be used. The consequence of this is that the rule cannot be directly applied to cases where a limit has a indeterminate form such as 0 · ∞, ∞ − ∞, 00 , ∞0 or 1∞ . In these situations, the limit expressions need to be rewritten before applying the rule. This can often be done by algebraic techniques (using a common denominator, rationalizing an expression with a radical or writing an expression as the reciprocal of its reciprocal, i.e., f (x) = 1/ 1f (x) ) or by using the properties of logarithms (ln ab = b ln a). It should be noted that 0∞ is not an indeterminate form, nor is a0 where |a| < ∞ and a 6= 0. Checklist of Key Ideas:  indeterminate forms of type 0/0 and ∞/∞  using L’Hˆopital’s rule  growth of exponential functions versus polynomials and rational functions  indeterminate forms of types 0 · ∞ and ∞ − ∞ and L’Hˆopital’s rule  indeterminate forms of types 00 , ∞0 and 1∞ and L’Hˆopital’s rule

125

6.6 Logarithmic Functions from the Integral Point of View PURPOSE: To define logarithmic functions using the FTC, part II. There is only one main idea introduced in this section. Here, the natural logarithmic function is defined as an integral. ln x =

Z x 1 1

t

recall: FTC, part II (see §4.6)

dt

The rest of the section is spent showing the usual properties of the natural logarithmic function and the natural exponential function in terms of integrals. All of the usual results can be found. For example, consider the following interesting result that comes from the basic properties of logarithms. The logarithm relationship ln (ab) = ln (a) + ln (b) written in terms of integrals means Z ab 1 1

t

dt =

Z a 1

t

1

dt +

Z b 1 1

t

dt.

This is not something that is obvious at first glance. Some other functions are introduced in this section that are defined in terms of integrals. An important function that is defined in this fashion is the error function. 2 erf(x) = √ π

Z x 0

2

e−t dt

One reason that this function is defined in terms of an integral is that it does not have a simple expression for its antiderivative. Checklist of Key Ideas:

 integral definition of ln x  properties of ln x  defining the natural exponential function  changing base of logarithms  some limits involving e  functions defined by integrals; the error function  functions as limits of integration; the chain rule

error function

126

6.7 Derivatives and Integrals Involving Inverse Trigonometric Functions PURPOSE: To define the inverse trigonometric functions and to give rules for their derivatives and integrals. sin−1 (x) sec−1 (x) tan−1 (x)

In this section, the inverse trigonometric functions are introduced. The three most important inverse trigonometric functions are sin−1 (x), tan−1 (x), and sec−1 (x). It is important to realize that these are inverse functions of the respective trigonometric functions and not reciprocal functions. Because they are inverse functions, the domain of sin x, tan x, and sec x have to be appropriately defined so that these inverse functions are well defined.

y = sin−1 (x) means “y is the angle whose sine is x”

The derivatives and integrals of the inverse trigonometric functions are defined using the inverse relationship and implicit differentiation together with some properties of right triangles and their angles. For example, y = sin−1 (x) which should be read “y is the angle whose sine is x.” In other words, x = sin (y), meaning that dy/dx may be found using implicit differentiation. The three inverse functions, sin−1 (x), sec−1 (x), and tan−1 (x), in√ √ trigonometric 2 2 volve the terms 1 − x , x − 1, and x2 + 1, respectively. It is important to know which one of these goes with which function. When first becoming familiar with these functions, it can be helpful to have this information written on a note card or in a notebook as the material is studied. Checklist of Key Ideas:  definitions of inverse trigonometric functions  domain restrictions of inverse trigonometric functions  derivative rules for inverse trigonometric functions  integration rules for inverse trigonometric functions

6.8 Hyperbolic Functions and Hanging Cables PURPOSE: To define the various hyperbolic trigonometric functions and to discuss their properties. sinh u =

eu − e−u 2

cosh u =

eu + e−u 2

tanh u =

sinh u cosh u

Hyperbolic functions are introduced and defined in this section. These functions are not really an application of integrals but rather are a new group of functions which are defined as combinations of exponential functions. Everything that is shown in this section can be found by remembering the definitions of each funcex − e−x tion. For example, since sinh x = then its derivatives and antiderivatives 2 can be found by simply knowing this information for ex and e−x . Once this type

127 of information is known for sinh x and cosh x, then similar results can be obtained sinh x for tanh x = . cosh x Other important features can be remembered by thinking of sinh x and cosh x in pieces: sinh x

=

cosh x

=

ex e−x − 2 2 ex e−x + 2 2

Recall that ex → 0 as x → −∞ and e−x → 0 as x → ∞. Then sinh x and cosh x become asymptotically close to ex /2 as x → ∞. On the other hand, sinh x becomes asymptotically close to −e−x /2 as x → −∞ and cosh x becomes asymptotically close to e−x /2 as x → −∞. Similar properties of tanh x and the other hyperbolic functions are found by either defining them in terms of ex and e−x or in terms of sinh x and cosh x. IDEA: All the definitions can be written in terms of sinh x and cosh x.

