135

LESSON 7-1 EXPONENTIAL FUNCTIONS An exponential function is a function represented by a constant base with a variable exponent. For example, � � 2xf x , xy e , and � � 2 53xg x � are exponential functions. These basic properties of exponents are used when working with exponential functions. For a and b positive real numbers and x and y any real numbers:

1. 0 1a 2. x y x ya a a � 3. x

x yy

aa

a�

4. � �yx xya a 5. ( )x x xab a b 6. x x

x

a ab b

§ · ¨ ¸© ¹

7. 1xxa

a� Note: ( )x x xa b a b� z �

When simplifying, do not leave answers with negative exponents. Examples: Simplify without using a calculator.

1. 4327 2.

01ee

§ ·�¨ ¸© ¹

3. 2

5 3

4

e ee

��§ ·¨ ¸© ¹

4. 3 25 25�� 5. Solve 9 27x without using a calculator. 6. Use a calculator to carefully graph 2 , 5 ,x xy y and xy e in the same coordinate plane. Do you see any similarities in the graphs? Graphs of Exponential Functions: If ( ) and 1xf x a a ! , then

1. The domain of f (x) is ( , )�f f . 2. The graph of f (x) is continuous, The range of f (x) is (0, )f . increasing, concave upward, and one-to-one (has an inverse function). 3. The x-axis is a horizontal asymptote 4. The y-intercept is (0, 1). to the left: lim ( ) 0

xf x

o�f Another key point is (1, a).

(Also, lim ( )x

f xof

f )

�� �� �� �� � ���

�����

�� �� �� �� � ���

�����

136

The letter e used as a base in Examples 2, 3, and 6, is not an unknown. It is a number called the natural base for exponential functions. It is the most common base in Calculus, because functions with base e are easier to differentiate and integrate than functions with other bases. By definition,

0

1lim(1 )x

xe xo

� . To three decimal places, 2.718e | . Example 7: Without using a calculator, sketch a graph of xy e .

Example 8: Using adjustments to the graph from Example 7, graph ( ) 1xf x e� � without using a calculator. Write an equation for the graph’s asymptote. ASSIGNMENT 7-1 Simplify without a calculator.

1. 238 2.

3225�

3. 0 03 5� 4. 0(3 5)�

5. 3

44

6. 2 1(3 )� � 7. 4 3(3 ) (9 )� 8. 2

3

84

9. 32

e

�§ ·¨ ¸© ¹

10. 22

2

ee�

§ ·�¨ ¸© ¹

11. 32 1

4

e ee

�

�

§ ·�¨ ¸© ¹

12. 2( 3)e�

Solve for x without a calculator.

13. 2 16x 14. 2 33 27x� 15. � �212

8x

16. 43 16x 17.

329

xe

ee

§ · ¨ ¸

© ¹ 18. (5 ) 1xe�

For Problems 19-24, sketch a graph without using a calculator. List all intercepts, and write an equation for each asymptote. Use a separate coordinate plane for each graph.

19. 2xy 20. 2 xy � 21. 2xy � 22. 12xy � 23. 2 1xy � 24. 2 xy

25. Find the average value of 1( )

2f x

x

� on the interval [�2,1].

26. Sketch the region bounded by 2 and 0x y y x � , and find the area of the region. (Show an integral set-up first.)

27. For 3( )f x x , find the value of c in > @2,0� such that ( ) ( )( ) f b f af c

b a�c �

.

�� �� �� �� � ���

�

�

�

�

x

y

�� �� � �

�

�

�

�

x

y

137

Evaluate the expressions or equations in Problems 28-33 without using a calculator.

28. t xt

ddt

xe dx�³ = 29.

2

3(1 sin )

xddx

t dt�³ = 30. 43

2ddx

xx

x�§ ·� ¨ ¸

© ¹

31. 2( ) 3 8f x x x � 32. 2 2

33 8t t dt� ³ 33. 4

3

2yy dyy

§ ·�� ¨ ¸

© ¹³

(2)f c

34. 12

dydx x and y = 0 when x = 2. Find an equation for the curve y. ( y = ______ ).

35. 1

3( ) 10 , (8) 50, and (1) 30f x x f f�

cc c . Find an equation for the curve f (x) 36. Use the acceleration graph at right to find the following for an object moving moving along a straight path.

a. a(1) b. a(6) c. a(9) d. a(12)

Suppose v(0) 2 m/sec. Find: acc. (in e. v(3) f. v(7) g. v(10) h. v(12) m/sec 2 )

i. At what time on [0,12] was the object moving the fastest? Justify your answer. j. At what time on [0,12] was the object moving the slowest? time (in sec) Justify your answer.

k. When was the object slowing down (speed decreasing)?

