Answers to Odd-Numbered Problems and Review Problems calculus... · 2010-08-24 · 628 Answers to Odd-Numbered Problems and Review Problems 31. V S; V increases by a factor of . 33.

Chapter 1, Section 11. f(0) � �2, f(�2) � 0, f(1) � 6

3. g(�1) � �2, g(1) � 2, g(2) �

5. h(2) � , h(0) � 2, h(�4) �

7. f(1) � 1, f(5) � , f(13) �

9. f(1) � 0, f(2) � 2, f(3) � 2

11. f(�6) � 3, f(�5) � �4, f(16) � 4

13. All real numbers x except x � �2

15. All real numbers x for which x � 5

17. All real numbers t

19. All real numbers x for which �x� � 3

21. All real numbers t except t � 1

23. f(g(x)) � 3x2 � 14x � 10

25. f(g(x)) � x3 � 2x2 � 4x � 2

27. f(g(x)) �

29. f(g(x)) � �x�

31. f(g(x)) � , g( f(x)) � or4

x2 �2

x� 1

2

x2 � x � 1

1

(x � 1)2

1

125

1

27

2�32�3

5

2

, f(g(x)) � g( f(x)) if x � 2.836 or x �

0.705

33. f(g(x)) � x, g( f(x)) � x, f(g(x)) � g( f(x)) for all realnumbers x except x � 1 and x � 2

35. f(x � 2) � 2x2 � 11x � 15

37. f(x � 1) � x5 � 3x2 � 6x � 3

39. f(x2 � 3x � 1) �

41. f(x � 1) �

Note: In 43 to 47, answers may vary.

43. h(x) � x � 1, g(u) � u2 � 2u � 3

45. h(x) � x2 � 1, g(u) �

47. h(x) � 2 � x, g(u) �

49. (a) f(2) � 46 (b) f(2) � f(1) � 26

51. (a) P(9) � ; 19,400 people

(b) P(9) � P(8) � ; 67 people

(c) Writing exercise; responses will vary.

53. (a) S(0) � 25.344 cm/sec(b) S(6 � 10�3) � 19.008 cm/sec

1

15

97

5

�u3 �4

u

1

u

x

x � 1

�x2 � 3x � 1

4 � 2x � x2

x2

Answers to Odd-Numbered Problemsand Review Problems

Answers to Odd-Numbered Problems and Review Problems 625

55. (a) All real numbers x except x � 200(b) All real numbers x for which 0 � x � 100(c) f(50) � $50 million(d) f(100) � f(50) � $100 million(e) f(x) � 37.5 million implies that x � 40%

57. (a) Year C(t) CEI

1975 2463 1.000

1976 2633 1.069

1977 2825 1.147

1978 3072 1.247

1979 3399 1.380

1980 3861 1.568

1981 4332 1.759

1982 4791 1.945(b) and (c) Answers vary.

59. (a) C[q(t)] � 625t2 � 25t � 900(b) C(3) � $6,600(c) C(t) � $11,000 implies that t � 4; after 4 hours

61. P(�4.0) � 1,540, P(�3.5) � 861, P(�3.0) � 435, P(�2.0) � 66, P(�1.0) � 1, P(1, 0) � 15, P(1.5) �66, P(2.0) � 190

63. Not defined at x � 1 or at x � �1.5 (where 2x2 � x� 3 � 0)

65. f(g(2.3)) � 6.31

Chapter 1, Section 21. f(x) � x

(0, 0) (1, 1)

3. f(x) � x3

5. f(x) � �x3 � 1

7. f(x) � 2 � 3x

x

y

(0, 2)

( , 0)23

(0, 1)

(1, 0)

(0, 0)

(2, 8)

626 Answers to Odd-Numbered Problems and Review Problems

9. f(x) �

11. f(x) � (x � 1)(x � 2)

13. f(x) � �x2 � 2x � 15

(0, 15)

(–5, 0) (3, 0)

(1, 0)

(0, –2)

(–2, 0)

(0, 1)

(0, –1)

�x � 1

x � 1 if x � 0

if x � 015. f(x) � 6x2 � 13x � 5

17. y � 3x � 5 and y � �x � 3;

19. y � 3x � 8 and y � 3x � 2; no points of intersection;the lines are parallel

x

y

(0, 8)

(– , 0)83

(– , 0)53

(– , )12

72

(0, 5)

(0, 3)

(3, 0)

��1

2,

7

2�

( , 0)13

(– , 0)52

(0, –5)


21. y � x2 and y � 6 � x; (2, 4); (�3, 9)

23. y � x2 � x and y � x �1; (1, 0)

25. P(x) � (x � 40)(120 � x); optimal price is $80 perrecorder

x

y

(80, 1600)

x

y

(1, 0)12, _( 1

4)(0, 0)

(–3, 9)

(2, 4)

(6, 0)

(0, 6)

27. (a)

(b) E( p) � �200p2 � 12,000p � �200p( p � 60)(c)

(d) If the units are free ($0), the expenditure is 0. At$60 per item, no units will be sold.

(e) $30

29. Average cost �

(0, 5)

(1, )436

x

x = 0

x

C(x)

C(x)

C(x)

x�

x3 � 12x � 30

6x

(0, 0) (60, 0)

(30, 180,000)

(60, 0)

(0, 12,000)D(p)

p


31. V � S ; V increases by a factor of .

33. (a) Each y value for y � �x2 is the negative of thecorresponding y value of y � x2. Hence the pointson the graph of y � �x are reflections across the xaxis of the points of the graph of y � x2.

(b) If g(x) � �f(x), the graph of g(x) is the reflectionacross the x axis of the graph of f(x).

35. (a) x 2 5 7 10

C(x) 132 195 237 300

(b) C(x) � 90 � 21x(c)

x

y

30

1

x

y

2�2��S

6 � 37. The added term is shifting the vertex of the graph tothe right and down.

The added term is shifting the vertex of the graph tothe right and down. The higher power on the addedterm is producing a more pronounced shift than theinitial term involving x to the first power.

x

y

x

y


39. In a right triangle c � . But, a � x2 � x1

and b � y2 � y1, therefore c �

(a) 5(b) 7

41. Writing exercise; responses will vary.

43. All real numbers x except x � �1.6 and x � 0.6.

Chapter 1, Section 3

1. m � �

3. m � �1

5. m is undefined.

7. m � 3, b � 0

x

y

7

2

x

y

2

–1 1

�(x2 � x1)2 � (y2 � y1)2

�a2 � b2 9. m � 3, b � �6

11. m � � , b � 3

13. m � , b �

(0, )45

(– , 0)43

x

y

4

5

3

5

(2, 0)

(0, 3)

x

y

3

2

(2, 0)

(0, –6)

x

y


15. m � � , b � 5

17. m is undefined; no y intercept

19. y � x � 2

21. y � � �

23. y � 5

25. y � �x � 1

27. 45x � 52y � 43

29. y � 5

31. y � �2x � 9

1

2

1

2x

(–3, 0)x

y

(2, 0)

(0, 5)

x

y

5

233. y � x � 2

35. y � C(x) � 60x � 5,000

37. (a) y � f(t) � 35t � 220 (b) 325 (c) 220

39. f(t) � �150t � 1,500

(10, 0)

(0, 1500)

t

y

(0, 220)

t

y

(0, 5000)

x

y


41. (a) y � f(t) � �4t � 248(b) f(8) � 216 million gallons

43. (a) F � C � 32° (b) 59°F (c) 20°C

45. (a) v(1930) � $800; v(1990) � $51,200; v(2000) �$102,400

(b) No, it was not linear.

47. (a) For each of x ounces of Food I, we have 3 gm ofcarbohydrates and 2 gm of protein. Food Icontains 3x gm of carbohydrates and 2x gm ofprotein. For each of y ounces of Food II, we have5 gm of carbohydrates and 3 gm of protein. FoodII contains 5y gm of carbohydrates and 3y gm ofprotein. The blend will contain 3x � 5y � 73 gmof carbohydrates and 2x � 3y � 46 gm of protein.

(b) At (11, 8); both dietary requirements are satisfiedif 11 ounces of Food I are mixed with 8 ounces ofFood II.

y

800

200

10

9

5

(62, 0)

(0, 248)

t

y

49. The two lines are not parallel; they do not have thesame slope.

51. (a)

(b) y � 60 � 5t, t � 0(c)

(d) 31.25 hours (31 hours, 15 minutes)

(0, 60)

(31.25, 216.25)

t

y

Hours rented 2 5 10 t

Total cost $70.00 $85.00 $110.00 60 � 5t

x

y

2

–2 2

3x + 5y = 73

2x + 3y = 46

x

y


53. Federal employment is increasing at approximately0.027 million employees per year.

55. The slope of L1 is m1 � and that of L2 is m2 � .

By hypothesis L1 L2 and OA � and OB � . AB � b � c. By the Pythagoreantheorem,

Therefore, � �1 and � �

Chapter 1, Section 41. A � 2w(500 � w)

3. P � x(18 � x)

5. R � x(35x � 15)

7. A � x(160 � x); 80 m by 80 m

9. V � x�1,000 �x2

2 �

(0, 0) (160, 0)

(80, 6400)

x

A

1

m2m1m1m2

�b

a��c

a� � m1m2

bc

a2 � �1

�2bc � 2a2

b2 � 2bc � c2 � 2a2 � b2 � c2

(a2 � b2) � (a2 � c2) � (b � c)2

�a2 � a2�a2 � b2�

c

a

b

a

11. V � r(60 � r2)

13. C � 0.08

15. R � kP; R � rate of population growth; P � size ofpopulation

17. R � k(T0 � Te); R � rate of temperature change; T0

� temperature of object; Te � temperature ofsurrounding medium

19. R � kP(T � P); R � rate of implication; P � numberof people already implicated; T � total number ofpeople involved

21. C � ; R � speed of truck

23. (a)

(b) Slopes are 0.15, 0.28, and 0.31

25. C � 4x2 �

27. V � x(18 � 2x)2; x � 3

(3, 432)

9

V

x3

1,000

x

(53500, 12107)

(22100, 3315)

115,000

y

x0

k1

R� k2R

�r2 �2

r�

f(x) � �0.15x

3,315 � 0.28(x � 22,100)

12,107 � 0.31(x � 53,500)

if 0 x � 22,100

if 22,100 x � 53,500

if 53,500 x � 115,000


29. P(x) � [200 � 20(15 � x)](x � 3)

The optimal selling price is $14.

31. D(t) �

�

33. Y � (60 � n)(400 � 4n); 80 trees

35. p � $40; q � 360 units

S( p)

D( p)

p

(40, 360)

40

100

y

y

n

(20, 25600)

(100, 0)

30�5t2 � 20t � 100�(60t)2 � (300 � 30t)2

x

y

500

3 14 25

37. (a) p � $80; q � 70 units(b)

(c) S(10) � 0; manufacturers will not supply anyunits unless the market price exceeds $10.

39. 2 hours, 45 minutes after the second plane leaves

41. R � (240 � 3t)(1.00 � 0.01t); 10 days

43. (a) a � 0, b 0, c 0, d � 0

(b) p �

(c) As a increases, p decreases (i.e., as supplyincreases, equilibrium price decreases). As dincreases, p increases (i.e., as demand increases,equilibrium price increases).

45. (a) 10 kayaks(b) 17 kayaks

47. (a) x 2,000 4,000 6,000 8,000

C(x) 85,200 96,200 107,200 118,200

d � b

a � c

R

t10 100

S( p)D( p)

p

(80, 70)

(10, 0)


(b) x 2,000 4,000 6,000 8,000

R(x) 39,000 78,000 117,000 156,000

(c) y � 5.5x � 74,200(d) y � 19.5x(e)

(f) (5,300, 103,350)(g) Approximately 4,360 books must be sold for a

revenue of $85,000. The profit is $13,160.

Chapter 1, Section 51. f(x) � b

3. f(x) � b

5. Limit does not exist.

7. 4

9. 7

11. 16

13.

15. Limit does not exist.

17. 2

19. 7

3

4

limxfi a

limxfi a

x

y

(5300, 103350)

21.

23. 5

25.

27.

29. f(x) � 15

f(x) � 0

31. x 1 0.1 0.01 0.001 0.0001

1,000(1 � 0.09x)1/x 1,090 1,093.73 1,094.13 1,094.17 1,094.17

1,000(1 � 0.09x)1/x � 1,094.17

33. (a) [10, 15] Answers will vary.(b) The growth rate is constant.(c) The growth rate begins to decrease at T � 45.

R(T) � 0

(d) Writing exercise; responses will vary.

35. For problem 7

3x2 2 5x 1 2 3.33 3.9303 3.9930

x 1.9 1.99 1.999

For problem 8

x3 2 2x2 1 x 2 3 �6.249 �6.9205 �6.9920

x �0.9 �0.99 �0.999

37. No

Chapter 1, Section 61. Yes

3. Yes

limTfi 50�

limxfi 0�

limxfi 3�

limxfi 3�

1

4

1

4

5

3


5. No

7. No

9. No

11. Yes

13. f(x) is continuous for all real numbers x.

15. f(x) is continuous for all real numbers except x � 2.

17. f(x) is continuous for all real numbers except x � �1.

19. f(x) is continuous for all real numbers except x � �3and x � 6.

21. f(x) is continuous for all real numbers except x � 0and x � 1.

23. f(x) is continuous for all real numbers x.

25. (a) W (20) � 3.7497W (50) � �7

(b) v � 25 mph [Note: at v � 99 mph, the wind chilltemperature is �7 at T � 30°F, which is out ofrange.]