For example,

tanh x =

sinh x cosh x

sechx =

1 cosh x

coth x =

cosh x sinh x

cschx =

1 sinh x

IDEA: Make notecards with the various definitions and properties of the hyperbolic functions to help remember them.

If the reader is not familiar with the material in this section, then it is suggested that the reader make a few notecards with any new information and have them handy as they attempt the exercises at the end of the section. Checklist of Key Ideas:  ex as a sum of even and odd functions  hyperbolic functions  curvilinear asymptotes  catenary; hanging cables  hyperbolic identities  hyperbolic inverses and derivatives  logarithmic forms of inverse hyperbolic functions

128

Chapter 6 Sample Tests Section 6.1

(b) log 22 log 24 (c) 5 (d) log 44

1. 2−5 = (a)

1 10

(b) − (c)

9. Solve log10 (x + 5) = 1 for x.

1 10

(a) 5

1 32

(d) −

1 32

2. Use a calculating utility to approximate decimal places.

(b) −5 (c) 0

(d) no solution √ 6 29. Round to four

10. If log10 x5/2 − log10 x3/2 = 4 then find x. (a) 4

(a) 1.7528

(b) 40

(b) 5.3852

(c) 1, 000

(c) 1.7530

(d) 10, 000

(d) 5.3854

11. Solve 4−2x = 6 for x to four decimal places. (a) 0.6462

3. Use a calculating utility to approximate log 28.4. Round to four decimal places.

(b) −0.6462 (c) 1.2925

(a) 3.3464

(d) −1.2925

(b) 3.3462 (c) 1.4533 (d) 1.4535

12. Solve for x if 3ex − xex = 0. (a) 3

4. Find the exact value of log3 81.

(b) −3 1 (c) 3 1 (d) − 3

(a) 12 3 (b) 4 1 (c) 4 (d) 4

y

7

5. Use a calculating utility to approximate ln 39.1 to four decimal places. (a) 1.5920 (b) 3.6661 (c) 1.5922 (d) 3.6663 6. Answer true or false. a3 b ln 2 = 3 ln a + ln b − 2 ln c. c √ 7. Answer true or false. log (5x x − 2) = (log 5)(log x)(log1/2 (x − 2)). 8. Rewrite 5 log 2 − log 12 + log 24 as a single logarithm. (a) log 64

-6

13.

6

-3

This is a graph of

x

129 (a) 4 − ln (2 + x)

(d)

(b) 4 + ln (2 + x) (c) 4 − ln (x − 2)

(d) 4 + ln (x − 2)

14. Use a calculating utility and the change of base formula to find log3 4.

2 ln x2 p x 3 + ln2 x2

dy 3 = . 4. Answer true or false. If y = ln (x3 ) then dx x   1 5. Find dy/dx if y = ln . sin x (a) cos x sin x

(a) 0.2007

(b) cot x

(b) 0.7925

(c) tan x

(c) 1.2619 (d) 0.4621 15. The equation Q = 6e−0.052t gives the mass Q in grams of a certain radioactive substance remaining after t hours. How much remains after 6 hours? (a) 4.3919 g (b) 4.3920 g (c) 4.3921 g (d) 2.3922 g

Section 6.2 1. If y = ln 6x then find dy/dx. (a) (b) (c) (d)

1 6x 6 x 1 x 6 ln 6x x

2. If y = ln (cos x) then find dy/dx. (a) tan x (b) − tan x 1 (c) cos x 1 (d) − cos x p 3. If y = 3 + ln2 x2 then dy/dx = 2 p x 3 + ln2 x2 2 (b) p 3 + ln2 x2 1 (c) p 3 + ln2 x2 (a)

(d) − cot x r x+2 dy then find by logarithmic differentiation. 6. If y = 8 x+3 dx   1 x + 2 −7/8 (a) 8 x+3 1 (b) 8(x + 3)2  r 1 1 1 8 x+2 (c) − 8 x+2 x+3 x+3 r x+2 (d) 7 x+3 7. If y = ln (kx) then

dny = dxn

1 k n xn (−1)n (b) n n k x (−1)n (c) (k − 1)! xn 1 (d) x (a)