For Problems 37-40, let R be the region bounded by 4y x x � and the x-axis. 37. Find the area of Region R. (Show an integral set-up first, and do not use a

calculator.) 38. Find the volume of the solid formed by revolving region R about the x-axis.

(Show an integral set-up, and then use a calculator to integrate.) 39. Set up (but do not integrate) an integral for finding the volume of the solid formed

by revolving region R about the line y = 2. 40. Set up (but do not integrate) an integral for finding the volume of the solid formed

by equilateral triangle cross sections with bases perpendicular to the x-axis.

Remember: 234

A s

� � � � �� ��

��

��

��

�

�

�

�

�

�

t

a(t)

138

LESSON 7-2 DIFFERENTIATING AND INTEGRATING EXPONENTIAL FUNCTIONS

Use your calculator to graph xy e and its derivative in the same coordinate plane. What do you notice? e is the only base for which the basic exponential function and its derivative are the same. For other bases, a logarithmic “hook-on” is required.

Differentiating Exponential Functions:

x xddx

e e u uddx

e e uc (Chain rule form), where u is a function of x.

lnx xddx

a a a lnu uddx

a a u a c ( uc and ln a are the “hook-on factors”)

Examples: Differentiate 1.

2 3xxy e � 2. 3

( ) tg t e�

3. ( ) 3 vf v 4. Find the relative extrema of

22( ) xf x xe� � . List as points and do not use a calculator. Integrating Exponential Functions:

x xe dx e C �³ u ue u dx e Cc �³ (Reverse Chain Rule form)

ln

xx a

a dx Ca

�³ ln

uu aa u dx C

ac �³

Examples: Find

5. 2 1

0

12 4x x dx�³ 6. 2

2

3

xedx

x³ 7. � �4

x x

x x

e edx

e e

�

�

�

�³

�� �� �� � �

�

�

�

�

x

y

�� �

�

�

( is the "hook-on" factor) uc

139

ASSIGNMENT 7-2 1. Sketch the graph of xy e� , and draw tangent lines at (0, 1) and ( 1, )e� . 2. Find the slope, and write an equation for each tangent line in Problem 1. a. (0, 1) b. ( 1, )e� Differentiate in Problems 3-10.

3. 5 1( ) xf x e � 4. 5( ) 2 yg y � 5. 2 2( ) 5t th t �

6. 2xy e � 7. 2xy e 8. 2 3( ) 2 xf x x e

9. 2( )1

xeg x

x

� 10. 2 3( ) ( 1)xh x e� �

Find dy

dx for Problems 11-12.

11. 10x yye xe x� 12. 2 0xye x� 13. Write an equation for the line tangent to the graph of

2 4 3( ) x xf x e � � at the point where 1x .

14. Find the domain, x- and y-intercepts, extrema, and points of inflection for the

graph of x xy xe e � . Then sketch its graph without using a calculator. For Problems 15-24, evaluate each integral.

15. 4 3

02 t dt³ 16.

1

1

1 2xe dx�

�³ 17. 22

0( )

xe x dx�³

18. 2

2

ue duu³ 19. � �2

(2 1) 10x xx dx��³ 20. 3 5 3( 2)y ye e dy� ��³

21. 2

x

xe

dxe

�

��³ 22.

2bx a x dxe �³ , 23. � �3

x x

x x

e edx

e e

�

�

�

�³

24. 2

2

x x

xxe e

dxe�

³

25. Approximate

2

0

xxe dx³ using

a. a Left Hand Riemann Sum with four equal subdivisions. b. the Trapezoidal Rule with four equal subdivisions.

where and are constants a b

140

26. Find the function whose derivative is 2( ) 6 2 3g x x xc � � � and whose graph contains the point (1, 4� ).

27. A paintball is shot vertically upward from a position 6 feet below ground level with an initial velocity of 128 ft/sec. (Remember: 2

0 0( ) 16 )s t t v t s � � � . a. At what time does the paintball reach its maximum height? b. What is the maximum height of the paintball? c. What is the paintball’s velocity when 3t sec? d. At what time(s) does the paintball reach ground level? Note: Make sure to express all answers using correct units. 28. Sketch the region bounded by 4yx � , 0, and 0.x y Then set up an

integral for the area of the region, and use a calculator to find the area. Set up (but do not integrate) integrals for computing the volumes of the solids described in Problems 29-31, where R is the region bounded by ln( 1), 0, and 4y x y x � . 29. The solid is formed by revolving region R about the x-axis. 30. The solid is formed by revolving region R about 2y � . 31. The volume is formed by semicircular cross sections in Region R, whose bases are

perpendicular to the x-axis.