(c) W is continuous at v � 4 or at v � 45 only when T � 91.4.

27. p(x) is discontinuous at 1, 2, 3, 4, and 5.

x

y

50

1 2 3 4 5 6

29. The graph is discontinuous when t � 6 and when t �12. The company could be restocking at those times.

31. A � �1

33. f(x) is continuous on the open interval 0 x 2, butf(x) is not continuous on the closed interval 0 � x � 2since it is not continuous at x � 2.

35. Let f(x) � . Since f iscontinuous at f(0) � 1 and f(1) � �1, there is a rootbetween 0 and 1.

37. At birth, weight in pounds will be less than height ininches. Later in life, weight in pounds will exceedheight in inches. Weight and height are both continu-ous measures and must therefore intersect at sometime.

39. Since height is a continuous measure and Nan hasgone from taller to shorter than her brother, theirheights must have been equal at some time during the16-year time period.

Chapter 1, Review1. (a) All real numbers x

(b) All real numbers x except x � 1 and x � �2(c) All real numbers x for which �x� � 3

2. (a) $45(b) $1(c) 9 months from now(d) P(x) → $40 as x → �

3. (a) g(h(x)) � x2 � 4x � 4

(b) g(h(x)) �

(c) g(h(x)) �

4. (a) f(x � 2) � x2 � 5x � 10

(b) f(x2 � 1) �

(c) f(x � 1) � f(x) � 2x � 1

�x2 � 1 �2

x2

��2x � 3

1

2x � 5

3�x � (x2 � 2x � 1)


5. Note: answers will vary.(a) g(u) � u5; h(x) � x2 � 3x � 4

(b) g(u) � (u � 1)2 � ; h(x) � 3x � 2

6. (a) Q[p(t)] �(b) � 4.93 units(c) 4 years from now

7. c � �4

8. (a)

(b) y

x(–.58, 0) (2.58, 0)

(0, 3)

y

x(–4, 0) (2, 0)

(0, –8)

�24.3�23.4 � 0.1t2

5

2u3

(c)

9. (a)

(b) E(p) � �50p(p � 16)(c) Optimal price � $8 per unit.

10. (a)

(b) 5 weeks(c) 20 weeks

(100, 20)

(50, 5)

t

D(p)

(0, 800)

(16, 0) p

y

x

(–1, )(1, 1)

12


11. (a) m � 3, b � 2

(b) m � , b � �5

(c) m � � , b � 0

(2, –3)

y

x

3

2

(4, 0)

(0, –5)

y

x

5

4

(– , 0)23

(0, 2)

y

x

(d) m � � , b � 8

12. (a) y � 5x � 4(b) y � �2x � 5

(c) y �

(d) y � �2x � 14

(e) y �

13. y � 3x � 1

14.

v

D

3

1

�3

5x �

12

5

2

9x �

2

3

(0, 8)

(12, 0)

y

x

2

3


15. (a) P(t) � 0.02t � 1.10

(b) $1.10 per gallon(c) $1.28 per gallon

16. (a) C(t) � 400t � 3,200(b) 5,200

17. (a) (3, �4)

(3, – 4)

y

x

(0, 3200)

C(t)

t

(5, 5200)

(0, 110)

(9, 128)

P(t)

t

(b) No intersection

(c) (1, 0) and (�1, 0)

(d) (3, 9) and (�5, 25)

(–5, 25)

(3, 9)

y

x

(–1, 0) (1, 0)

y

x

y

x


(e) (2, 6)

18. P(x) � (x � 80)(100 � x); optimal price � $65

19. P(x) � 2(100 � x)(x � 50); optimal price � $75

20. V(r) � 20r � r33

4

(72, 1250)

50 100

P(x)

x

(90, 100)

80 100

y

x

(2, 6)

y = 0

x = 0

y

x

21. For x machines C(x) � 80x �

minimum cost when x � 12

22. Discontinuous over intervals (9, 10), (15, 16), and (23, 24)

23. (a) 150 units(b) $1,500 profit(c) 180 units

24. A � �2; C � �4; S(p) � D(p) at p � 5; S(6) � D(6) � 13

25. Publisher A if n � 12,000 copies

26. R(x) � k(n � x) where n is the total number of rele-vant facts in the subject’s memory and x is the num-ber of facts that have been recalled

5

x

y

12

minimum1920

C

x

11,520

x


27. C(x) �

28. C(x) � 1,500 � 2x, C(x) is continuous for all realnumbers x for which 0 � x � 5,000

29. 16.36 minutes after 3 o’clock

30. (a) T � C � 38

(b) 185 chirps, 52.8°F

31. (a) � (b) (c) 0 (d) �12

32. y

x

3

2

2

3

1

5

(0, 1500)

x

C(x)

5�x2 � 810,000 � 4(3,000 � x) 33. (a) Not continuous for x 0(b) Not continuous for x � �3

(c) Not continuous for x � 2 and x � �

(d) Continuous for all real numbers x

34. (a) A � 6(b) A � 2

35. (a) After 9 minutes(b) Because f(1) � 10 0 and f(7) � 10 � 0 and

f(x) � 10 is continuous on [1, 7].

36. B � A(4,000)3

37. f(x) is undefined at x � 1 and x � �2.

x = –2 x = 1

y = 3

y

x

w

4,000 x

3

2


38. The two lines are not parallel; they do not have thesame slope.

39. (a) f (g(�1.28)) � 3.898 (b) g( f ( )) � 26.071

40.

41. The function is defined for all real numbers x exceptx � 1.

(4, 0)

(–3, 0)

(0, –10)

x

y

2

y

x

�2

(0, )8435

(0, )5410

y

x

Chapter 2, Section 11. f �(x) � 5, m � 5

3. f �(x) � 4x � 3, m � �3

5. g�(t) � � , m � �8

7. f �(x) � , m �

9. f �(x) � 2x � 1, y � 5x � 3

11. f �(x) � � , y � �48x � 36

13. � 0

15. f �(x) � 1 � 2x; f �(�1) � 3

17. (a) msec � 3.31(b) mtan � 3

19. f �(x) � 3x2 � 6x, so f �(0) � f �(�2) � 0; there existsa horizontal tangent line at (�2, 4) and (0, 0).

21. (a) P�(x) � �800x � 6,800(b) P�(8.5) � 0; profit is maximized at x � 8.5.

(–2, 4)

(–3, 0) (0, 0)

y

x

dy

dx

6

x3

1

6

1

2�x

2

t2


23. highest point; (0, 1)

25. t � � 0.449 sec. H � � 0.988 meters.

27. (a) f �(x) � 3(b) y � 3x � 2(c) The tangent to a line at a point on the line is the

original line itself.

29. (a) � 2x � 3

(b) � 2x; � 3

(c) The derivative in (a) is the sum of the derivativesin (b)

(d) f �(x) � g�(x) � h�(x)

31. The difference quotient DQ is the change in f dividedby the change in x. The derivative f � is the limit ofDQ as x → 0. With f � 0 and x � 0, this meansthat the graph of the function is rising. If f �(x) 0,the graph of the function falls.

dy

dx

dy

dx

dy

dx

242

245�22

49�22

49

x

y

(0, 1)

33.

35. See table at bottom of this page.

37. f(x) is not defined at x � 1. Therefore, there can be nopoint of tangency at x � 1.

x

y

x

y

–2 3

Dx �0.02 �0.01 �0.001 0 0.001 0.01 0.02

x 1 Dx 3.83 3.84 3.849 3.85 3.851 3.86 3.87

f(x) 4.37310 4.37310 4.37310 4.37310 4.37310 4.37310 4.37310

f(x 1 Dx) 4.35192 4.36251 4.37204 4.37310 4.37415 4.38368 4.39426

1.059 1.059 1.06 undefined 1.05 1.058 1.058f (x� x) � f (x)

x

Table for answer to problem 35



1. � �4x

3.

5. � 2x � 2

7. f �(x) � 9x8 � 40x7 � 1

9. � � �

11. f �(x) �

13.

15. � 5x4 � 18x2 � 14x

17. y � �9x � 4

19. y � � 15

21. y � 4x � 14

23. y � 3x � 3

25. f �(2) � 9

27. f �(1) � �

29. f �(x) � 0 at lowest point; f �(x) � 2x � 4 � 0 when x � 2; (2, �9) is the lowest point.

3

2

�11

2x

dy

dx

dy

dx� �

x

8�

2

x2 �3

2x1/2 �

2

3x3 �1

3

3

2 �x �

3

2�x5

1

2�t3

2

t3�1

t2dy

dt

dy

dx

dy

dt� �

9

2�t3

�5dy

dx

31. a � ; b � �

33. y � 6x and y � �14x

35. (a) E(p) � p(�200p � 12,000)

(b) p � $30; E(30) � 180,000

37. (a) C�(t) � 200t � 400(b) C�(5) � 1400; increasing(c) 1,500 issues

39. (a) f �(x) � �3x2 � 12x � 15(b) f �(1) � 24; 24 radios/hr(c) 26 radios

41. D�(t) � 100 km/hr

p

E(p)

60

180,000

30

16

3

8

9

x

y

–22

(2, –9)


43. P�(x) � 2 � (a) P�(9) � 20 persons per month(b) 0.39%

45. (a) T�(6) � $280 per year(b) 17.95% per year

47. (a) f(t) �

(b) 7.69%(c) The percentage rate of change approaches zero.

49. (a) v(t) � 2t � 2; a(t) � 2(b) t � 1

51. (a) v(t) � 3t2 � 18t � 15; a(t) � 6t � 18(b) t � 1, t � 5

53. (a) v(t) � 8t3 � 6t � 36; a(t) � 24t2 � 6(b) t � 1.5

55. H(t) � �16t2 � 144H�(t) � �32t(a) H(t) � 0; t � 3(b) H�(3) � �96 ft/sec

57. Mars

59. If y � mx � b, then � m.

61. (a) The rate of change of cost with respect to the

number of units produced; dollars

units

dy

dx

x

y

0.1

1

11 � t

6x1/2 (b) The rate of change of units produced with respect

to time;

(c) The rate of change of cost with respect to time;

63. Answers will vary.

Chapter 2, Section 31. f �(x) � 12x � 1

3. � �300u � 20

5. f �(x) � (5x4 � 6x2)

7. � �

9. f �(t) �

11. � �

13. f �(x) �

15. f �(x) �

17. y � 17x � 4

19. y � 3x � 2

21. 70.5

23.13

64

2(x2 � 2x � 4)

(x � 1)2

11x2 � 10x � 7

(2x2 � 5x � 1)2

3

(x � 5)2

dy

dx

�(t2 � 2)

(t2 � 2)2

3

(x � 2)2

dy

dx

1

3

dy

du

dollars

hour

units

hour


25. y � � x � 5

27. y � x �

29. (a) and (b) � 4x � 5

31. t � hours; max. pop. � 18,000

33. 66.67%

35. (a) P�(t) � thousand per year

(b) 1,500 per year(c) 1,000(d) 60 per year(e) The rate of growth approaches zero.

37. (a) N�(3) � 108 people per week(b) This disease does not reach epidemic proportions

during the 8-week period for which this equationis accurate.


39. (a) 1946; 1974(b) 1943; 12.5 (Answers may vary.)

(c) 5.8% (Answers may vary.)

41. (a)

(b) � �24x2 � 44x � 7dy

dx

fgdh

dx� fh

dg

dx� gh

df

dx

f �gdh

dx� h

dg

dx � (gh)df

dx�

d

dx( fgh) �

d

dx( f(gh)) � f

d

dx(gh) � (gh)

df

dx�

6

(t � 1)2

2

3

dy

dx

1

2

1

2

1

343.

, since c is a constant

45. According to the power rule with n � �p, (x�p) �

�px�p�1. Applying the quotient rule to y � x�p �

shows that

� �pxp�1�2p

� �px�p�1

47.

x

y

–2 –1 1

2

–3

2

(–2.633, –19.798)

(.633, –.202)

�xp � 0 � 1 � pxp�1

x2p

dy

dx

1

xp

d

dx

� c �df

dx

� c �df

dx� f � 0

� c �df

dx� f �

dc

dx

d

dxcf �


49. Minima (3, 0); maxima (1.8, 8.4); f �(x) �x2(x � 3)(5x � 9)

Since f(x) is a polynomial function and therefore hassmooth turns at low and high points, there will be hor-izontal tangents at these points, in particular, x � 0, x � 3, and x � 9/5. Thus the first derivative will beequal to zero at these locations and have x interceptsthere. Note that f(x) does not have to have a high orlow point when f�(x) � 0.

Chapter 2, Section 41. (a) C�(3) � $499.70

(b) C(4) � C(3) � $500.20

3. (a) C�(x) � x � 4, R�(x) � 9 �

(b) C�(3) � $5.20

x

2

2

5

x

yy = f '(x)

(0, 0) 1.8 3

(2.5, –11)

(1.1, 8)

(b)

x

y

(1.8, 8,4)

y = f (x)

(0, 0) (3, 0)

(a)

(c) C(4) � C(3) � $5.40(d) R�(3) � $7.50(e) R(4) � R(3) � $7.25

5. (a) C�(x) � x � 2, R�(x) � �3x2 � 8x � 10

(b) C�(3) � $4

(c) C(4) � C(3) � � $4.33

(d) R�(3) � $7(e) R(4) � R(3) � $1

7. (a) C�(x) � , R�(x) �

(b) C�(3) � $1.50(c) C(4) � C(3) � $1.75

(d) R�(3) � � $2.06

(e) R(4) � R(3) � $2.05

9. 2.1

11. 20.6%

13. Cost will increase by approximately $50.08.

15. 0.5

17. 6%

19. f � 6 radios

21. Q � �12,000 units

23. (a) $241 (b) $244

25. P � 1,500 people

27. The area is accurate to within 2.26%.

29. 100 � 0.67% Q

Q

33

16

2x2 � 4x � 3

(1 � x)2

x

2

13

3

2

3


31. S � 0.08r2, V � 0.04r3

33. 28.37 cubic inches

35. (a) The volume of blood increases by approximately20%.