8. Suppose that f (x) = (cos (x))π . What is dy/dx? (a) π (cos (x))π −1 (b) (cos (x))π −1 (c) π (− sin (x))π −1 (d) −π sin (x) (cos (x))π −1

9. If y2 + ln (xy) = y3 then dy/dx = x (a) 1 2y + y − 3y2 (b) (c)

−1 x(2y + 1y − 3y2 ) x+y
xy 2y − 3y2

(d) none of the above 10. If y = sec (ln x) then y′ =

130 1 sec (ln x) tan (ln x) x (b) sec (ln x) tan (ln x)     1 1 tan (c) sec x x 1 (d) sec2 (ln x) x dy 1 then = 11. If y = ln x dx (a) x 1 (b) − (ln x)2 1 (c) − x (ln x)2 x (d) − (ln x)2

(a) ln x +C

(a)

x2 (x2 + 1)3 sin x then which of the following exprescos (3x) y′ sions is ? y

12. If y =

(a) (b) (c) (d) 13. If y = (a) (b) (c) (d) 14.

Z

6x 2 + + cot x + 3 tan (3x) x x2 + 1 3 1 + + csc x − sec (3x) 2x x2 + 1 −3 sin (3x) 4x2 (x2 + 1) cos x 1 2 2x + + 2 x x + 1 tan (−2x) ln x then dy/dx = x 1 − 2 x 1 x2 ln x 1 − 2 x2 x 1 − 3 x

5 dx = x 5 (a) +C 2x2 5 (b) − 2 +C 2x (c) −5 ln x +C (d) 5 ln x +C

15.

Z

ln x dx = x

(b) (ln x)2 +C (ln x)2 +C 2 (d) 2(ln x)2 +C (c)

16.

Z e 1 1

x

dx =

(a) 1 (b) e 1 (c) e (d) 0 17. Answer true or false.

Z 2 1

dx ln 7 − ln 4 = . 3x + 1 3

Section 6.3 1. Answer true or false. If y = x8 e7x then dy/dx = 56x7 e7x . 2. If y = (ln x)e2x then dy/dx = (a) 2(ln x)e2x +

e2x x

(b) 2(ln x)e2x (c) 2(ln x)e2x (d)

e2x−1 x

e2x x

3. Answer true or false. If x + exy = 2 then 4. Let f (x) = 5x . Find

dy −yexy − 1 = . dx xexy

d [ f (x)]. dx

(a) 5x ln 5 (b) 5x−1 (c) x ln 5x (d) 5x ln x 5. Answer true or false. If f (x) = π sin x+cos x then dy/dx = (sin x + cos x)π sin x+cos x−1 . 6. Answer true or false. If y = xcos x then  dy  cos x = − sin x ln x xcos x . dx x 7. The equation y′ x = −yx is satisfied by (a) y = ex

(b) y = cos x (c) y = sin x (d) y = e−x

131

Section 6.4

3h − 1 = h→0 2h

8. Evaluate the limit. lim

1. Answer true or false. If the position of a particle along a line is given by s(t) = lnt for t > 0, then the limiting velocity of the particle as t → ∞ is zero.

(a) 1 (b) 0 (c) ∞ ln 3 (d) 2 9.

Z

2. If y = ln (x) − x then y has a horizontal tangent line at which of the following x-values? (a) x = −1

(b) x = 0

3ex dx =

(c) x = 1 (a) 3ex +C

(d) y has no horizontal tangents.

3e2x (b) +C 2

   dz dy dx . = dt dt dt 4. The largest open interval over which f is concave up for 4 f (x) = ex is

3. Answer true or false. If z = x ln y then

ex+1 +C x+1 (d) None of these. (c)

(a) (−∞, 0)

dy = ex and y(0) = 2. 10. Find y(x) given that dx (a) y(x) = ex + 1

5. f (x) = ln (x2 + 2) has

y(x) = ex

(a) a relative maximum only

(d) y(x) = ex − 2

(b) a relative minimum only

(c)

Z

(c) both a relative maximum and minimum

2

2xex dx =

(d) no relative extrema

2

(a)

6. Use a graphing utility to generate the graph of f (x) = x2 e3x , then determine the x-coordinates of all relative extrema on (−10, 10) and identify them as relative maxima or minima.

ex +C 2x 2

(b) 2ex +C

(a) There is a relative maximum at x = 0.

2

(c) x2 + ex +C

(b) There is a relative minimum at x = 0.

2

(d) ex +C 12. Answer true or false. For ex − 2.

Z

ex dx a good choice for u is ex − 2

13. Answer true or false. To evaluate good choice for u is u = ex . 14. Answer true or false. To evaluate choice for u is u = 1 + ex . 15.