32. Find the average value of 4

1(4 )

yx

�

on [2, 3].

33. A rectangle is inscribed under the curve 2

41

yx

�

and is bounded below by the x-axis as shown in the figure at right. a. Find the point ( , )x y as shown in the figure that produces the rectangle having the most area. b. Find the area of the rectangle.

x

y

(x,y)

141

LESSON 7-3 DERIVATIVES OF INVERSE FUNCTIONS At right are the graphs of a function ( )f x and its inverse 1( )f x� . Remember that if the graph of f contains the point (a, b), then the graph of 1f � contains the point (b, a). Also, the graph of 1f � is the reflection of the graph of f across the line y x . From the graphs above, do you see a relationship between the slope of the graph of f at (a, b) and the slope of the graph of 1f � at (b, a)?

Example 1: Let ( )f x x . a. Sketch the graph of ( )f x . b. Find 1( )f x� . Hint: You must list a domain restriction.

c. Sketch the graph of 1( )f x� in the same coordinate plane as the graph of ( ).f x d. Differentiate both ( )f x and 1( )f x� . e. Find the slope of the graph of ( )f x at (4, 2) and the slope of the graph of 1( )f x� at

(2, 4).

f. What conclusion can you make about these slopes?

Since slope ym

x'

'

, it should make sense that switching x and y (for inverse

functions) should produce reciprocal slopes for inverse functions.

Derivatives of Inverse Functions:

If (a ,b) is a point on f, then (b ,a) is a point on 1f � , and 1 1( ) ( )( )

f bf a

� c c

or if f and g are inverse functions, then � �

1( )( )

g xf g x

c c

.

Derivatives of inverses have reciprocal slopes at “image points” (points reflected across y = x). (a, b) and (b, a) are image points.

� � � �

�

�

�

�

x

y

y x

� �f x

� �1f x� x

y (a,b)

(b,a)

142

Note: When finding derivatives of inverse functions, do not use the same x-value for both f and 1f � . This hardly ever works. (It only works when the x- and y-values of the ordered pairs are the same.) Example 2: Let f and g be inverse functions such that:

( 1) 1f � (0) 2f (1) 5f 32

( 1)f c � (0) 2f c 12

(1)f c

From the given information, find each of the following if possible. Hint: Make a table or chart to organize your data. a. (1)gc b. (2)gc c. (3)gc d. (0)gc e. (5)gc If a function f has an inverse function 1f � , then f is one-to-one and must be either strictly increasing or strictly decreasing (strictly monotonic) on its entire domain. We can use f c to find out where f is increasing and where f is decreasing. Example 3: a. Use ( )f xc to show that 3( ) 6f x x x � is not one-to-one on its entire domain. b. Find the largest interval containing 0x for which f is one-to-one. c. Find the largest interval containing 2x � for which f has an inverse function.

143

ASSIGNMENT 7-3

1. 3( ) 1f x x � . Let 1( ) ( )g x f x� . a. Find ( )g x . b. Graph ( )f x and ( )g x in the same coordinate plane. c. Find ( ) and ( )f x g xc c . d. Find (1) and (0)f gc c . e. What is the relationship between the slopes in Part d?

2. ( ) 3 1f x x � . Let 1( ) ( )g x f x� a. Find ( )g x . b. Graph ( )f x and ( )g x in the same coordinate plane. c. Find ( ) and ( )f x g xc c . d. Find (1) and (2)f gc c . e. What is the relationship between the slopes in Part d? 3. ( ) 1f x x � . Let 1( ) ( )g x f x� a. Find ( )g x . b. Graph ( )f x and ( )g x in the same coordinate plane. c. Find ( ) and ( )f x g xc c . d. Find (3) and (2)f gc c . e. What is the relationship between the slopes in Part d? 4. Let f and g be inverse functions such that:

4 13 5

( 1) 0, (0) 1, and (1) 3

( 1) , (0) , and (1) 2

f f f

f f f

� ­°® c c c� °̄

Find each of the following (if possible).

a. ( 1)gc � b. (0)gc c. (1)gc d. (2)gc e. (3)gc

5. If (2) 3f and (2) 4f c , find � �1 (3)f � c . 6. If (1, 2) is a point on 3( ) 2 1f x x x � � , find � �1 (2)f � c .

7. If � �3 14( ) ( 0), find (6)x

f x x x f � � ! c .

f and g are inverse functions in Problems 8-10. Find gc at the given value.