(b) Writing exercise; responses will vary.

37. 8% increase

39. (a)

(b) 1.4656(c) They are essentially equal; the accuracy would

depend on the rounding.

41. (a) , f (x) � x2 � N, f �(x) � 2x


�1,265 � 35.5668

�1

2�xn �N

xn�

�1

2�x2n � N

xn�

�2x2

n � x2n � N

2xn

xn�1 � xn �x2

n � N

2xn

x � �N

x

y

(1.46557, 0)


1. � 6(3x � 2)

3. �

5. � �

7. � �

9. �

11. �160

13.

15. �16

17. f �(x) � 8(2x � 1)3

19. f �(x) � 8x2(x5 � 4x3 � 7)7(5x2 � 12)

21. f �(t) �

23. g�(x) �

25. f �(x) �

27. h�(s) �

29. f �(x) � (x � 2)2(2x � 1)4(16x � 17)

15(1 � �3s)4

2�3s

24x

(1 � x2)5

�4x

(4x2 � 1)3/2

�2(5t � 3)

(5t2 � 6t � 2)2

2

3

�2x

(x2 � 1)2

dy

dx

x

(x2 � 9)3/2

dy

dx

4x

(x2 � 1)3

dy

dx

x � 1

�x2 � 2x � 3

dy

dx

dy

dx


31. G�(x) � � (3x � 1)�1/2(2x � 1)�3/2

33. f �(x) �

35. f �(y) �

37. y � �48x � 32

39. y � �12x � 13

41. x � 0, x � �1, x � �

43. x � �

45. x � 2

47. f �(x) � 6(3x � 5)

49. (a) $2,280 per year(b) 10.3% per year

51. Demand will be decreasing by six blenders permonth.

53. Demand will be decreasing by 12%.

55. E� � [0.074v2 � 122.65]

57. (a) ; F decreases as C increases

(b) %

59. (a)2

x

50

A � C

dF

dC� �

kD2

2�A � C

1

v2

2

3

1

2

5 � 6y

(1 � 4y)3/2

(x � 1)4(9 � x)

(1 � x)5

5

2(b) �

(c) �

(d)

61. y � h(x)[h(x)]2

� h(x) � 2h(x) � h�(x) � [h(x)]2 � h�(x)

� 2[h(x)]2 � h�(x) � [h(x)]2 � h�(x)

� 3[h(x)]2h�(x)

63. f �(0) does not exist; f �(4.3) � 16.626; f(x) has onlyone horizontal tangent line at x � 0.51.

Chapter 2, Section 61. f �(x) � 450x8 � 120x3

3.

5. f �(x) � 180(3x � 1)3

7. � 80(t2 � 5)6(3t2 � 1)

9. f �(x) � (1 � x2)�3/2

11.

13. f �(x) � 16(2x � 1)2(5x � 1)

15.

17. (a) v(t) � 15t4 � 15t2, a(t) � 60t3 � 30t

(b) a(t) � 0 when t � 0, �2

2

d 2y

dt2�

2(1 � 2t)

(t � 1)4

d 2z

dx2 �4(3x2 � 1)

(1 � x2)3

d 2h

dt2

d 2y

dx2 � �5

4x3/2 �18

x4 �1

4x5/2

dy

dx

3

(2x � 1)(1 � x)

1

2x

1

x


19. (a) v(t) � �3(1 � t)2 � 4(2t � 1)a(t) � 6(1 � t) � 8

(b) a(t) � 0 when t �

21. (a) 28 units per hour(b) 10 units per hour per hour(c) 2.5 units per hour(d) 2.3125 units per hour

23. (a) 75,000 per year(b) 0 per year(c) Decrease of 500 people(d) Decrease of 521 people

25. (a) a(t) � �

(b) Speed is decreasing by kilometers per hour per hour.

(c) Speed decreases by 2 kilometers per hour.

27. (a) p�(x) �

(b) p�(x) �

� 0 when x �


29. f 4(x) � 120x � 48

31. f �(x) � � 48x�5

33. a(t) � (7t � 5)(4 � t)2(71 � 35t) 0 � t � 5

y = a(t)

y = v(t)

04

557

7135

t

�15

8�3x�7/2

m��m � 1

m � 1�B

mAxm�1[�B � Bm � (1 � m)xm]

(B � xm)3

AB � Axm(1 � m)

(B � xm)2

11

3

4

3t

20

3

7

3

(a) a(t) � 0 on (0.71, 2.03)a(t) 0 on (0, 0.71) and on (2.03, 4) and (4, 5)

(b) maximum velocity at t � 0(c) maximum acceleration at t � 1.22;

v(1.22) � 272.14(d) 774.27 � (�5680) � 6454.27


1.

3.

5.

7. � 2

9.

11. y �

13. y � � � 2

15. y �

17. (a) None(b) (9, 0)

19. (a) None(b) None

21. (a) (1, �2), (�1, 2)(b) (�2, 1), (2, �1)

13

12x �

11

12

1

2x

1

3x �

4

3

dy

dx�

y � 5x(x2 � 3y2)4

15y(x2 � 3y2)4 � x

dy

dx�

1

3(2x � y)2

dy

dx�

3 � 2y2

2y(1 � 2x)

dy

dx�

y � 3x2

3y2 � x

dy

dx� �

x

y


23. (a)

(b)

25. (a)

(b)

27. (a)

(b)

29. y � �1.704

31. � 1.74 units/month

33. � 20 mm/min

35. � 15.419 units per month

37. � 75 mph

39. � �$400 per week

41. 4 feet per second

43. � (�1.68 � 10�5) mm/min

45. y � 0.057 units

K

L

dv

dt

dK

dt

ds

dt

dx

dt

dr

dt

dx

dt

dy

dx� �

3

(x � 1)2

dy

dx�

y � 1

1 � x�

�3

(x � 1)2

dy

dx�

x(x � 4)

(x � 2)2

dy

dx�

2x � y

x � 2�

x2 � 4x

(x � 2)2

dy

dx� �

3

(x � 2)2

dy

dx� �

y

x � 2 47. � 1; � 0; 2b2x � 2a2yy� � 0,

y� . At P(x0, y0), m � �

so the equation of the tangent line is

y � y0 � � (x � x0)

a2yy0 � a2y20 � �b2xx0 � b2x0

2

b2xx0 � a2yy0 � b2x02 � a2

2y02

which must equal 1 since P(x0, y0) lies on the curveand thus satisfies the equation of the curve.

49. ft/sec2

51. Let y � , then ys � xr and � ,

. But ys�1 � , so

�

�

�r

s� xr/s�1

r

s� xr�1�r/s�r

r

s� xr�1 �

xr/s

xr

dy

dx

ys

xr/s �xr

yr/s

dy

dx�

rxr�1

sys�1

rxr�1sys�1 dy

dxxr/s

8

5

x0x

a2 �y0y

b2 �x0

2

a2 �y0

2

b2

b2x0

a2y0

b2x0

a2y0� �

2b2x

2a2y� �

b2x

a2y

2x

a2 �2yy�

b2

x2

a2 �y2

b2


53. Two horizontal tangent lines at (.226, �1.241) and(�.226, 1.241).

55. Horizontal tangent lines at (.84, 1.23) and (.84, �1.23)

Chapter 2, Review1. (a) f �(x) � 2x � 3

(b) f �(x) �

2. (a) f �(x) � 24x3 � 21x2 � 2

(b) f �(x) � 3x2 �5

3x6 �1

�x�

3

x2 �4

x3 �3

x4

1

(x � 2)2

x

y

.5

.5

y � 1.23

y � �1.23

y � 1.241

y � �1.241

x

y

(c)

(d) � 2(2x � 5)2(x � 1)(5x � 8)

(e) f �(x) � 20(5x4 � 3x2 � 2x � 1)9(10x3 � 3x � 1)

(f) f �(x) �

(g)

(h)

(i) f �(x) �

(j) f �(x) �

(k)

3. (a) y � �x � 1 (b) y � �x � 1

(c) y � x (d) y � � x �

4. (a) 31 (b) 0

5. (a)

(b)

(c)

6. (a) � 3(30x � 11) (b) � �

7. (a) 2 (b)3

2

4

(2x � 3)3

dy

dx

dy

dx

�100

t � 1

1002(3 � 5t)

t(3 � 2t)

50(4t3/2 � 6t1/2 � 1)

t5/2 � 3t3/2 � t

5

3

2

3

dy

dx�

�7

2�(1 � 2x)(3x � 2)3

3(3x � 1)2(3x � 7)

(1 � 3x)5

9(3x � 2)

�6x � 5

dy

dx�

4(x � 1)

(1 � x)3

dy

dx� 2�x �

1

x��1 �1

x2� �5

2x�3x

x

�x2 � 1

dy

dx

dy

dx�

�14x

(3x2 � 1)2


8. (a) f �(x) � 120x3 � 24x � 10 �

(b) � 24(3x2 � 2)2(21x2 � 2)

(c) f �(x) �

9. (a) (b)

(c)

(d)

10. (a) � (b) �28

11.

12. (a) t � 10 sec(b) v(10) � �160 ft/sec(c) t � 5 sec, S(5) � 400 ft

13. Cost is increasing at a rate of $1,663.20 per hour.

14. (a) v(t) � 6t2 � 42t � 60 � 6(t � 5)(t � 2), and a(t) � 12t � 42 � 6(2t � 7). The objectadvances when 1 t 2 and when 5 t 6.The object retreats when 2 t 5. The object isdecelerating

when t and accelerating when t � .

(b) 49

15. (a) v(t) � ,

a(t) � . The object is

advancing and decelerating when 0 t 3.

(b)1

4

2(2t3 � 3t2 � 72t � 12)

(t2 � 12)3

�2(t � 4)(t � 3)

(t2 � 12)2

7

2

7

2

d 2y

dx2 �6y2 � 9x2

4y3 ��9

2y3

2

3

dy

dx� �

1 � 10y3(1 � 2xy3)4

4 � 30xy2(1 � 2xy3)4

dy

dx�

1 � 10(2x � 3y)4

15(2x � 3y)4

dy

dx� �

2y

x

dy

dx� �

5

3

2(x � 5)

(x � 1)4

d2y

dx2

2

x3

16. (a) Use was increasing by 1,652 people per week.(b) Use increased by 1,514 people per week.

17. (a) Output will increase by approximately 12,000 units.

(b) Output will increase by 12,050 units.

18. Population will be increasing by 0.3% per month.

19. Output will decrease by approximately 5,000 units per day.

20. 1.35%

21. Pollution will increase by approximately 10%.

22. Labor should be increased by approximately 1.5%.

23. Decrease input y by approximately 1.7 units.

24. 240 feet

25. 16.27%

26. (a) 27 units per hour(b) 12 units per hour per hour(c) Production will be increased by approximately

1.2 units per hour.(d) Production increase by 1.17 units per hour.

27. 5.5 seconds; 242 feet

28. �2.16 centimeters per second

29.

� 2(0.03) � 0.066% accuracy

30. �

31. (a) 1980; approximately 14%(b) 1972–1974; 1980–1982(c) Writing exercise; responses will vary.

C2dP

dt �3t2

T 2 �2t3

T 3�dV

dt� [C1 � C2P(t)]� 6t

T 2 �6t2

T3 �

�dA

A � � �2rdr

r2 � � 2�dr

r �


32. The lantern is 10 feet above ground when t � 2.96sec. At that time, the shadow is L � 14.8-feet longand is lengthening at the rate of 285 ft/sec.

33. Distance is decreasing by 3.288 feet per second.

34. (a) t � 6.65 sec(b) v(6.65) � �125.8 ft/sec(c) t � 2.72 sec, h(2.72) � 247.27 ft

35. f �(x) � 24x3 � 30x2 � 30x � 13 � 0, when x ��1.78, �0.35, 0.88

36. f �(x) � is never 0.

37. Tangents:

y � 0.96x � 0.192 at �1

2,

�3

6 �

y

x

y = f (x)

y = f '(x)13

3

–1.5

23

–

11

(1 � 3x)2

y

x

y = f (x)

y = f '(x)

0.88–1.2

0.35–0.35–1.78

y � �0.96x � 0.192 at

Vertical asymptote at x � 2. Horizontal tangent at theorigin.

38. (a) for 0 � t � 2v(t) � t3/2(3.29t2 � 10.85t � 6.75)a(t) � t1/2(11.5t2 � 27.13t � 10.13)

(b) v(t) � 0 when t � 0.83, and s(0.83) � 0.4(c) amin occurs at t � 1.28;

s(1.28) � �0.128, v(1.28) � �2.511, a(1.28) � �6.514.