(c) (−∞, ∞) (d) nowhere

(b) y(x) = ex + 2

11.

(b) (0, ∞)

Z 1 0

e−3x dx =

(a) 0.216 (b) 0.148

Z 1 0

ex (1 + 5ex )12 dx, a

Z 1 0

(c) There is a relative minimum at x = 0 and relative maxima at x = −1 and x = 1.

(d) There are no relative extrema.

7. Find the displacement of a particle between t = 0 and t = 2 if v(t) = et + t 2 . (a) 7.437

ex dx, a good 1 + ex

(b) 9.056 (c) 11.389 (d) 12.389 8. Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = x + ex , y = 0, x = 0 and x = 1 about the x-axis. (a) 5.53

(c) 0.317

(b) 11.05

(d) 0.519

(c) 17.37

132 (a) (5.17581, 0.14168)

(d) 34.73 9. Use cylindrical shells to find the volume of the solid when 2 the region enclosed by y = 2ex , x = 1, x = 2 and y = 0 is revolved about the y-axis.

(b) (6.60199, 0.09556) (c) (6.72727, 0.09091) (d) (3.90865, 0.19543)

(a) 103.8π (b) 12.970π

Section 6.5

(c) 6.485π (d) 25.940π 10. Answer true or false. The arc length of the curve y = ex + e2x Z 4q 1 + (e3x )2 dx. from x = 0 to x = 4 is given by 0

11. Answer true or false. The arc length of the parametric curve Z 3√ x = 3et and y = et for 0 ≤ t ≤ 3 is given by 4et dt. 0

12. Use a CAS or a calculator with integration capabilities to approximate the arc length of the curve y = −xex from x = 0 to x = 2.

x→0

sin 7x = sin x

(a) 7 1 (b) 7 (c) −7 1 (d) − 7 2. lim

x2 − 9

x→∞ x2 − 3x

(a) 21.02

=

(a) 1

(b) 4.17

(b) ∞

(c) 15.04

(c) −∞

(d) 19.71 13. Answer true or false. The area of the surface generated by revolving x = ey+2 , 0 ≤ y ≤ 1 about the y-axis is given by Z 1 p ey 1 + e2y+4 dy. 2π e2 0

14. Answer true or false. The area of the surface generated by revolving the parametric curve x = t 2 and y = et for 0 ≤ t ≤ 1 Z 1 p about the x-axis is given by 2π et e2t + 4t 2 dt. 0

15. The average value of y =

1. lim

1 on the interval [1, 10] is x

(a) ln 10 (b) ln 9 1 (c) ln 10 9 (d) 99/100 16. Find the volume of the solid that results when the region enclosed by the curves x = −ey , x = −1 and y = 1 is revolved about the y-axis. (a) 6.894

(d) 0

tan2 x = x→0 x

3. lim

(a) 1 (b) ∞ (c) −∞

(d) 0 4. lim+ x→0

ln (x + 1) = ex − 1

(a) 0 (b) 1

(c) ∞ (d) −∞

ex = x→∞ x4 (a) 1

5. lim

(b) 0

(b) 3.195

(c) ∞

(c) 10.205

(d) −∞

(d) 32.060 17. A laminar region is bounded by the curves x = 1, x = 10, y = 0, and y = 1/x with δ = x2 + 1. Find the center of gravity of the region.

6. lim e−x ln x = x→∞

(a) 0 (b) 1

133

Section 6.6

(c) ∞ (d) −∞

1/x

7. lim (1 + 3x) x→0

1. Simplify: e3 ln x = =

(a) x3

(a) 3

(b) 3x

(b) ∞

(c) x/3

(c) ln 3

(d) e3 2. Simplify: ln (e−6x ) =

(d) e3 8. lim+ x→0

(a) x6

sin x = ln (x + 1)

(b) x−6 (c) −6x

(a) 10

(d) −6 − x

(b) 1

3. Simplify: ln (xe4x ) =

(c) ∞ (d) −∞

(a) 4

9. Answer true or false. lim

x→0

cos cos

  1 = 10. lim e−x − x→∞ x

1 x
2 x




1 = . 2

(b) 4 + ln x (c) 4x + ln x (d) 4x2 4. Which of the following expressions is equivalent to ln (7/2)?