8. 2

3

(2) 5

(2)

(5)

f

f

g

�

c

c

9. 5 3( ) 2 1(1) 2(2)

f x x xfg

� �

c

10. 2( )(1)

xf x eg

� c

144

For Problems 11-13, use ( )f xc to find the largest interval on which ( )f x has an inverse function.

11. 2( ) 3 10

(Interval contains 1)f x x x

x � �

12.

3( ) 12(Interval contains 3)f x x x

x �

13. 3 2( ) 3 3f x x x x � �

Differentiate in Problems 14-17.

14.

2

2

5 1

d ydx

y xx

�

15. � �426 3xddx

� 16. 2

( )

( )

x

xf x

ef x

c

17. 5( ) (2 1)( )

g t tg t

�cc

Evaluate the integrals in Problems 18-21 without using a calculator.

18. 2 5

2

34

x dx�³ 19.

30

1

( 2)t

te

dte�

� �³ 20.

4

13y

dyy�

³ 21. � �21x dx�³

22. Sketch the region bounded by 2 4x y � and 2x y � . Then set up an integral for the area, and find the area without using a calculator.

23. Set up (but do not integrate) an integral for the volume of the solid formed by revolving the region from Problem 22 about 4x � .

24. An object moves along a vertical path with its position at time t (in seconds), according to the equation 1( ) ty t te � (where y is measured in centimeters (cm)).

a. Find the object’s position at time 2t � sec. b. Find the equation for the object’s velocity. c. Find the object’s velocity at time 2t � sec. d. Find an equation for the object’s acceleration. e. Find a ( 2)� . f. For what interval(s) of time is the object moving downward? g. Find the object’s minimum position. Justify your answer. h. Use a calculator to find the total distance traveled by the object on [ 2,2]� .

Use a calculator to evaluate problems 25-27.

25. (1.237)

for ( ) 5x

f

f x e

c

�

26. (1.237)

for ( ) 5x

f

f x e

cc

�

27. 1.237

05xe dx�³

28. Use the Trapezoidal Rule (with 4n ) to approximate 2

30

11

dxx�³ .

For Problems 29-32, find each limit (if it exists).

29. 3

2 1lim3 4x

xxo�

��

30. 2

22

4lim3 2t

tt to

�� �

31. 21

1lim1x

x

xo�

�

� 32.

2

1lim1x

x

xo�

�

�f

145

LESSON 7-4 LOGARITHMIC FUNCTIONS

Since ( ) xf x e is one-to-one (continuous and increasing), it must have an inverse. However, if you switch x and y in the equation xy e to get yx e , you cannot isolate the new y by using algebraic methods. So, we must define

1( )f x� for the function ( ) xf x e . For ( ) xf x e , 1( )f x� is called the natural logarithmic function, and we write

1( ) lnf x x� (so that and lnyx e y x must be equivalent). In general, if 1( ) ( 0), then ( ) logx

af x a a f x x� ! (so that and logy

ax a y x must be equivalent).

Note: loge x is usually written as ln x and 10log x is usually written simply as log x . Graphs of Logarithmic Functions: If ( ) log and 1af x x a ! , then 1. The domain of ( ) is (0, )f x f . 2. The graph of ( )f x is continuous, The range of ( ) is ( , )f x �f f . increasing, concave downward, and one-to-one (has an inverse function). 3. The y-axis is a vertical asymptote 4. The x-intercept is (1, 0). downward:

0lim ( )x

f xo

�f Another key point is (a, 1).