Chapter 3, Section 11. f �(x) � 0 for �2 x 2; f �(x) 0 for x �2 and

x � 2

3. f �(x) � 0 for x �4 and 0 x 2; f �(x) 0 for �4 x �2, �2 x 0, and x � 2

5. f(x) is increasing for x � 2; f(x) is decreasing for x 2

7. f(x) is increasing for x �1 and x � 1; f(x) isdecreasing for �1 x 1

2.2

0.770.4

–6.4

0.22

1.361

2

y

t

y = a(t)

y = s(t)y = v(t)

0.47 0.83 1.27 1.89

0 2

12

�1

2,

��3

6 �


9. g(t) is increasing for t 0 and t � 4; g(t) is decreas-ing for 0 t 4

11. (0, 2) relative minimum; (1, 3) neither

13. (�1, 3) neither

15. (1, 0) neither

17. (�1, 0) neither; (0, �1) relative minimum; (1, 0) neither

19. (0, 1) neither, (1, 0) relative minimum

21.

23.

x

y

(–.5, .19)

2

–2

(.6, 4.3)

(1.9, –2.4)

x

y

2

1

–2

(2, –4)

25.

27.

29.

31. Critical Numbers Classification

�5 Relative maximum

0 Relative minimum

Neither72

x

y

(–1, 6)

(1, 2)2

4

6

1–1

x

y

(–1, 3)(0, 5)

y

t


33.

Maximum of 85.81% at 23.58°C.

35. R(x) � x(10 � 3x)2; � (10 � 3x)(10 � 9x)

Revenue is maximized when x � units.

37. Maximum concentration occurs when t � 0.9 hours.

(0.9, 0.083)

y

t

( , 49.38)109

109

103

dRdx

R(x)

y

x

10

9

dR

dx

(30, 64)

(15, 46.75)

(23.58, 85.81)H(t)

t

39.

41.

43. a � � , b � , c � 3

45.

47. y � ax2 � bx � c, � 2ax � b

� 0 when 2ax � b

� 0 ⇒ 2ax � �b ⇒ x � �b

2a

dy

dx

dy

dx

(0, 0)

y

x

18

5

9

25

(–5, 4)

(1, –1)

y

x

1 20


49. f �(x) � 0 when x � 0, 1.276, and �3.526

51.

f �(x) � 0 at x � �2.2, 0, 0.19, and 2

53. (a) is the graph of f(x); (b) is the graph of f�(x). Thekey is that the graph in (a) is falling (rising) wherethe graph in (b) is below (above) the x-axis.

x

y

y = f (x)

y = f �(x)

(0, 4)

(1, –5)

(0, 0)–5

5

y

x

y = f'(x)

y = f(x)

55. h(x) � (x � 3)3 � 3(x � 3)2 � 5(x � 3) � 13

57.

Chapter 3, Section 21. f �(x) � 0 for x � 2, f �(x) 0 for x 2

3. Increasing at x �3 and x � 3; decreasing at �3 x 3; concave upward at x � 0; concavedownward at x 0. Maximum at (�3, 20); minimumat (3, �16); inflection at (0, 2).

inf(0, 2)

(–3, 20)

(3, –16)

y

x

f (x)g(x)

(0, –11)

The graph of gis obtained bycompressingthe graph of f.

y

x

f (x)

g(x)

(0, 11)

y

x


5. Increasing at x � 3; decreasing at x 3; concave up-ward at x 0 and x � 2; concave downward at 0 x 2. Minimum at (3, �17); inflection at (0, 10)and (2, �6).

7. Increasing at all x; concave upward at x � 2; concavedownward at x 2. Inflection at (2, 0).

9. Increasing at x � 0; decreasing at x 0; concave up-ward at , �1 x 1, x � ; concavedownward at � x �1 and 1 x .Minimum at (0, �125); inflection points at (� , 0), ( , 0), (�1, �64), and (1, �64).

inf(–1, –64)

(0, –125)

y

xinf

(–√5, 0)inf

(√5, 0)inf

(1, –64)

�5�5

�5�5�5x ��5

(0, –8)

inf(2, 0)

y

x

inf(0, 10)

(3, –17)

inf(2, –6)

y

x

11. Increasing at x � �1; decreasing at x �1; concaveupward at x �4 and x � �2; concave downwardat �4 x �2. Minimum at (�1, �54); inflectionat (�4, 0) and (�2, �32).

13. Increasing at all real x; concave upward at x �1;concave downward at x � �1; inflection at (�1, 0).

15. Increasing at x � �1; decreasing at x �1; concaveupward at all real x. Minimum at (�1, 0).

(–1, 0)

(0, 1)

y

x

inf(–1, 0) (0, 1)

y

x

(–1, –54)

y

x

inf(–4, 0)

inf(–2, –32)


17. Increasing at x � 0; decreasing at x 0; concave up-ward at all real x. Minimum at (0, 1).

19. Increasing for �1 x 1; decreasing for x �1and x � 1; concave up for �1.53 x �0.35 and x � 1.88; concave down for x �1.53

and �0.35 x 1.88. Maximum at ; minimum

at (�1, �1); inflection at (�1.53, �0.84), (�0.35,�0.49), and (1.88, 0.29).

21. f �(x) � 6(x � 1); maximum at (�2, 5); minimum at(0, 1)

23. f �(x) � 12(x2 � 3); maximum at (0, 81); minimum at(3, 0) and (�3, 0)

25. f �(x) � ; maximum at (�3, �11); minimum at

(3, 13)

27. f �(x) � 12x2 � 60x � 50; maximum at ;

minimum at (0, 0) and (5, 0)

�5

2,

625

16 �

36

x3

13

x

y

inf.(–1.5, –0.8)

(–1, –1)

inf.(–0.4, –0.5)

inf.(1.9, 0.3)

(1, )

�1, 1

3�

(0, 1)

y

x

29. f �(t) � ; maximum at (0, 2)

31. f �(x) � ; maximum at (�4, �13.5). Test

fails for x � 2 (there is an inflection point at (2, 0)).

33. (a) Increasing for x 0 and x � 4; decreasing for 0 x 4

(b) Concave upward for x � 2 and concavedownward for x 2

(c) Relative minimum at x � 4, relative maximum atx � 0; inflection point at x � 2.

(d)

35. (a) Increasing at � x ; decreasing at x �and x �

(b) Concave upward for x 0 and concavedownward for x � 0

(c) Relative maximum at x � and relative mini-mum at x � � ; inflection point at x � 0

(d)

x

y

–√5 √5–1 1

�5�5

�5�5�5�5

x

y

2 4

24(x � 2)

x4

4(3t2 � 1)

(1 � t2)3


37. A typical graph is shown.

39. f(x) is increasing for x � 2.f(x) is decreasing for x 2.f(x) is concave upward for all real x.f(x) has a relative minimum at x � 2.f(x) has no inflection points.

41. f(x) is increasing for x � 2.f(x) is decreasing for x 2.f(x) is concave upward for x �3 and x � �1.f(x) is concave downward for �3 x �1.f(x) has a relative minimum at x � 2.f(x) has inflection points at x � �3 and x � �1.

x

y

–3 2–1

x2

y

–1

2 3

y

x

43. C(x) � 0.3x3 � 5x2 � 28x � 200(a) M(x) � C�(x) � 0.9x2 � 10x � 28

(b) Only inflection number of C(x) is x � 5.56. Itcorresponds to a minimum on the graph of M(x).

45. Output rate is Q�(t) � �3t2 � 9t � 15.(a) Rate is maximized at t � 1.5 (9:30 A.M.)(b) Rate is minimized at t � 4 (noon).

47. Rate of growth is P�(t) � �3t2 � 18t � 48.(a) Rate is largest when t � 3 years.(b) Rate is smallest when t � 0 years.(c) Rate of growth of P�(t) is P �(t) � �6t � 18,

which is largest when t � 0.

49. � A � BP(t); P�(t) � ;

P�(t) � [P(t)(A � BP(t))(A � 2BP(t))]; P�(t) �

0 when P(t) � or P(t) � . These are inflection

points. P(t) is changing most rapidly at these points.

51. y � ax2 � bx � c; � 2ax � b; � 2a. The

function y is concave upward if and only if � 0;

that is, a � 0. It is concave downward only when

a 0.

d2y

dx2

d2y

dx2

dy

dx

A

2B

A

B

1

1002

P(t)(A � BP(t))

100

100P�(t)

P(t)

C(x)

M(x)

200

28

5.56 x


53. f(x) � 3.7x4 � 5.03x3 � 2x2 � 0.7(a)

(b) f(x) � 3.7x4 � 5.03x3 � 2x2 � 0.7; f�(x) �14.8x3 � 15.09x2 � 4x; f �(x) � 44.4x2 � 30.18x� 4

c f(c) f9(c) f 0(c)

�4 1,300.42 �1,204.64 835.12

�2 106.74 �186.76 241.96

�1 10.03 �33.89 78.58

0 �0.7 0 4

1 �0.03 3.71 18.22

2 26.26 66.04 121.24

(c) x intercepts at approximately (�0.4, 0) and (1, 0)(d) Relative minimum at (0, �0.7)(e) f(x) is increasing ( f�(x) � 0) for x � 0.(f) f(x) is decreasing ( f�(x) 0) for x 0.(g) Inflection points at (0.50, �0.60) and (0.18,

�0.66)(h) f(x) is concave upward ( f �(x) � 0) for x 0.18

and x � 0.5.( i ) f(x) is concave downward ( f �(x) 0) for

0.18 x 0.5.( j ) Answers will vary.(k) Absolute maximum at (�4, 1300); absolute mini-

mum at (0, �0.7)

Chapter 3, Section 31. ��

3. ��

(0, �0.7)(0.18, �0.66)

(0.5, �0.6)

y

x

5.

7. 0

9. ��

11. Vertical asymptote, x � 0; horizontal asymptote, y � 0

13. Vertical asymptote, none; horizontal asymptotes are y � �1, y � 1

15. Vertical asymptote, x � �2, x � 2; horizontalasymptote, y � 1

17. Vertical asymptote, x � �2; horizontal asymptote, y � 3

19. Vertical asymptote, x � �1, x � 1; horizontalasymptote, y � 1.

21. Vertical asymptote, t � 2, t � 3; horizontalasymptote, y � 1.

23. Vertical asymptote, x � 0, x � 1; horizontal asymp-tote, y � 0.

25.

x

y

–3 –1 1

(0, –2)

inf.(–1, 0)

(–2, 2)

–2

1

2


27.

29.

31.

x

yx = –1.5

( , 0)x

y

25

inf.x = –1

inf.x = 1.5

(–2, –125)

(2.25, –48.2)

12

x

y

inf.x = –0.4inf.

x = –1.6

(–2, 0)

(–1, 0)

(0, 0)

33.

35.

37.

x

y

2

inf.x = –0.58

inf.x = 0.58

(0, –9)

x

y

(0, –0.11)

x � �3 x � 3

x

y


39. Answers may vary.


43. (a) f(x) is increasing for x �3 and x � 0; f(x) is de-creasing for �3 x 0.

(b) Relative minimum at x � 0; relative maximum atx � �3

45. B � � ; A � �10

x

y

2

y � 4

x � 2

5

2

x

y

x

y


49. p � $40

51. (a) I�(S) � ; I�(S) � so the

graph is always rising and is always concave down.


53. Relevant portion is 0 � x � 100.

x

y

300

100 300

x

y

5

4

3

2

1

00 100 200 300 400 500

�2ac

(S � c)3 0ac

(S � c)2 � 0

x

y


55. The cost is minimized when x � 200.

57. (a)

(b)

P�(Vc) � 0, therefore

and .

Then P �(Vc) � 0, so

and

Thus,

and solving, we get .

(c) Pc �nRTc

2b�

a

9b2 , Tc �8a

27nRb

Vc � 3b

2(Vc � b)2

Vc3 �

3(Vc � b)3

V c4

nRTc

a�

3(Vc � b)3

V c4

2nRTc

(Vc � b)3 �6a

V4c

nRTc

a�

2(Vc � b)2

V c3

nRTc

(Vc � b)2 �2a

V c3

P�(V) �2nRTc

(V � b)3 �6a

V 4

P�(V) � �nRTc

(V � b)2 �2a

V3

P(V) �nRTc

V � b�

a

V2

�P �a

V2�(V � b) � nRT

x

y

800

200

(200, 800)

59. (a) f�(x) � , so f(x) is decreasing for x 1 and

increasing for x � 1. There is a relative minimum where x � 1, at (1, �3).

(b) f �(x) � satisfies f �(x) � 0 for x �2 and

x � 0 (graph is concave up) and f �(x) 0 for �2 x 0 (graph is concave down). There areinflection points at x � �2 and x � 0.

(c) x intercepts at x � 0 and x � 4; y intercept at theorigin; no asymptotes

(d)

61. (a) f�(x) � . f(x) is decreasing on

�16.7 x �5.6 and �5.6 x �2.1. f(x) is increasing on �� x �16.7 and �2.1 x 4.5 and 4.5 x. f(x) has a relativemaximum at (�16.68, 0.031) and a relative mini-mum at (�2.12, 0.32)

(b) f �(x) � .

The graph of f(x) is concave downward on �24.3 x �5.6 and x � 4.5, and is concaveupward for x �24.3 and �5.6 x 4.5.There is an inflection point at (�24.3, 0.028).

(c) x intercept at x � �9.4; y intercept at y � 0.4; thegraph has vertical asymptotes at x � �5.6 and x � 4.5, and a horizontal asymptote at y � 0.