(a) ∞

(a)

(b) −∞

Z 7 1 0

(c) 1

(b)

t

1

(d) 0 (c)

11. lim+ (1 − ln x)x =

dt

Z 7 1

dt

Z 14 1

dt

1

x→0

dt

2t

Z 7/2 1

2t

ln 7 ln 2 5. Which of the of the following expressions is equivalent to ln (7)? (d)

(a) 0 (b) 1 (c) ∞ (d) −∞

(a)

sin 3x 12. Answer true or false. lim = 1. x→0 1 − cos x

2x3 − 2x2 + x − 3

= 2. x3 + 2x2 − x + 1 p  14. Answer true or false. lim x2 − 2x − x = 0. x→∞   sin x 1 15. lim+ = − x x x→0

13. Answer true or false. lim

x→∞

(b)

0

(c)

dt t (d) ln 28 − ln 21 6. Which of the following expressions is equivalent to ln (6.5)? Z 65 dx

(a)

1 10

(b)

Z 13 dx

(c)

(d) −∞ 16. Answer true or false. lim+ x→0

√




x ln x = 0.

(d)

x

1

x

2

(c) ∞

dt

t

Z 2 7 1

(a) 0 (b) 1

t

2

Z 7 1

Z 5 13 dx

2

x

1

Z 65 dx 10

x

134 7. Let f (x) = e−3x . Which of the following is the exact value of f (ln 2)?

1. If y = sin−1 (2x) then find dy/dx.

(a) 6−6 (b) 1/8 (c) −8 (d) −1/8 8. Answer true or false. If F(x) = 9. Answer true or false. If F(x) = 10. Answer true or false. lim

x→∞



Z x2 2 1

t

2 dt, then F ′ (x) = . x

Z x3 2

6 dt then F ′ (x) = . x

1 x

= 0.

1+

1

t

3x

11. Answer true or false. lim (1 + 5x)1/(5x) = e. x→0

12. Answer true or false. lim (1 + 2x)1/(2x) = 0. x→0

13. Which of the following expressions is equivalent to ln 9.8?
1 ln 100 − ln 2 10 1 ln 98 (b) 10 (c) ln 2 + ln 49 − ln 10 2 ln 49 (d) ln 10 (a)

14. Which of the following expressions is equivalent to ln 42? (a)

Z 7 dx

x

1

(b)

+

Z 50 dx

Z 40 dx

x

1

(d) 3

Z 6 dx

x

1

+

Z 14 dx

15. If F(x) =

Z 2 dx 1

x

Z e2x 1

is FALSE?

1

3. If y = esin

−1

x

then dy/dx =

−1

esin x (a) √ 1 − x2

(b) −(sin−2 x)esin 1 (c) (cos x)esin x (d) −

(a)

−1

(cos−1 x)esin √ 1 − x2

x

−1

x

t

dt then which of the following statements

(a) F(x) is always increasing. (b) F(x) has no critical points. 2 e2x (d) F(x) has no points of inflection. (c) F ′ (x) =

1 x cos−1 x √

1 − x2 √ −x + (cos−1 x) 1 − x2 √ (1 − x2 ) cos−1 x − x 1 − x2 (c) x(1 − x2 ) cos−1 x 1 (d) x p 5. Let y = sin−1 x. Then dy/dx = (b)

x

1

2 (a) √ 1 − 2x2 2 (b) √ 1 − x2 1 (c) √ 1 − 4x2 2 (d) √ 1 − 4x2 √ 2. If y = tan−1 x then find dy/dx. √ x (a) 2x(1 + x) x (b) 1+x x (c) 2(1 + x2 ) r 1 (d) 1 + x2

4. Find dy/dx if y = ln (x cos−1 x).

x

8

(c)

Section 6.7

(a) y = √ 4

1 − x2

1 p √ 2( sin−1 x)( 1 − x2 ) 1 (c) y = √ 4 2 1 − x2

(b) y = 16. Answer true or false. Z x Z x3 dt dt 3 = for all x > 0. t 1 t 1

1

135 (d) y = − 6. If

1 p √ 4 −1 2( sin x)( 1 − x2 )

x2 − sin−1 y = ln x (a)



1 − 2x x

then find dy/dx. q 1 − y2

13. Answer true or false. tan−1 (1) + tan−1 (2) = tan−1 (−3). 14.

Z

(a) cos−1 (x) +C √ (b) 2 1 − x2 +C

 q 1 1 − y2 (b) − + 2x x

(c) ln |sin (x)| +C 1  −1 2 (d) sin (x) +C 2

sin−2 y − 2x2 (c) x sin−2 y

(d)

− sin−2 y + 2x2 x sin−2 y

7. Answer true or false. To have an inverse, a trigonometric function must have its domain restricted to [0, 2π ].

sin−1 (x) √ dx = 1 − x2

15. Find the volume of the solid that is generated by revolving the curve y = (1 + x2 )−1/2 about the x-axis between x = 0 and x = 1. (a)

π2 4

(b) π /2

(b)

π2 2

(c) π

(c) π ln 2

8. Find the exact value of sin−1 (1). (a) 0

(d) 3π /2

(d)

9. Find the exact value of cos−1 (cos (3π /4)). (a) 3π /4

16. Evaluate

(b) π /4 (c) −π /4

10. Use a calculating utility to approximate x if sin x = 0.15 and π /2 < x < 3π /2. (a) 0.1506 (c) 3.2932

17.