(Also, lim ( )x

f xof

f )

Compare these graphical characteristics of ( ) logaf x x to those of ( ) xf x a from Lesson 7-1 (page 135). Example 1: Use a calculator to graph lny x and logy x in the same coordinate plane. Do you see any similarities in the graphs? Example 2: Without using a calculator, sketch a graph of ln( 2)y x � . Write an equation for the graph’s asymptote.

y x

� � xf x e

1( ) lnf x x�

x

y � �1,e

� �,1e

� � � � ��

��

��

�

� � � � ��

��

��

�

� � �

��

��

�

�

x

y

146

For changing forms of an equation involving exponentials or logarithms, we use the following Change of Form Definition:

ln

Exponential form Logarithmic formlog

y

ya

x e y x

x a y x

­ ½ l ° °® ¾

l ° °¯ ¿

Example 3: Change the following equations from exponential form to logarithmic form or vice versa.

a. 43 81 b. 0 1e c. log(.1) 1 � Example 4: a. Since 0 1, ln1e b. Since 1 , lne e e

c. Because the natural exponential function and the natural logarithmic function are inverses, lnln n ne e

Example 5: Use the inverse idea from Example 4c. to simplify.

a. 2ln e b. ln(3 )xe c. log210 d. 2

2log 2x

Properties of Logarithms:

1. ln( ) ln lnab a b � These properties work for any bases,

2. ln ln lnaa b

b � but only if 0 and 0a b! !

3. ln lnna n a

Example 6: Expand using Logarithm Properties 1-3 above. a. 5

8ln b. 3 2ln 1x �

Example 7: Condense into a single logarithm. ( 0 and 0)x y! !

a. 3ln 5lnx y� � b. 12

ln ln( 1) 3lnx x y� � �

147

Example 8: Solve for x.

a. 2 5 6xy e � � b. 2 2log log ( 8) 3x x� �

Change of Base Formula: logloglog

ba

b

xx

a

Since the only two logarithmic bases on your calculator are 10 (log key) and (lne key), you will change bases on your calculator in one of two ways:

log lnlog or loglog lna a

x xx x

a a

Example 9: Use your calculator to find 7log 112 to 3or more decimal places. Example 10: a. Find an exact value for x, if 23 6x� . b. Use your calculator to find a decimal value for your answer from Part a. to 3 or more

decimal places.

ASSIGNMENT 7-4 Decide whether each statement in Problems 1-8 is true or false for 0 and 0a b! ! . (Check your answers before working on the rest of the assignment.)

1. log( ) log loga b a b� � 2. ln( ) ln lna b a b� �

3. loglog loglog

aa b

b� 4.

logloglog

a ab b

5. 3(ln ) 3lnx x 6. 3ln 3lnx x 7. 2ln 2ln , for all x x x 8. 2ln 2ln , for 0x x x !

148

For Problems 9-12, change each equation from exponential form to logarithmic form or vice versa. 9. 3 1

1255� 10. 17xe

11. 3log 729 6 12. log 2x � Simplify each expression in Problems 13-16.

13. ln(2 1)xe � 14. ln a be � 15. 5log 5 p 16.

23log3 m

For Problems 17-20, solve for x without using a calculator. Simplify your answers.

17. 2log 3x 18. ln 1x � 19. 2

31 log 27x � 20. log 64 3x For Problems 21-26, sketch a graph without using a calculator. List all x-intercepts, and write an equation for each asymptote. Use a separate coordinate plane for each graph. 21. 2logy x 22. 2log ( 3)y x � 23. 2log ( )y x � 24. 2logy x 25. 2logy x In Problems 26 and 27, ( )f x is given. Without using a calculator, find 1( )f x� , and graph both f and 1f � in the same coordinate plane.

26. 2( ) xf x e 27. ( ) ln( 1)f x x �

Remember that your graphs should be reflections of each other across y x . Use Properties of Logarithms to expand the expressions in Problems 28-30. (All variables represent positive quantities.)

28. ln abc

29. � �2log xy 30. 2( )ln a b

c�

Use Properties of Logarithms to condense the expressions in Problems 31-33 into single logarithms. (All variables represent positive quantities).

31. log 2logx y� 32. 12

3ln lnx y� 33. ln (2ln ln )a b c� �

For Problems 34-36, solve for t without using a calculator.

34. 2

ln 6t te � 35. 2 1 3 0te � �

36. 2 2log log ( 2) 3t t� �

149

Use a calculator to solve for x in Problems 37 and 38. (Express answers to 3 or more decimal place accuracy.)

37. 1 23 5xe x� � � 38. ln(.5 ) .2 xx e � Find the values of the logarithms in Problems 39 and 40. (Express answers to 3 or more decimal place accuracy.) 39. 3log 20 40. 5log (.02) 41. Use an f c number line to determine whether or not 3 2( ) 9 27 36f x x x x � � � is

strictly monotonic (strictly increasing or strictly decreasing). 42. If 5 3( ) 2 1f x x x � � , find � �1 ( 2)f � c � . 43. If � �1( ) 4, find (2)f x x f � � c .