2(x3 � 28.2x2 � 106.02x � 273.874)

(25 � 1.1x � x2)3

x2 � 18.8x � 35.34

(25 � 1.1x � x2)2

x

y

4

(1, –3)

inf.x = –2

inf.x = 0

4(x � 2)

9x5/3

4(x � 1)

3x2/3


(d)

Chapter 3, Section 41. Absolute maximum at (1, 10); absolute minimum at

(�2, 1)

3. Absolute maximum at (0, 2); absolute minimum at

5. Absolute maximum at (�1, 2); absolute minimum at(�2, �56)

7. Absolute maximum at (�3, 3125); absoluteminimum at (0, �1024)

9. Absolute maximum at ; absolute minimum at

(1, 2)

11. Absolute minimum at (1, 2)

13. f(x) has no absolute maximum or minimum for x � 0.

15. Absolute maximum at (0, 1).

17. (a) R(q) � 49q � q2; R�(q) � 49 � 2q; C�(q) � q � 4;

P(q) � q2 � 45q � 200;

P�(q) � q � 45 � 0 when q � 20�9

4

�9

8

1

4

�3, 10

3 �

�2, �40

3 �

(b) A(q) � q � 4 � ; A(q) is minimized at

q � 40.

19. (a) R(q) � 180q � 2q2; R�(q) � 180 � 4q; C�(q) �3q2 � 5; P(q) � �q3 � 2q2 � 175q � 162; P(q)is maximized at q � 7.

C�(q)

R�(q)

P(q)

(0, 5)

(7, 152)

(0, –162)

(7, 622)

(0, 180)

7 q

(0, 4)

C�(q)

A(q)

(40, 14)

40 q

200

q

1

8

(0, 49)

(0, 4)(20, 9)

(20, 250)

C�(q)

R�(q)

P(q)

q

x

y

(–16.7, 0.03)

–2 2 4

(–2.12, 0.32)inf.x = –24

x = –5.58 x = 4.48


(b) A(q) � q2 � 5 � ; A(q) is minimized at

q � 4.327

21. (a) R(q) � 1.0625q � 0.0025q2;

R�(q) � 1.0625 � 0.005q; C�(q) � ;

P(q) � ;

P�(q) � ;

profit P(q) is maximized when P�(q) � 0; when q � 17.3 units.

y = C�(q)

y = R�(q)

y = P(q)

q

y

17.3

�0.005q3 � 0.033q2 � 0.33q � 10.56

(q � 3)2

�0.0025q3 � 0.055q2 � 3.1875q � 1

(q � 3)

q2 � 6q � 1

(q � 3)2

(0, 5)

C�(q)A(q)

4.327 x

162

q (b) A(q) � is minimized when q �

� 1.3874.

23. R�(q) � �4q � 68; A(q) � � �2q � 68 �

(a) R�(q) � A(q) when q � 8

(b) A increasing (A� � 0) if 0 q 8 and

decreasing (A� 0) if q � 8

(c)

25. The slope f �(x) � 4x � x2 has its largest absolute

value when x � �1. The graph is steepest at .

The slope of the tangent is �5.

27. (a) x � 4 or in 1982. The membership in 1982 was46,400.

(b) x � 11 or in 1989. The membership in 1989 was12,100.

��1, 7

3�

y � A(q)

y � R�(q)68

2 17 32 q

y

8

128

q

R(q)

q

y = C�(q)y = A(q)

q

y

1

1.4

1 � �10

3

q2 � 1

q(q � 3)


29. v �

31. The speed of the blood is greatest when r � 0, i.e., atthe central axis.

33. (a) v � 40.7 km/hr(b) Writing exercise; responses will vary.

35. Sensitivity is greatest when D � C.

37. R � r

39. V(T ) � �6.8 � 10�8T3 � 8.5 � 10�6T2

�6.4 � 10�5T � 1(a) V(T ) is minimized at T � 3.95, V(3.95) �

0.999876


41. A(r) � � [(1 � r2)2 � kr2]�1,

A�(r) � �2r[(1 � r2)2 � r2]�2[�2(1 � r2) � k] � 0,

when 2r2 � 2 � k, or r1 � , (k � 2).

It can be shown that since

k � 2, and since r1 is the only critical number for r �0, it must correspond to an absolute maximum by thesecond derivative test.

43. (a) Since q � qc satisfies C�(qc) � ,C(qc)

qc

A(r1) �64(k � 2)

k2(k � 4)2 0

�2 � k

2

1

(1 � r2)2 � kr2

3.95

V

T

4� w2

3�2AScomputation shows that A(qc) � . So,

A(qc) � 0 if and only if C (qc) � 0.

(b) Stated condition means C (q) � 0 for sufficientlylarge q. Thus, A(qc) � 0 in part (a) and q � qc isa minimum by the second derivative test.


1.

3. x � 25, y � 25

5. $40.83 � $41.00

7. 80 trees

9. Make the playground square with side S � 60 m.

11. Let x be the length of the field and y the width. The area is A � xy square units. The perimeter is 2(x � y) � 2p. Thus y � p � x and the area

A � px � x2

A� � p � 2x � 0

at x � . A 0, so the maximum area corresponds

to x � and y � p � x � , namely a square.

13. 6 by 2.5

15. 2 by 2 by meters

17. 2 hours after the initial time; minimizing the squareof x will also minimize x when x � 0.

19. The most economical route is to run the cable 2,000meters under water and 400 meters over land.

4

3

p

2

p

2

p

2

1

2

C(qc)

qc


21. r � 1.51 inches; h � 3.02 inches

23. 27 cubic inches

25. r � h

27. (a) 10 machines(b) $400(c) $200

29. (a) 200 bottles(b) every three months

31. (a) � � �

(b) At p � 6, � � � , so �� . Since

�� 1, demand is elastic (i.e., as price increases,revenue decreases).

(c) $5.77

33. (a) Demand is of unit elasticity when p � 20.Demand is elastic for p � 20. Demand is inelasticfor p 20.

(b) Revenue function is increasing for 0 � p 20.Revenue function is decreasing for 20 p �

. Revenue function is maximized at p � 20.

(c) R(p) � 120p � 0.1p3 ⇒ R�(p) � 120 � 0.3p2;R�(p) � 0 when p � 20. Since R(p) � �0.6p 0, this represents a maximum for R(p).

(d) (20, 1600)

(0, 120) q

R

p(√1,200, 0)

�1,200

��9

8� �9

8

9

8

2p2

100 � p2

2

3

35. 11,664 cubic inches

37. C(x) � 1,200 � 1.20x � ; x � 6

39. 5 years from now

41. The consumer expenditure is

E(p) � px(p) � 72p1/2

This function is always increasing. The revenueincreases when the price is increased.

43. 10 days from now

45. (a) F(r) � a r4(r0 � r)

(b) r � r0

47. Let v be the truck’s speed. Then the cost is C� �k2v

for constants k1, k2. This is minimized when C� �

� k2 � 0 or � k2v (wages � cost of fuel).

49. (a) Let x be the number of machines and t the numberof hours required to produce Q units. The setupcost is Cs � xs and the operating cost (for all xmachines) is Co � pt. Since n units can be produced

per machine per hour, Q � nxt or t � . The to-tal cost is

C � xs �

C� � s � � 0

when x � . Since C � 0, this corresponds

to a minimum.

(b) , Co �pQ

nx� �pQs

nCs � xs � �pQs

n

�pQ

ns

pQ

nx2

pQ

nx

Q

nx

k1

v

�k1

v2

k1

v

4

5

100

x2


51. S � Kwh3 � Kh3 ;

S�(h) � � 0 when h � 13 in.; w � 7.5 in.

53. (a) P(x) � x � x2 � 5x � 100 � tx;

P�(x) � � x � 10 � t � 0 when x � (10 � t)

(b) t � 5(c) The monopolist will absorb $4.25 of the $5 tax

per unit. $0.75 will be passed on to the consumer.(d) Writing exercise; responses will vary.

Chapter 3, Review1. f(x) is increasing for �1 x 2. f(x) is decreasing

for x �1 and x � 2.

2. f(x) is increasing for x �0.793 and x � 1.682. f(x)is decreasing for �0.793 x 1.682.

(–0.793, 22.505)

(0, 17)

(1.682, –0.225)

y

x

(–1, –12)

(2, 15)

(0, –5)

y

x

2

5

5

2

7

8�15 �3

8x�

675h2 � 4h4

�225 � h2

�225 � h2 3. f(x) is increasing for x �2 and x � 2. f(x) isdecreasing for �2 x 2.

4. f(x) is increasing for x �2 and x � 0. f(x) isdecreasing for �2 x �1 and �1 x 0.

5. f(x) is increasing for x �2 and x � 2. f(x) isdecreasing for �2 x 0 and 0 x 2.

x = 0

(2, 10)

(–2, –6)

y

x

t = –1

(–2, –4)

(0, 0) t

(–2, 64)

(2, –64)

– , 0203( )

, 0203( )

y

x


6. f(x) is increasing for x and x � . f(x) is decreas-

ing for x .

7. Relative maximum at x � 0; relative minimums at

x � �1, x � 7; neither at x � .

8.

9. f(x) is increasing for x � 3 and is decreasing for x 3.The graph is concave upward for all real x.

(0, 1)

(3, –8)

y

x

–6 –3 2

5

y

x

3

2

(3, 0)

( , – )

( , 0)12

y

x

272

32

3

2

1

2

3

2

1

210. f(x) is increasing for x 0 and x � 2. f(x) is decreas-

ing for 0 x 2. f(x) is concave upward for x � 1.f(x) is concave downward for x 1.

11. f(x) is increasing for x �1 and x � 3. f(x) is de-creasing for �1 x 1 and 1 x 3. f(x) is con-cave upward for x � 1. f(x) is concave downward forx 1.

x = 1

(3, 6)

(–1, –2)(0, –3)

y

x

(0, 2)

inf(1, 0)

(2, –2)

y

x


12. f(x) is increasing for �3 x �1. f(x) is decreasingfor x �3, �1 x 0, and x � 0. f(x) is concaveupward for �4.7 x �1.3 and x � 0. f(x) is con-cave downward for x �4.7 and �1.3 x 0.

13. Relative maximum at (2, 15); relative minimum at(�1, �12).

14. Relative maximum at (�2, �4); relative minimum at(0, 0).

15. Relative maximum at ; relative minimum at

.

16. Relative maximum at ; no relative

minimum.

17. Absolute maximum at (�3, 40); absolute minimumat (�1, �12).

18. Absolute maximum at (2, 6); absolute minimum at (3, �37).

19. Absolute maximums at ; absolute

minimum at (0, 0).

20. No absolute maximum; absolute minimum at (2, 10).

��1

2,

1

2�; �1, 1

2�

��3

2, �

4

3�

�3

2, 0�

�1

2, 2�

x = 0

y = 0

(–1, 0)

(–3, – )427

y

x

infx = –4.7

infx = –1.3

21. � 2 � 0 � 2

22.

does not exist because of 0 in the denominator.

23. � 0 � � � ��

24.

25.

� ��

26.

� ��

27.

28.

29. � ��x �1

xlimxfi 0�x�1 �

1

x2� �limxfi 0

1 �1

x�

1

x2

x2 � 3x � 1� 0lim

xfi ��

1 �3

x

7

x2 � 1

� �1� limxfi ��

x(x � 3)

7 � x2limxfi ��

limxfi ��

x �3

x�

2

x2 �7

x3

1 �1

x2 �1

x3

limxfi ��

x4 � 3x2 � 2x � 7

x3 � x � 1

� limxfi ��

1 �3

x2 �5

x3

2

x2 �3

x3

limxfi ��

x3 � 3x � 5

2x � 3

limxfi ��

x

x2 � 5� lim

xfi ��

1

x �5

x

� 0

limxfi 0�

�x3 �1

x2�

limxfi 0

�2 �1

x3�

limxfi ��

�2 �1

x2�


30.

�

31. The maximum speed is 52 miles per hour at 1:00 P.M.and 7:00 P.M. The minimum speed is 20 miles perhour at 5:00 P.M.

32. R � kN(P � N) so R� � k(P � 2N) and R� � 0 when

N � . Since R � �2k 0, there is a

maximum at N � .

33. $6.50 per lamp.

34. p � $8. per card � $8

35. Each plot should be 50 meters by 37.5 meters.

36. (a) 80 feet by 80 feet(b) 80 feet by 160 feet

37. r � h

38. After 2 hours and 20 minutes on the job.

39. 77 or 78 houses

40. 12 machines

41. 11:00 A.M.

42. 17 floors

43. Row all the way to town.

2

3

P

2

�P

2�P

2

�x2 � x � 0limxfi 0��x2�1 �

1

x� �limxfi 0�

limxfi 0�

x�1 �1

x

44. A � �3, B � 9, C � �1; f(x) � �3x3 � 9x2 � 1

45. Rectangle: 3.8956 feet by 4.1566 feet; side oftriangle: 3.8956 feet

46. (a) 8 units(b) 13.156 units

47. square units

48. 400 floppsies and 700 mappsies

49. 4.0249 miles across water; 5.3167 miles on shore

50.

51. (a) f(x) is increasing for 0 x 1 and x � 1. f(x) isdecreasing for x 0.

(b) f(x) is concave upward for x and x � 1. f(x) is

concave downward for x 1.

(c) f(x) has a relative minimum at x � 0. f(x) has in-

flection points at x � and x � 1.1

3

1

3

1

3

5�3 by 5�6

64

7

(2, 11)

(1, 5)

(0, 1)

y

x


(d)

52. 4,000 maps per batch

53. Hint: If x units are ordered, C � k, x �

54. Hint: If v is the truck’s speed, C � � k2v.

55.

x

y

y = 1

x = –1 x = 1

k1

v

k2

x.

x

y 56.