Z

(d) no solution 1 11. Answer true or false. cos−1 x = for all x. cos x 12. A ball is thrown at 5 m/s and travels 245 m horizontally before coming back to its original height. Given that the acceleration due to gravity is 9.8 m/s2 , and air resistance is negv2 ligible, the range formula is R = sin 2θ , where θ is the 9.8 angle above the horizontal at which the ball is thrown. Find all possible positive angles in radians above the horizontal at which the ball can be thrown. (a) 1.5708 (b) 1.5708 and 3.1416 (c) 0.7854 and 1.5708 (d) 0.7854

Z

sin (x) dx. 1 + cos2 (x)

(a) ln |sin (x)| +C

(d) 5π /4

(b) 2.9910

π ln 2 2

(b) ln |1 + cos2 (x)| +C
(c) − tan−1 cos (x) +C
(d) − sin−1 cos (x) +C x √ dx = 1 − x2

(a) sec−1 (1/x) +C 1 2 −1 x sin (x) +C 2 √ (c) − 1 − x2 +C √ (d) x2 1 − x2 +C

(b)

18.

Z

dr √ = er 1 − e−2r (a) r + 2 ln (1 − e−2r ) +C (b) sec−1 (e−r ) +C (c) − sin−1 (e−r ) +C √ (d) −2e−r 1 − e−2r +C

136

Section 6.8

(a)

(b) 8 cosh8 x +C

1. Evaluate sinh (7).

(c) 7 cosh6 x +C

(a) Not defined. (b) 551.1614

(d)

(c) 548.3161 (d) 549.4283

8.

2. Evaluate cosh−1 (2).

Z

(a)

(b) 1.3165

(b) 10 sinh10 x +C

(c) 1.3152

(c) 9 sinh8 x +C

(a) (5x + 1) cosh (5x + 1)

(d)

(d) −5 cosh (5x + 1)

4. Find dy/dx if y = sinh (3x2 ). (a) 6x cosh (3x2 ) (b) −6x cosh (3x2 ) (c) 6 cosh (6x)

p 5. Find dy/dx if y = 2 sech(x + 5) − x3 . (a)

(b)

−sech(x + 5) tanh (x + 5) − 3x2 p sech(x + 5) − x3 (x + 5) cosh (x + 5) − 3x2 p sinh (x + 5) − x3

− cosh (x + 5) + 3x2 (c) p sinh (x + 5) − x3

(d)

sech(x + 5) tanh (x + 5) + 3x2 p sech(x + 5) − x3

sinh (3x + 6) dx =

(a) 3 cosh (3x + 6) +C 1 (b) cosh (3x + 6) +C 3 (c) −3 cosh (3x + 6) +C 1 (d) − cosh (3x + 6) +C 3 7.

cosh7 x sinh x dx =

x 6

1 (a) √ 36 + x2 1 (b) √ 6 36 + x2 1 (c) √ 36 − x2 1 (d) √ 6 36 − x2

(c) −(5x + 1) cosh (5x + 1)

(d) −6 cosh (6x)

1 sinh8 x +C 8

9. Find dy/dx if y = sinh−1

(b) 5 cosh (5x + 1)

Z

1 sinh10 x +C 10

(a) 1.3170

3. Find dy/dx if y = sinh (5x + 1).

Z

1 cosh6 x +C 6

sinh9 x cosh x dx =

(d) 1.3174

6.

1 cosh8 x +C 8


.

10. Answer true or false. If y = − coth−1 (x + 3) when |x| > 0, 1 . then dy/dx = − 2 x + 6x + 8 11.

Z

dx √ = 1 + 16x2 1 sinh−1 (4x) +C 4 1 (b) coth−1 (4x) +C 4 1 (c) cosh−1 (4x) +C 4 1 (d) tanh−1 (4x) +C 4 (a)