44. If � � � �4 7 and 4 3f f c � , use the equation of a tangent line to approximate � �13

4f .

45. A conical tank, as shown at right, has a hole in its bottom and is leaking water at the rate of 1 cubic foot per minute. Find the rate of change in the height, h, of the water in the tank when h = 4 ft? V = 21

3r hS

Write appropriate units for your answer. Hint: Find a relationship between r and h. r ___ h

46. Use the graph of f c shown at right to graph f cc and a possible graph of f.

y

f c

10 ft.

150

LESSON 7-5 DIFFERENTIATING LOGARITHMIC FUNCTIONS We can find the derivative of the natural log function by using the formula for the Derivative of Inverse Functions from Lesson 7-3 (page 141).

� �

1( )( )

g xf g x

c c

where f and g are inverse functions.

Since xy e and lny x are inverse functions, let ( ) and ( ) lnxf x e g x x .

Then, ln

1 1 1( ) , so ln ( )(ln )

xx

ddx

f x e x g xf x e x

c c c

.

Differentiating Logarithmic Functions:

1lnddx

xx

1ln (Chain rule form)ddx

uu u

u uc

c

1loglna

ddx

xx a

loglna

ddx

uu

u ac

(Chain rule form)

Examples: Differentiate.

1. ln(5 )y x 2. � �2( ) ln 3f t t t � 3. ( ) lnh x x x Example 4: a. Graph y = ln x in the coordinate plane at right. b. For the right half of the graph the absolute value has

no effect, so 1lnddx

xx

for 0x ! .

What is lnddx

x when 0x � ?

Plot some slopes and think about it. lnd

dxx for 0x � .

When differentiating the natural log function, ln and lnx u the absolute value can be ignored. In these cases, absolute value may change the domain of the function – but not the derivative.

�� �

��

xy

151

Example 5:

Find 3ln 5 2ddy

y�

When possible, simplify logarithmic functions before differentiating them.

Example 6: Differentiate 2

2 1ln1

x xy

x�

�

.

First, rewrite as y = Then, yc

Example 7: Use the Change of Base Formula to show that 1loglna

ddx

xx a

.

(See page 150 – Differentiating Logarithmic Functions) Example 8: If � �2

2log 1 , find (2)y x yc �

152

ASSIGNMENT 7-5 Differentiate in Problems 1-13.

1. 5lny x 2. � �2ln 5y x x � 3. � � 3ln 2f x x x �

4. 4( ) (ln )g y y 5. 3( ) lng t t t � 6. ( )f x 25log 1x �

7. ( ) ( 1) lnh x x x � 8. 2

ln1

xy

x

� 9. ( )

lnt

f tt

10. � �ln ln( 1)y x � 11. 23 1( ) xf x e � 12.

23 15 xy �

13. lny x x

In Problems 14 - 16, find dydx

by using implicit differentiation.

14. ln ln(2 1)y x x � 15. 2 lny x y 16. ln( ) 10x xy�

17. For

3 22

22 , find 3

xxe d y

y edx

�

� .

18. Let ( ) lnf x x . a. Sketch the graph of ( ) lnf x x . b. On your graph, draw a tangent line at (1, 0). c. Write an equation for the tangent line that you drew. 19. Find equations for the tangent and normal lines to 2 ln( 1) 1y x x � � � at the point (0, 1). 20. Find the domain, extrema, and points of inflection for the graph of lny x x � .

Then sketch the graph without using a calculator. Evaluate the integrals in Problems 21-23.

21. 2

2

1 2 3x x

x

e edx

e� �

³ 22. 2( 2 3)x x xe e e dx� � �³ 23. 2(2 4) ( 2)te t dt� �³

153

For Problems 24 and 25, let R be the region bounded by 2 , 0, 0,x

y e y x�

and x k where k is some positive constant. 24. Sketch Region R, set up an integral for its area, and find the area (in terms of k). 25. Find the volume (in terms of k) of the solid formed by revolving Region R about

the x-axis. 26. Set up one or more integrals which could be used to find the total area of the

region(s) bounded by 2 2 3( ) and ( ) 3 3f x x g x x x x � � . You may use a calculator to graph the equations, and you do not need to actually integrate.

27. The graph of ( )f x is shown at right.

Approximate 12

12( )f x dx

�³

a. by using a trapezoidal approximation with equal subintervals of width 4x ' . b. by using Left hand rectangles of equal width, where n = 6.