57.

58.

59. (a) � �2x � 68 �

(b) R�(x) � �4x � 68

10

x

R(x)

x

x

y

y = –

(1, –1)

(0, 0)

12

x

y

x

y

y = 1

x = 4.3x = .7


(c)

60. (a) A(x) � � 0.9x2 � 9x � 47 � is

minimized when x � 6.47.

(b) P(x) � �0.9x3 � 7x2 � 21x � 120 is maximizedwhen x � 6.4.

y

x

y = P(x)

6.4

y

x

43.45

6.47

y = A(x)

100

x

C(x)

x

x

y

65

60

55

50

45

0 1 2 3 4 5 6

61. An example is shown.

62. f �(x) does not exist at x � 1. An example is shown.

63. (a)

so

(b) If , then

so

dP

dt

A

B��P �B

2�2

0

A��1 �P

B�P �1

4B

dP

dt� A�1 �

P

B�P � H

H � 1

4AB

d2P

dt2 � 0 when P �B

2

� �A �2AP

B dP

dt

d2P

dt2 � A�1 �P

B�dP

dt� A��

1

B

dP

dt �P

1

y

x

2 4 5 7

y

x



64. (a) E �


Chapter 4, Section 11. e2 � 7.389, e�2 � 0.135, e0.05 � 1.051, e�0.05 �

0.951, e0 � 1, e � 2.718, � 1.649, � 0.607

3.

5. (a) 9(b) Approximately �18.379(c) 12(d) �1

7. 3

9.

11. 243

13. (a) $1,967.15(b) $2,001.60(c) $2,009.66(d) $2,013.75

1

5

(1, 3)

(0, 1)

y = 4x

y = 3x

y = 0

(1, 4)

y

x

1

�e�e

dQ

Q100

dI

I100

�I

Q

dQ

dI

15. $3,534.12

17. (a) $6,361.42(b) $6,342.19

19. (a) 50,000,000(b) 91,105,940

21. 400

23. 320

25. The initial investment will be quadrupled.

27. 20,480 bacteria

29. 4,000

31. (a) 12,000 people per square mile(b) 5,959 people per square mile

33. 204.8 grams

35. (a) 0.5488(b) 0.1215(c) 0.1813

37. (a) B � P

B � P(1 � x) simple interest formulawith t � 1

P(1 � x) � P

x � �1 �r

k�k

� 1

1 � x � �1 �r

k�k

�1 �r

k�k

compound interest for-mula with t � 1

�1 �r

k�k


(b) B � Per continuously compounded formulawith t � 1

B � P(1 � x) simple interest formula with t � 1

Per � P(1 � x)

er � 1 � x

x � er � 1

39. 8.2% compounded quarterly

41. (a) 310(b) No. Prediction of 169 is close enough to actual

167 and is a prediction, not an absolute.(c) Writing exercise; responses will vary.

43. $1,206.93

45. (a) No. A fair monthly payment is $166.07.(b) Writing exercise; responses will vary.

47. As n → ��, → e � 2.71828

49. � ��


1. ln 1 � 0, ln 2 � 0.693, ln e � 1, ln 5 � 1.609, ln �

�1.609, ln e2 � 2. ln 0 and ln �2 are undefined; ex

cannot be negative or equal to zero.

3. 3

5. 5

7.

9. � 11.552ln 2

0.06

8

25

1

5

limnfi ��

�2 �5

2n�n/3

�1 �1

n�n

11. � 0.402

13. e�C�t/50

15. 4

17. � 1.820

19. � 1

21. 3.4657

23. 10.5729

25. �5.5

27. � 11.55 years

29. � 5.33%

31. 5,614.06 years

33. Q(t) � 6,000e0.0203t

35. Q(t) � 500 � 200e�0.1331t

37. 10,523.15 years

39. 24.84 years ago; 95.8%

41. (a) 45%(b) 2.34%

43. Scélérat; Wednesday morning at 1:27 A.M.

45. (a) 2 � 108

(b) 101.2 � 15.85 times more intense

ln 2

13

ln 2

0.06

ln b

ln a

2

ln 3

ln 5

4


47. The year 2095

49. (a) k � � 0.999; x � � � 4.61

meters(b) Writing exercise; responses will vary.

51. (a) Since x � , k � and Q(t) � Q0e�(ln 2/�)t

(b) Q0(0.5)kt � Q0e�(ln 2/�)t

kt ln 0.5 � � t

So, k �

53. (loga b)(logb a) � � 1

55. 10x and log10 x are reflections about y � x.

57. x � �5.059 (using a graphing utility)

59. x � 2.277 (using a graphing utility)

61. 69.92 mg; 93 hours later

Chapter 4, Section 31. f �(x) � 5e5x

3. f �(x) � 2(x � 1)

5. f �(x) � �0.5e�0.05x

ex2�2x�1

x

y

10

8

6

4

2

–2–2 2 4 6 8 10

y = log x

y = 10 x

�ln b

ln a��ln a

ln b�

1

�

�ln 2

� �

ln 2

�

ln 2

k

ln 0.01

k

ln 0.05

�3

7. f �(x) � (6x2 � 20x � 33)e6x

9. f �(x) � �6ex(1 � 3ex)

11. f �(x) �

13. f �(x) �

15. f �(x) � 2x ln x � x

17. f �(x) � e

19. f �(x) � �

21. y � x

23. y � e2

25. y � x �

27. f �(x) � f(x)

29. f �(x) � f(x)

31. f �(x) � f(x)(2x ln 2)

33. (a) C�(x) � 0.2e0.2x

(b) � ; Marginal average cost �

(c) R(x) � xe�3x, R�(x) � e�3x(�3x � 1)(d) x � 0.205(e) x � 5

e0.2x(0.2x � 1)

x2

e0.2x

x

C(x)

x

� 3

x � 1�

2

6 � x�

2

3(2x � 1)�

� 5

x � 2�

1

2(3x � 5)�

1

2

1

2

2

(x � 1)(x � 1)

2x/32

3

3

x

3

2�3xe�3x


35. (a) C�(x) � 2x

(b) ; Marginal average cost � 1 �

(c) R(x) � , R�(x) �

(d) x � 0.18(e) x �

37. (a) Population is increasing at the rate of 1.22 millionor 1,221,403 people per year.

(b) Constant rate of 2% per year

39. (a) Value is decreasing at the rate of $1,082.68 peryear.

(b) Constant rate of �40% per year

41. (a) Approximately 406 copies(b) 368 copies

43. (a) 462.10 years(b) 1,072.96 years

45. (a) 8.8%, 2 hours after consumption(b) t � 6.9 hours after consumption

47. f�(x) �

49.

51. The percentage rate of change of f with respect to x is

100 or 100

Since [ln f(x)] � � [ f(x)] �

100 � 100 [ln f(x)]d

dx

f�(x)

f(x)

d

dxf(x)

f(x)

d

dx

1

f(x)

d

dx

�d

dxf(x)

f(x) �f�(x)

f(x)

1 � ln x

ln 10

2x(x ln 2 � 1)

x2

�2

x � 3 ln (x � 3)

(x � 3)2

x ln (x � 3)

(x � 3)

2

x2

C(x)

x� x �

2

x

53. The population of the town x years from now will be

P(x) � 5,000

From problem 51, the percentage rate of change xyears from now will be

Hence the percentage rate of change 3 years fromnow will be

12.5% per year

55. f(x) � (3.7x2 � 2x � 1)e , f�(�2.17) ��428,640

Chapter 4, Section 41. f(t) is increasing for all real t. f(t) is concave upward

for all real t. There is a horizontal asymptote at y � 2.

(0, 3)

y = 2

y

t

�3x�2

100(3 � 2)

32 � 12 � 19�

�100(x � 2)

x2 � 4x � 19

� 100d

dx [ ln 5,000 �

1

2 ln (x2 � 4x � 19)]

100d

dx ln 5,000�x2 � 4x � 19

�x2 � 4x � 19


3. g(x) is decreasing for all real x. a(x) is concave down-ward for all real x; y � 2 is a horizontal asymptote.

5. f(x) is increasing for all real x. f(x) is concave down-ward for all real x; y � 3 is a horizontal asymptote.

7. g(t) is increasing for all real t. g(t) is concave down-ward for all real t; y � 5 is a horizontal asymptote.

(0, 2)

y = 5

y

t

(0, 1)

y = 3

y

x

(0, –1)

y = 2

y

x

9. f(x) is increasing for all real x. f(x) is concave upwardfor x 0.549. f(x) is concave downward for x �0.549. Inflection point is (0.549, 1), and y � 2 and y � 0 are horizontal asymptotes.

11. f(x) is increasing for x � �1. f(x) is decreasing for x �1. f(x) is concave upward for x � �2. f(x) is con-cave downward for x �2. Relative minimum is

and inflection point is ; the

x-axis (y � 0) is a horizontal asymptote.

13. f(x) is increasing for x 1. f(x) is decreasing for x � 1. f(x) is concave upward for x � 2. f(x) isconcave downward for x 2. Relative maximum is(1, e), inflection point is (2, 2); the x-axis (y � 0) is ahorizontal asymptote.

(1, e)

(0, 0)(–1, – ) 1

e

y

x

inf x � �2

��2, �2

e2��1, �1

e�

y = 2

y = 0

infx � 0.549

y

x


15. f(x) is increasing for 0 x 2. f(x) is decreasing forx 0 and x � 2. f(x) is concave upward for x 0.6and x � 3.4. f(x) is concave downward for 0.6 x 3.4. Relative minimum is (0, 0); relative maximum is

; inflection points are (0.6, 0.2) and (3.4, 0.4).

The x-axis is a horizontal asymptote.

17. f(x) is increasing for all real x. f(x) is concave upwardfor x 0. f(x) is concave downward for x � 0. Inflec-tion point is (0, 3). The x-axis (y � 0) and (y � 6) arehorizontal asymptotes.

inf (0, 3)

y = 6

y = 0

y

x

(0, 0)y = 0

infx � 1

y

x

(2, ) 4e2

infx � 3.4

�2, 4

e2�

(1, e)

y = 0 (0, 0)

inf(2, 2)

y

x

19. f(x) is increasing for x � 1. f(x) is decreasing for x 1. f(x) is concave upward for x e. f(x) is concavedownward for x � e. Relative minimum is (1, 0); in-flection point is (e, 1). The y-axis (x � 0) is a verticalasymptote.

21. (a) As t → �, f(t) → 1

(b) 0.741 (c) 0.0888

23. 18.75°C

25. (a)

(b) 500 (c) 1,572 (d) 2,000

27. 37.5 units per day

y = 2

0.5

y

x

y = 1

y

t

(1, 0)

inf(e, 1)

y

x


29. (a) Approximately 403 copies(b) 348 copies

31. Q(t) � 80(40 � 76e�1.2t)�1

Q�(t) � 80(�1)(4 � 76e�1.2t)�2(76)e�1.2t(�1.2)

� 80 � 76 � 1.2

After 2 weeks (at the end of the second week)

Q(2) � 80 � 76 � 1.2

� 5.576 or 5,576 people

The disease is spreading most rapidly in the middleof the third week, after about 2.45 weeks.

33. (a)

(b) 1 (100%)

35. (a) P(x) � 1,000e�0.02x(x � 125)

(b) $175

37. 69.44 years from now


41. (a) t � ln

“In the long run,” the concentration approaches 0.

�a

b�1

a � b

x

y

500

100

C �p0

1 � p0

e�2.4

(4 � 76e�2.4)2

e�1.2t

(4 � 76e�1.2t)2

(b)


43. (a) N(0) � 15 employees; N(5) � 482 employees;2.10 years; 500 employees

(b)

45. (a) Q(t) � 1.139e0.06t, Q(7) � 1,734 staff members,Q(53) � 27,389 staff members

(b) t � 9.38 (about 1957), t � 11.5 years(c) Writing exercise; responses will vary.

47. (a) P

(b)


x

y

λe

1λ

�1

��

e

y

t

15

y = 500

y = F(t)

16,667

y = N(t)

x

y

0.3

0.25

0.2

0.15

0.1

0.05

00 1 2 3 4

y � 25xe�5x


49. ; ��


Chapter 4, Review1. (a)

(b)y = 5

(0, 3)

y

x

y = 0

(1, 1.8)

(0, 5)

y

x

y

xy = 0

0.52

0.758

limxfi ��

f(x) �limxfi ��

f(x) � 0 (c)

(d)

2. (a) 391 (b)

(c) (d)

3. $8,000

4. (a)

(b) 10,000 units(c) 32,027 units(d) $9,808.29(e) 50,000 units

y

y = 50

(0, 10)

x

6

5

65

2

100

3

y = 3

y = 2(0, 2.5)

y

x

y = 1

y = –2

(0, –1)

y

x


5. 60 units per day

6. (a)

(b) 10 million(c) 17.28 million (17,283,507)(d) 30 million

7. (a) 5(b) 2(c) 32(d) ln 27 � 3.296

8. (a) � 34.657

(b) 0(c) e2 � 7.389

(d) � 1.864

9. 14.75 minutes

10. (a) f�(x) � 6e3x+5

(b) f�(x) � �xe�x(x � 2)

(c) g�(x) �

(d) h�(x) � 2(1 � ln x)

(e) f�(t) �

(f) g�(t) � (1 � ln t)2

ln 3

ln 2t � 1

(ln 2t)2

x � 2

x2 � 4x � 1

3

ln 5

ln 4

0.04

y = 30

t

y

11. (a) 11.57 years(b) 11.45 years

12. 8.20% per year compounded continuously

13. (a) $1,072.26(b) $1,070.52

14. (a) $4,323.25(b) $4,282.09

15. r � 6%

16. 5.83%

17. (a) 0.13 parts per million per year(b) Constant rate of 3%

18. $140 per camera

19. (a) f(x) is increasing for x ; f(x) is decreasing for

x � ; f(x) is concave upward for x � 1; f(x) is

concave downward for x 1. Relative maximum

; inflection point is .