12. Answer true or false.

Z

4dx = 4 sinh−1 (ex ) +C 1 + e2x

13. Answer true or false.

Z

√

ex dx 1 + e2x

= sinh−1 (e2x ) +C

14. Answer true or false. lim (cosh x)2 = 0. x→∞

15. Answer true or false. lim (coth x)2 = 1. x→−∞

137

Chapter 6 Test

(d) ∞   1 x 8. Answer true or false. lim 8 + = e8 . x x→0

1. If y = ln (5x2 ) then find dy/dx. 2 x 2 (b) 2 x 2 (c) 5x2 1 (d) 2 x

2xy − yexy 9. Answer true or false. If exy = yx2 then y′ = xy . xe − x2   1 1 − 10. Evaluate the limit: lim+ x sin x x→0

(a)

(a) ∞ (b) 0 (c) 2

2. Answer true or false. If y = 3 ln xe3x then dy 3e3x = + 9 ln xe3x . dx x d 3. If f (x) = 8x then find [ f (x)]. dx

(d) Cannot be determined. 11. If y = ln (2x ) then

1 2 x (b) 2 (c) ln (2) ln (2) (d) 2x 12. Answer true or false. If y = sin (arctan (ex )) then dy = ex (1 + e2x )−1/2 − e3x (1 + e2x )−3/2 . dx 13. Evaluate the limit: lim (cos x)sin x (a)

(a) 8x ln x (b) x ln 8x (c) x8x−1 (d) 8x ln 8 √ 4. Find dy/dx if y = tan−1 4 x. 1 √ √ 4 4( x3 + x 4 x) √ x √ (b) 1+ x √ x (c) 2(1 + x) r 1 (d) 4 1+x (a)

x→0

(a) 0 (b) 1 (c) ∞ (d) Cannot be determined. √

sin 9x = x→0 sin 8x (a) 1 (b) ∞

(c) −∞ 9 (d) 8 sin 2x = 7. lim x→0 2x

14. Evaluate the limit: lim (ln (3x − 2) − ln x) x→∞

cos−1 x + 1

5. Answer true or false. If y = −1 dy √ = √ . dx 2( cos−1 x + 1)( 4 x2 − 1) 6. lim

then

(a) 0 (b) ln (2) (c) ln (3) (d) ∞ 2x − 1 = e2 x x→0

15. Answer true or false. lim

9

−1 16. Evaluate the limit: limx→1 xx−1

(a) 0 (b) 8 (c) 9 (d) ∞

(a) 0 (b) 1 1 (c) 2

dy = dx

17. If cos (a)

y
x = ln

x y

  x y

then

dy = dx

138 y−1 y sin (y/x) y (c) − x y (d) x

26. Answer true or false.

(b)

27. Answer true or false. u = x2 .

18. Answer true or false. The function y = xe−x has no horizontal tangent lines. 19. Evaluate the limit: lim+ x ln x x→0

(b) −1

(c) −∞

(d) ∞

(b) (c)

=

(c) 693 (d) 2, 178

+ ln x)

1 ab3 21. Answer true or false. log √ = log a + 3 log b − log c c 2 22. If 52x = 8 then solve for x. (a) 1.292

0

32. Use a CAS or a scientific calculator with numerical integration capabilities to approximate the area of the surface generated by revolving the curve y = ex+1 , −1 ≤ x ≤ −0.5 about the x-axis. (b) 9.27

(c) 0.204

(c) 1.48

(d) 0.102

(d) 6.78

23. Suppose that the number of bacteria present in a bacteria culture at time t is given by N = 10, 000e−t/10 . Find the smallest number of bacteria in the culture during the time interval 0 ≤ t ≤ 50. (a) 67

33. Answer true or false. The area of the surface generated by revolving the parametric curve x = t 2 and y = et for 0 ≤ t ≤ 1 Z 1 p t 2 e2t + 4t 2 dt. about the y-axis is given by 2π 0

34. Use a CAS to find the surface area of the solid that results when the curve y = −ex , 0 ≤ x ≤ 0.5 is revolved about the x-axis.

(b) 10, 000 (c) 3, 679

(a) 18.54

(d) 73, 891

(b) 9.27

dx = 5x

(c) 1.48 (d) 6.78

(a) ln (5x) +C ln x (b) +C 5 (c) 5 ln x +C

35. Use cylindrical shells to find the volume of the solid when the 3 region enclosed by y = 2 , x = 1, x = 2 and y = 0 is revolved x about the y-axis.

(d) ln x +C 25. Answer true or false. For √ u is u = 2 + x.

31. Answer true or false. The arc length of the parametric curve Z 2√ x = e3t and y = e3t for 0 ≤ t ≤ 2 is given by 3et dt.

(a) 18.54

(b) 0.646

24.

28. Answer true or false. dx dz ex dy = + ex ln y . If z = ex ln y then dt y dt dt 29. Answer true or false. The function f (x) = ex ln x has a relative minimum on (0, ∞).