��� �� �� � � ��

���

��

��

��

x

y

y = f(x)

154

LESSON 7-6 INTEGRATION INVOLVING THE NATURAL LOG FUNCTION Differentiation and integration are inverse operations.

So if 1ln ,ddx

xx

then 1 ln ,dx x Cx

�³ and if ln , then ddx

uu

uc

lnudx u C

uc

�³ .

Log Rules: 1 lndx x Cx

�³ and lnudx u C

uc

�³

Note: Although it is true that both 1 1ln and ln ,d ddx dx

x xx x

1 lndx xx

³ only. Why?

Examples: Integrate

1. 3dx

x�³ 2. 2 1

PdP

P �³ 3. 2

3 2

9 6t tdt

t t��³

When integrating a fraction where the degree of the numerator t the degree of the denominator, you will have to use long division (or creative thinking) to “split the fraction.”

Example 4: Integrate 2

2

4 22

x xdx

x� ��³

When integrating functions that contain logarithms, you usually want to think about doing a “Reverse Chain” integration, where lnu x (or something containing ln x )

1and ux

c (or something similar).

Example 5: Integrate ln xdx

x³ Example 6: Integrate 3

1(2 ln )

dxx x�³

155

Example 7: Integrate 11

dxx �³

ASSIGNMENT 7-6 Evaluate each integral in Problems 1-14.

1. 1

2edx

x³ 2. 3

1

42 1

dtt �³ 3.

1 1

14x dx�

�³

4. 2 2

0

xe dx�³ 5. 2 1xe x dx�³ 6. 2 1

xdx

x �³

7. 2 1x

dxx�

³ 8. 2

4 63 2

ydy

y y�

� �³ 9. 11

xdx

x��³

10. 3

1( 1)

dxx��³ 11.

23

31

udu

u �³ 12. ln x

dxx³

13. 5

2(1 ln )

dxx x�³ 14. 4

2dx

x�³

For Problems 15 and 16, find the inverse of the given function, and then sketch the function and its inverse in the same coordinate plane. 15. ( ) ln( )f x x � 16. 2( ) 1xg x e � Simplify the expressions in Problems 17 and 18 without using a calculator.

17. 3127

log 18. 10

3ln ee

19. Use Log Properties to expand � �ln 1x y � .

156

20. Use Log Properties to condense 2log 3log logp q r� � into a single logarithm. Solve for x without a calculator:

21. 2 3 5 0xe � � 22. 1 3ln 5x� � Differentiate in Problems 23-27.

23. 23( ) ln( 1)f x x � 24.

2

lnt

yt

25. ( ) ln (1 ln )g y y �

26. � �ln 2 1 lny x x � 27. 2 2lnx y y�

dydx dy

dx

28. List the domain and range for each of the following without using a calculator.

a. ln( 2)y x � b. � �2ln ( 2)y x � 29. Is � �2ln ( 2) 2ln( 2)x x� � ? Why or why not?

30. For the function ln( ) x

x xg x

e , use a calculator to find each of the following:

a. the average rate of change of ( )g x on the interval [1, 4]. b. the average value of ( )g x on the interval [1, 4]. c. the instantaneous rate of change of ( )g x at 1x . d. the x-value(s) where ( )g x has a horizontal tangent. 31. Region R is bounded by , 1, 1, and 3xy e y x x . a. Without using a calculator sketch Region R. Set up (but do not integrate) integrals which could be used to find the following

areas or volumes. b. the area of Region R. c. the volume of the solid formed by revolving Region R about the x-axis. d. the volume of the solid formed by square cross sections in Region R which are

perpendicular to the x-axis. e. the volume of the solid formed by revolving Region R about the line 1y . 32. Find a, b, and c for 2( )f x ax bx c � � , such that (1) 10f , and ( )f x has a

relative minimum at ( 1,2)� .

157

UNIT 7 SUMMARY Exponential and Logarithmic Graphs:

Graph of ( ) xf x e Graph of ( ) lng x x

0

11e

e e

ln1 0ln 1e

All basic exponential � �( ) xf x a and logarithmic � �( ) logag x x graphs with 0a ! are similar to the graphs shown above.

( ) and ( ) lnxf x e g x x are inverse functions, so lnln x xe e x .

Change of Form Definition: Exponential form Logarithmic formlnlog

y

ya

x e y x

x a y x

­ ½ l ° °® ¾

l ° °¯ ¿

Properties of Logarithms: only true when 0 and 0 a b! !