(0, 0)

inf

y = 0

(1, ) 1e2

( , )12e

12

y

x

�1, 1

e2��1

2,

1

2e�

1

2

1

2


(b) f(x) is increasing for all real x. f(x) is concave up-ward for x � 0. f(x) is concave downward for x 0. Inflection point is (0, 0).

(c) f(x) is increasing for all real x. f(x) is concave up-ward for x 0. f(x) is concave downward for x �0. Inflection point is (0, 2)

(d) f(x) is increasing for x � 0. f(x) is decreasing for x 0. f(x) is concave downward for x �1 andx � 1. Inflection points (1, ln 2) and (�1, ln 2).

(0, 0)

(–1, ln 2)

(1, ln 2)

y

x

(0, 2)

y = 0

y = 4

y

x

(0, 0)

y

x

20. 51.02 years

21. 0.1629R0

22. (a) 78.9% (b) 662.37 years old (about 1326 A.D.)

23. k � 0.091; T0 � 145°

24. (a) D(10) � 0.00195; D(25) � 0.000591(b)

25.

26. 20.09 years

27. 0.6065 years

( , 2.89)1

√e

(1, )inf

4

√π

y

x

r 0.04 0.06 0.09 0.10 0.12

Actual 17.33 11.55 7.70 6.93 5.78

Rule of 69 17.25 11.5 7.67 6.90 5.75

Rule of 70 17.50 11.67 7.78 7.00 5.83

Rule of 72 18.00 12.00 8 7.20 6.00

y

(0, 0.008)

t


28. Bronze age began about 5,000 years ago (3,000 B.C.);maximum percentage is 55%.

29. (a) k � � 0.0040

(b) (multiplied by 100)

(c) Writing exercise. Responses will vary.

30. 0.8110 minutes � 48.66 seconds; �8.64°C perminute

31. (a)


32. ln

33. (a) V(5) � $207.64

(b) V�(t) �

(c) 100 ln

34. f(t) � 30 � Ae�kt so Ae�kt � 30 � f(t)f �(t) � �Ae�kt(�k) � kAe�kt

� k[30 � f(t)]

35. 10�1.6 � 0.0251

�1 �2

L�V0�1 �

2

L�t

ln�1 �2

L�

�k1

k2� �

E0

R � 1

T2�

1

T1�

(0, 0.25)

y

x

ke�kt

1 � e�kt

ln (0.992)

�2

36. (a) 1790 3,867,0871800 5,256,5501830 12,956,7191860 30,207,5001880 50,071,3641900 77,142,4271920 108,425,6011940 138,370,6071960 162,289,8231980 178,782,4991990 184,566,6532000 189,034,385

(b) The model predicts that the population wasincreasing most rapidly around 1915.


37.

(2, 4)

(1, 2)(–1, 2)

(0, 1)

(–2, 4)

2–x 3–x 5–x (0.5)–x

(–1, 5)

y

x

200

150

100m

illio

ns

50

00 4 8 12 16 20


38.

39. (1.24, 3.90)

40. x � 1.0654

41. Graphs intersect at (1.166, 0.858)

x

y

y � ln (1 � x2)

1.166

y � 1–x

3x

(e, 3.5)

y = 4 – ln √x

(1, 4)

y

x

3–x

(–2, 3) (2, 3)

(0, 1)

√3–x √3x

(–1, √3) (1, √3)

(–1, 3)

y

x

42.

( ) � ( )


1. � C

3. � � C

5. 5x � C

7. t3 � t3/2 � 2t � C

9. 2y3/2 � � ln � y� � C

11. � x5/2 � C

13. ln �u� � � e2u � � Cu3/2

3

3

2u

1

3

2

5

ex

2

1

y2

2�5

3

1

x

x6

6

�n�n � 1�n � 1�n

n ( ) ( )

8 22.63 22.36

9 32.27 31.62

12 88.21 85.00

20 957.27 904.84

25 3,665 3,447

31 16,528 15,494

37 68,159 63,786

38 85,679 80,166

43 261,578 244,579

50 1,165,565 1,089,362

100 1.12 � 1010 1.05 � 1010

1,000 2.87 � 1047 2.76 � 1047

�n�n � 1�n � 1�n


15. x � ln x2 � � C

17. � x4 � x3 � x2 � C

19. t7/2 � t3/2 � C

21. f(x) � 2x2 � x � 1

23. f(x) � � � 2x �

25. 10,128 people

27. 20 meters

29. N(4) � N(2)

31. 15 units

33. $436

35. $2,300

37. v(r) � a(R2 � r2)

39. (a) 504(b) Writing exercise; responses will vary.

41. f �(x) is maximized when x � 10(a) Seven items per minute(b) f(x) � x � 0.6x2 � 0.02x3

(c) 100 items

43. The spy takes seconds to stop, and the car travels

199.9 feet. The camel is toast.

22

7

1

2

5

4

2

x

x4

4

2

3

2

7

11

3

5

4

1

x45. H(x) must be a constant. Let H(x) � F(x) � G(x).

Then H�(x) � F�(x) � G�(x). But F�(x) � G�(x) � 0because F�(x) � G�(x). Therefore H�(x) � 0 and H(x)is a constant. So F(x) � G(x) � C.

47.

49. (a) v(t) � �23t � 67; s(t) � � t2 � 67t

(b)

(c) t � 2.9 seconds; s(2.9) � 97.6 feet; 49.2 feet persecond


1. (2x � 6)6 � C

3. (4x � 1)3/2 � C

5. �e1�x � C

7. � C

9. (t2 � 1)6 � C

11. (x3 � 1)7/4 � C4

21

1

12

1

2ex2

1

6

1

12

(0, 67)

(2.9, 97.6)

(2.9, 0)

s(t)

v(t)

t

23

2

bx dx � ex ln b dx �bx

ln b� C


13. ln �y5 � 1� � C

15. (x2 � 2x � 5)13 � C

17. ln �x5 � 5x4 � 10x � 12� � C

19.

21. (ln 5x)2 � C

23. � � C

25. [ln (x2 � 1)]2 � C

27. f(x) � (x2 � 5)3/2 � 1

29. (a) x(t) � � (3t � 1)3/2 �

(b) x(4) � �16.4(c) t � 0.4

31. (a) x(t) � ln �t � 1� � � 1

(b) x(4) � ln 5 � � 1� 0.8094

(c) t � 53

33. 2.3 meters

35. 61.07 million � 61,070,138

1

5

1

t � 1

40

9

4

9

1

3

1

2

1

ln x

1

2

�3

2 � 1

u2 � 2u � 6� � C

3

5

1

26

2

537. (a) L(t) � 0.03 � 0.07; 3:00 P.M.;

0.37 parts per million(b) The ozone level at 11:00 A.M. (t � 4) is L(4) �

0.345. The same level occurs at t � 12 (7:00 P.M.).

39. (a) V(t) � 4,800e�t/5 � C(b) $1,049.61

41. $3.52 per kilogram

43. (a) p(x) � 30 ln �3 � x� � � 27.75 � 30 ln 3

� 30 ln � � 27.75

(b) p(4) � $10.53(c) 784

45. R�(x) � , C�(x) � 2 � x � x2

Let 14 � x � u, du � �dx. Then,

� 6�14 � x �2

3(14 � x)3/2 � C1

� ��2

3u3/2 � 6u1/2�

� � (u1/2 � 3u�1/2) du

R(x) � � u � 3

�udu

11 � x

�14 � x

90

3 � x�3 � x

3 �90

3 � x

40 12 15

(15, 0.28)

t

0.25

0.345

��t2 � 16t � 36


We also have

C(x) � (2 � x � x2) dx � 2x � � � C2

P(9) � P(5) � R(9) � C(9) � [R(5) � C(5)] ��295.54 � (�64.167) � �231.373


1. y � x3 � x2 � 6x � C

3. y � Ce3x

5. y � �ln (C � x)

7. y2 � x2 � C

9. � x3/2 � C

11. y � C�x � 1�

13. ln(y � 3)10 � �(2x � 5)�5 � C

15. x � C for 2t � 1 � 0

17. y � e5x �

19. y3 � x2 � 21

21. y �

23. y � 2 ln �y � 1� � ln �t� � 2(1 � ln 3)

6

4(4 � x)3/2 � 3

3

2

4

5

1

5

e(2t�1)/4

4�2t � 1

1

3�y

5

2

x3

3

x2

225. If Q � B � Ce�kt, then B � Q � Ce�kt.

� �Ce�kt(�k) � kCe�kt � k(B � Q)

27. Let Q denote the number of bacteria.

� kQ

29. Let Q denote the investment.

� 0.07Q

31. Let P denote the population.

� 500

33. Let N denote the total population and Q(t) the numberof people who have caught the disease.

� kQ(t)[N � Q(t)]

35. (a) �

,

since Q is the amount (in pounds) of fluoride inthe reservoir, 200 is the total gallons of water (aconstant) and �4 is the rate of water flowing outof the reservoir (a constant rate). Only Q willchange.

(b) Q(t) � Ce�t/50

Q(0) � 1,600

So, Q(t) � 1,600e�t/50

dQ

dt� �

Q

50

� Q

200�(�4) � �Q

50

dQ

dt� � Q

200��4�

�million gallonsper day �� pounds per

million gallons�Poundsper day

dQ(t)

dt

dP

dt

dQ

dt

dQ

dt

dQ

dt


37.

39. Writing exercise; responses will vary.

41. (a) A � ; B �

(b) dp

� ln �P� � ln �k � mP� � C

(c) ln � t � C

ln � kt � C

� Cekt

P(1 � Cmekt) � Ckekt

P �

�

where E � and D � .1

Cm

k

m

E

1 � De�kt

Ckekt

Cmekt � 1

P

k � mP

� P

k � mP�� P

k � mP�1

k

1

k

1

k

� 1

kP�

m

k(k � mP)

m

k

1

k

p

p0

t

a – rb + s

limtfi ��

p(t) �a � r

b � s

p(t) �a � r � Ce�(b�s)kt

b � s

dp

dt� k[a � bp � (r � sp)]

Chapter 5, Section 41. �(x � 1)e�x � C

3. (2 � x)ex � C

5.

7. �5(v � 5)e�v/5 � C

9. x(x � 6)3/2 � (x � 6)5/2 � C

11. � � C

13. 2x � (x � 2)3/2 � C

15. �e�x(x2 � 2x � 2) � C

17. ex(x3 � 3x2 � 6x � 6) � C

19.

21. � (ln x � 1) � C

23. (x2 � 1) � C

25. (9x4 � 5) � C

27. f(x) � � � ln 2

29. 176.87

5

2�ln x �1

2�x2

4

(x4 � 5)9

360

ex21

2

1

x

x3

3 �ln x �1

3� � C

4

3�x � 2

(x � 1)9

9

(x � 1)10

10

4

15

2

3

1

2t2�ln 2t �

1

2� � C


31. $239.75

33. (a) u � xn v� � eax dx

u� � nxn�1 dx v � eax

xneax dx � xn � eax � eax � nxn�1 dx

� xneax � xn�1eax dx

(b) x3e5x dx � x3e5x �

� x3e5x �

� x3e5x � x2e5x

�

� x3e5x � x2e5x � xe5x

� � e5x � C

� e5x(125x3 � 75x2 � 30x � 6) � C

Chapter 5, Review

1. x6 � x3 � � C

2. x5/3 � ln �x� � 5x � x3/2 � C

3. (3x � 1)3/2 � C

4. (3x2 � 2x � 5)3/2 � C

5. (x2 � 4x � 2)6 � C1

12

1

3

2

9

2

3

3

5

1

x

1

6

1

625

1

5

6

125

6

125

3

25

1

5

�1

5xe5x �

1

5 e5xdx�6

25

3

25

1

5

�1

5x2e5x �

2

5 xe5x dx�3

5

1

5

x2e5x dx3

5

1

5n

a

1

a

1

a

1

a1

a

6. ln �x2 � 4x � 2� � C

7. � � C

8. (x � 5)13 � C

9. (x � 5)14 � (x � 5)13 � C

10. e3x � C

11. (3x � 1)e3x � C

12. �2(x � 2)e�x/2 � C

13. (x3 � 1) � C

14. 10(2x � 19)e0.1x � C

15. x2 ln 3x � x2 � C

16. x ln 3x � x � C

17. (ln 3x)2 � C

18. � (ln 3x � 1) � C

19. � C

20. (x2 � 1) [ln (x2 � 1) � 1] � C

21. f(x) � (x2 � 1)4 � 31

8

(x2 � 1)10

20�

(x2 � 1)9

18

1

x

1

2

1

4

1

2

ex31

3

5

9

5

3

5

13

1

14

1

13

3

4(2x2 � 8x � 3)

1

2


22. 11,250 people

23. 10,945 inmates

24. y � x4 � x3 � 5x � C

25. y � Ce

26. y � Ce�kx � 80

27. 2ey � e2x � C or y � ln (e2x � C)

28. y � x5 � x3 � 2x � 6

29. y �

30. y �

31. y � x2 � 3x � 5

32. $22,857.14

33. $2,265.80

34. (a) p1(x) � 0.2x � 0.001x3 � 250; p1(10) � $2.53per dozen

(b) p2(x) � 0.3x � 0.001x3 � 250; p2(10) � $2.54per dozen

p(x)

x

(0, 250)

(10, 253)

(10, 254)

y � P2(x)

y � P1(x)

2e1��1�x2

�2(x ln x � x � 5,001)

1

2

0.01x2

1

4

35. $1,000

36. meter

37. $75

38. 126 people

39. $87.57

40. 259.49 billion barrels

41. ; 81.2 pounds

42. P(t) � 10 � 6e�0.0263t

43. ; 640 days

44. � ; P � Ce�(k/x)t

45. P � C(P0)

46. ; 45 people

47. (a) S � S0 � CekAt


Chapter 6, Section 11. 1.95

3. 144

5. � ln 3 � 3.7653

7.2

9

8

3

dQ

dt� kQ(P � Q)

�t

�kP

x

dP

dt

dQ

dt� 18 �

18Q

5,000

dQ

dt �

�Q

50

2

3


9. 3.2

11.