(b) 2, 334

(d) Cannot be determined.

Z

2

ex dx can be easily solved by letting

(a) 574, 698 dy dx

(a) (x + 1)xx xx+1 (1 + 1x ln (x)xx+1

Z

1 + 3ex dx = ln |x| + 3ex +C x

30. Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = −e2x , x = 2, and y = −1 about the x-axis (round to the nearest whole number).

(a) 0

20. If y = xx+1 then

Z

Z

√

dx √ , a good choice for x(2 + x)

(a) 2.08π (b) 4.16π (c) 2.08π 2

139 (a) 4 tanh4 x +C

(d) 4.16π 2

(b) 5 tanh6 x +C

36. Evaluate the limit: lim+ x3/2 ln x = x→0

(c) 6 tanh6 x +C 1 (d) tanh7 x +C 7

(a) 0 (b) 1 (c) ∞ (d) The limit does not exist. 37. Which of the following expressions is equivalent to ln 12.8? (a)

Z 8 2

dt

t Z 8 2 dt (b) 10 t Z 128 1 (c) dt 10 t Z 12 Z 0.8 1 1 (d) dt + dt t t 1 1 1

38. Use a calculating utility to approximate x if sin x = 0.42 and 3π /2 < x < 5π /2.

41. Answer true or false.

43. Answer true or false. p 2 F ′ (x) = 2x 1 + e−x . x

G(x) = by

45.

Z

(d) sech2 (5x4 ) 40.

Z

tanh6 xsech2 x dx =

If F(x) =

Z π 0 ex

0

Z x2 p 2

1 + e−t dt then

(t + 1)2 − 1 dt between x = 0 and x = π is given

(x + 1) dx.

dx = 1 + e2x (a) ex ln |1 + e2x | +C

(b) ln |1 + e2x | +C (c) tan−1 (ex ) +C

(d) 2x +C

(d) 6.723

(c) 5x4 tanh (x5 )

= 4 cosh−1 (ex )

44. AnswerZtrueqor false. The arc length of the curve defined by

(c) 6.719

(b) −5x4 sech2 (x5 )

4dx e2x − 1

x→∞

(b) 6.717

(a) 5x4 sech2 (x5 )

√

42. Answer true or false. lim (coth x)2 = 1.

(a) 6.715

39. Find dy/dx if y = tanh (x5 ).

Z

46.

Z

dx √ = x 3x2 − 1 √ (a) sec−1 (x 3) +C √ √ (b) 3 sin−1 (x 3) +C (c) 2(3x2 − 1)1/2 ln |x| +C 1 (d) ln |x| + ln |3x2 − 1| +C 2

140

Chapter 6: Answers to Sample Tests Section 6.1 1. c 9. a

2. a 10. d

3. c 11. b

4. d 12. a

5. b 13. b

6. true 14. c

7. false 15. a

8. a

2. b 10. a

3. d 11. c

4. true 12. a

5. d 13. d

6. c 14. d

7. c 15. c

8. d 16. a

2. a 10. a

3. true 11. d

4. a 12. true

5. false 13. false

6. true 14. true

7. d 15. c

8. d

2. c 10. false

3. false 11. false

4. c 12. c

5. b 13. true

6. b 14. true

7. b 15. c

8. c 16. a

2. a 10. d

3. d 11. b

4. b 12. false

5. c 13. true

6. a 14. false

7. d 15. d

8. b 16. true

2. c 10. false

3. c 11. true

4. b 12. false

5. a 13. c

6. d 14. a

7. b 15. c

8. false 16. true

2. a 10. b 18. c

3. a 11. false

4. c 12. d

5. b 13. false

6. b 14. d

7. false 15. a

8. b 16. c

2. a 10. false

3. b 11. a

4. a 12. false

5. a 13. false

6. b 14. false

7. a 15. false

8. a

2. false 10. b 18. false 26. true 34. d 42. true

3. d 11. c 19. a 27. false 35. b 43. true

4. a 12. false 20. b 28. true 36. a 44. true

5. false 13. b 21. true 29. false 37. c 45. c

6. d 14. c 22. b 30. b 38. b 46. a

7. b 15. false 23. a 31. false 39. a

8. false 16. c 24. b 32. d 40. d

Section 6.2 1. c 9. b 17. true Section 6.3 1. false 9. a Section 6.4 1. true 9. a 17. b Section 6.5 1. a 9. false Section 6.6 1. a 9. true Section 6.7 1. d 9. a 17. c Section 6.8 1. c 9. a Chapter 6 Test 1. a 9. true 17. d 25. true 33. true 41. false