1. ln( ) ln lnab a b � 2. ln ln lnaa b

b � 3. ln lnna n a

Change of Base: lnloglna

xx

a

Differentiation and Integration Rules:

x xddx

e e u uddx

e e uc x xe dx e C �³ u ue u dx e Cc �³

lnx xddx

a a a lnu uddx

a a u ac ln

xx a

a dx Ca

�³ ln

uu a

a u dx Ca

c �³

1lnddx

xx

lnddx

uu

uc

1lnddx

xx

lnddx

uu

uc

1 lndx x Cx

�³ lnudx u C

uc

�³

1loglna

ddx

xx a

loglna

ddx

uu

u ac

Derivatives of Inverse Functions:

If f and g are inverse functions, then 1( )( )

f ag b

c c

where ( , )a b is a point on the graph

of f and ( , )b a is the “image point” on the graph of g.

x

y

(0,1)

(1,e)x

y

(e,1)

(1,0)

158

ASSIGNMENT 7-7 REVIEW 1. Without using a calculator, list the domain, two points that the graph of the function

contains, and the asymptote for the graph of the function. Then sketch each graph in the same coordinate plane.

a. xy e b. lny x 2. Using adjustments to the graph of xy e from Problem 1, sketch 3xy e� � . What

is the asymptote for this graph? 3. Find the inverse of 3xy e� � , and sketch its graph in the same coordinate plane

that you used for Problem 2. What is the asymptote for this graph? 4. Write an equation for the line tangent to 3 when 0xy e x� � . 5. Without differentiating, find the equations of the lines tangent to and normal to the

graph of the inverse of 3xy e� � (when 2x � for y-inverse). Note: This is the x-value for the inverse function – NOT for the original function. (Refer back to problem 4.) 6. Use an f c number line to show that ( ) lnf x x x � is not one-to-one and therefore

does not have an inverse. Write an interval for the largest domain for which ( )f x does have an inverse.

For Problems 7-10, solve for x without using a calculator. Simplify your answers. 7. 5log 2x � 8. ln( 2 3) 5xe � � 9. ln 2 1 0x�

10. 3 3 3log log (2 1) log ( 4) 1x x x � � � � 11. 3 2 14 8 x�

For Problems 12-16, find the indicated derivative.

12. 3( )

( )

tg t t e

d g tdt

�

13. 2 1

2

2

xy e

d ydx

�

14. 2ln 1d

dx

x

x

§ ·�¨ ¸ ¨ ¸© ¹

15. 3 ln

dydx

y x y x�

16. 17. � �� �3264 log 1

dydx

y x �

� �2 3ln 1y x x

y

�

c

159

If possible, differentiate each function in Problems 18-21. Assume C and c are constants. One example is not possible with techniques you have learned. Note the differences in each problem.

18. cy C 19. cy x 20. xy c 21. xy x Evaluate the integrals in Problems 22-28.

22. 1

20

31

xdx

x �³ 23. 0

1

2 2t t

te e

dte�

� �³ 24.

2 13

tdt

t�

³ 25. 3

2 1x

dxx �³

26. 3(ln )dy

y y³ 27. 2xe dx

x

�

³ 28. 1 3e

dxx

�

�³

29. Find the average value of 2(ln )( ) t

f tt

on the interval 2,e eª º¬ ¼ .

Find the limits in Problems 30 and 31.

30. 2

3

(2 3)lim10x

x xxo

��f

31. 3

(2 3)lim10x

x xxo

��f

32. Given � �2, 1� is a point on the graph of � � 3 11f x x x � � , find � � � �1 1f � c � .

33. If � � 3xg x e x � , find � � � �1 1g� c .

34. Region R is bounded by , 0, 2, and 0.xy e x x y Do not use a calculator. a. Sketch a graph and shade region R. b. Find the area of region R. c. Find the volume of the solid formed if the region is revolved about the x-axis. d. Set up an integral for the volume of the solid formed if the region is revolved about the line y = 9. e. Set up an integral for the volume of the solid with semicircular cross sections whose diameter lies in region R perpendicular to the x-axis.

35. If � � 12

xf x e�c and � � 2

122

fe

, find � �f x .

36. If � � ln xf x x ec � and � �2 3f , use a calculator to find � �4f . 37. If the area of the rectangle shown is increasing at

the rate of 4 2cm

sec , find dxdt

when x = e cm.

x cm

� �ln x cm