13.

15. e

17.

19. 1 � � 0.2642

21.

23. 15

25.

27.

29.

31.

33. (a)

(b) ; part of the area under the curve (x � 1)2 � y2 � 1

4

4

81

4

3

4

1

2

38

3

8

3

2

e

8

3

7

6

4

3

35.

Chapter 6, Section 21. a decrease of $1,870

3. $774

5. $75

7. 132 units

9. (a) 16 years(b) $2,090.67(c)

11. (a) 14.7 years(b) $582.22

(0, 306)

(0, 130)

(16, 386)

x

y

16

2.037

A � 4.2

2.34��2

5x2 � 2 � (x3 � 8.9x2 � 26.7x � 27)dx

y = x3 – 8.9x2 + 26.7x – 27

x

y

(4.2, 2.23)

(2.34, –0.44)

x2

5y2

2– = 1


(c)

13. (a) 11 years(b) $26,620(c)

15. (a) 10 weeks(b) $14,857(c) In geometric terms, the net earnings in part (b) is

the area between the curve y � R(t) and the hori-zontal line y � 676 from x � 0 to x � 10.

x

y

y = 675

5,000

4,000

3,000

2,000

1,000

00 2 4 6 8 10 12 14

t

y

11

(0, 3,620)

(0, 7,250)

(11, 5,072)

C'(t)

R'(t)

x

y

(0, 140)

(14.7, 391)

(0, 90)

14 15

17. $13,994.35

19. $4,511.88

21. $76,424.11

23. (a) $1,000(b)

The total amount, $1,000, that consumers are will-ing to spend for 15 units of the commodity is the

area under the curve D(q) � from q

� 0 to q � 5.

25. (a) $199.69(b)

The total amount, $199.69, that consumers arewilling to spend for 10 units of the commodity is

the area under the curve P(q) � from q �

0 to q � 10.

27. (a) $563.99

300

4q � 3

(10, 6.98)

(0, 100)

q

D(q)

q

300

(0.1q � 1)2

(5, 200)

(0, 300)

5 q

D(q)


(b)

The total amount, $563.99, that consumers arewilling to spend for 15 units of the commodity isthe area under the curve D(q) � 50e�0.04q from q � 0 to q � 15.

29. p0 � $30, CS � $468

31. p0 � $66.52, CS � $12.46

q

p

q0 = 3

75 p0 = 66.52

D(q)

q

p

150

p0 = 30

q0 = 6

D(q)

(15, 27.44)

(0, 50)

15 q

D(q) 33. p0 � $17.50, PS � $6.25

35. p0 � $28.80, PS � $2.84

37. (a) 36 units(b) CS � $1,122.07

39. (a) R(q) � 124q � 2q2

P(q) � �2q3 � 57q2 � 120q � 7,600(b) q � 20 units(c) CS � $400

41. (a) q0 � 2(b) CS � $3.09; PS � $0.67(c)

q

p

5

p0 = 1 p � D(q)

p � S(q)

13

CS

PS

q0 = 2

q

p

p0 = 28.80

q0 = 7

S(q)

q

p

15

q0 = 5

S(q)

p0 = 17.5


43. (a) $207,360(b) $207,360(c) Responses may vary.

45. $11,296.88

Chapter 6, Section 31. 2

3.

5. 1.228

t

y

–1 2

(0, 1)

1.228

f (t) = e– t

(–2, 0)

(–4, 4)

–4

43

f(x) = x2 + 4x + 4

(0, 4)

y

x

4

3

(0, 0)

2

4

(4, 4)

f (x) = x

y

x

7. 18.7°C

9. Approximately 493 letters per hour

11. 80 members

13. 4,207 members

15. 116,039 people

17. Let S(r) � k(R2 � r2) denote the speed of the bloodin centimeters per second at a distance r from thecentral axis of the artery of (fixed) radius R.

The area of a small circular ring at a distance rj is(approximately) 2 rj �r square centimeters, so theamount of blood passing through the ring is

V(r) � 2 rj �r[k(R2 � r j2)]

� 2 k(R2rj � r j3) �r

cubic centimeters per second.Hence, the total quantity of blood flowing

through the artery per second is

The area of the artery is R2 and the average velocityof the blood through the artery is

Vave �

The maximum speed for the blood occurs at r � 0, soS(0) � kR2. Thus

Vave � S(0)

19. 15 meters

1

2

kR4/2

R2 �kR2

2

� 2 k�R2r2

2�

r4

4 ��0

R

� kR4

2

� 2 k R

0 (rR2 � r3)dr

2 k(R2rj � rj3) �r�

n

j�1lim

nfi ��


21. (a) S(t) dt

(b) S(t) dt

(c) The average speed is equal to the total distancedivided by the total number of hours.

23. 19,566.55 pounds

25. T � ln 2

average concentration:

or approximately 0.203

27. P0 f(N) �

29. G �

31. G �

33. G � 1 � � 0.164

35. Writing exercise. Responses may vary.

37. The distribution of income for stock brokers is morefairly distributed, since the Gini index is smaller.

2(e � 2)

e � 1

7

30

1

3

N

0 r(t) f(N � t) dt

�c

a�

.2029c

a or

9c

64a ln 2

1.3863

a or

2

a

N

0

1

N N

0

39.

41.


1.

3. the integral diverges


7.

9.

11.1

9

5

2

1

10

1

2

x

y

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x

y

1.2

1.0

0.8

0.6

0.4

0.2

00.0 0.2 0.4 0.6 0.8 1.0


13. �, the integral diverges

15. � 2e�1

17.

19. 5e10

21. �, the integral diverges

23. 2

25. $20,000

27. $150,000

29.

�

�

31. 200 patients

33. 50 units

35. (a) 1

(b)

(c)1

4

3

4

0 � ��A

r2 e0 � 0 �B

r2 e0� �A

r�

B

r2

��A

re�rt �

Bt

re�rt �

B

r2 e�rt��0

N

limNfi ��

�A

re�rt � B��

t

re�rt �

1

r N

0 e

�rt dt�limNfi ��

(A � Bt)e�rt dtN

0lim

Nfi ��

2

9

2

e

37. (a) 1

(b)

(c)

39. (a) 1

(b) 1 � � 0.1813

(c) � 0.6065

41. (a) 1

(b) � � � 0.3298

(c) � 0.1991

43. (a) 0.0855(b) 0.7981(c) 0.0907

45. (a) 0.6321(b) 0.3012

47. 0.6

49. � 0.3679

51. The probability that the grenade has expired is0.6171. The probability that it is still good is 0.3829.

53. (a) Capitalized cost of Machine 1: $28,519. Capital-ized cost of Machine 2: $20,222. The companyshould buy Machine 2.


1

e

4

e3

2

e

3

e2

1

e0.5

1

e0.2

5

32

11

32


Chapter 6, Review1. 0

2.

3. 1,710

4. 1 �

5.

6.

7.

8.

9. (2e3 � 1)

10. 55e2 � 45


12. 1


14.

15.1

4

3

5

1

9

1

2

2

e

3

5

65

8

1

e

17

3

16.

17.

18.


20.

21. (a) 1 (b) (c)

22. (a) 1

(b)

23. (a) 1(b) 0.3694(c) 0.3679

24. (a) 1(b) 0.4762(c) 0.6250

25. 135.6629

26. 1.0396

27. 1.5940

28. 1.7647

29. $7,377.37

30. $7,191.64

31. 62 homes

1

3

1

3

1

3

1

3

1

ln 2

1

4

2

3


32. $7,040,000

33. 218,010 people

34. 116,039 people

35. 515.48 billion barrels

36. $4,081,077.42

37. 132 units

38. $565,056

39. 36

40.

41.

42. � 8 ln 2 � 10.88

43.

44.

45. (a) 3 units(b) $127.50(c) $31.50

13

2

3

10

16

3

9

2

3

4

(d)

46. $1.32 per pound

47. The population will increase without bound.

48. (a) 22.1%(b) 55.07%(c) 44.93%

49. 10,000 subscribers

50. $120,000

51.

52. 0.0498

53. (a) 0.7047(b) 0.1466

2

9

10

10

q

p

q

p

Consumers’ willingness to spend

Consumers’ surplus

1

1


54. (a) x1 � 0.0, x2 � 0.5, x3 � 1.0, x4 � 1.5, x5 � 2.0(b) Area � 4.2152(c) x1 � 0.0, x2 � 0.25, x3 � 0.5, x4 � 0.75, x5 �

1.0, x6 � 1.25, x7 � 1.5, x8 � 1.75, x9 � 2.0, area� 4.5559

55. 0.3130

56. e�1.1x dx � 0.9091

57. � ��, the integral diverges

58.

59. (4.911, 0.762)

60. The Gini index for teachers is G1 � 0.57 and for realestate brokers is G2 � 0.24. The distribution ofincome is more fairly distributed for real estatebrokers.

61. The graphs intersect at x � 1 and approximately x �0.41; area � 0.1692

62. The region bounded by the curves is between x ��4.66 and x � �1.82; the curves also intersect at x� 4.98. The area is approximately 3.

(0.406, –0.37)

(1, 0)

y = x ln x

y = –x3 – 2x2 + 5x – 2

y

x

�3

8,

2

5�

dx

x � 2��

0

��

0

Chapter 7, Section 11. f(2, �1) � �3, f(1, 2) � 16

3. g(4, 5) � 3, g(�1, 2) � � 1.7321

5. f(e2, 3) � 6.7258; f(ln 9, e3) � 0.7324

7. g(1, 2) � 2.5; g(2, �3) � 4.3333

9. f(1, 2, 3) � 6, f(3, 2, �1) � 6

11. F(1, 1, 1) � 0.2310, F(0, e2, 3e2) � 0.1048

13. All ordered pairs (x, y) of real numbers for which

y � x

15. All ordered pairs (x, y) of real numbers for which y � x2

17. All ordered pairs (x, y) of real numbers for which x � 4 � y

� 4

3

�3

x

y

x = 4.98

x = –1.82 y = √25 – x2

y = x – 2x + 1

x = –4.66


19.

21.

23.C = 2

C = 1C = –1

C = –2

y

x

C = –4

C = 5(0, 4)

(–1, 0)

(4, 4)

(2, 0) (5, 0)

(2, –9)

y

x

12

(0, )

32

(0, – )

(0, 1)

(–3, 0) (1, 0)

(2, 0)

C = 2

C = –3

C = 1

y

x

25.

27. (a) 160,000 units(b) Production will increase by 16,400 units.(c) Production will increase by 4,000 units.(d) Production will increase by 20,810 units.

29. (a) R(x1, x2) � 200x1 � 10x2 � 25x1x2 � 100x2 �10x2

2

(b) $1,840

31. (a) If a � b � 1, production is more than doubled.(b) If a � b 1, production is increased (but not

doubled).(c) If a � b � 1, production is doubled.

33. R(x, y) � 60x �

35. (a) S(15.83, 87.11) � 0.5938

(b) Height � 90.05 cm

200

100

50

30

200 20 40 60 80 100 120

s = 0.789

s = 0.5938

x2

5�

xy

10� 50y �

y2

10

C = 1C = e

y

x


(c) 197.75%(d) Writing exercise; responses will vary.

37. (a) 70 units

(b) y � � x � 35

(c)

(d) Unskilled labor should be decreased by threeworkers.

39. 260

41. (a) 0.866 cm/sec(b)

20

15

10

5

00 2 4 6 8 10

x

y

x = –1

y = –2

(0, 258)

(129, 0)

(x + 1)(y + 2) = 260

x

y

x

y(0, 35)

703

( , 0)

3

2

43. (a)

(b) 159.76°C

45. (a) 1,437,480

(b) 425,920


47. (a)

K 277 311 493 718 554

L 743 823 1,221 3,197 1,486

Q 33,093 36,780 55,478 125,448 66,186

(b) Q(3(277), 3(743)) � 99,279 � 3(33,093)Q(0.5(277), 0.5(743)) � 16,546.5 � 0.5(33,093)

Q(2(277), 0.5(743)) � 23,400.3 �

For this last Q,

Q(2K, 0.5L) � 2�1/2(57)K1/4L3/4 � Q(K, L)

Chapter 7, Section 21. fx � 2y5 � 6xy � 2x, fy � 10xy4 � 3x2

3. � 15(3x � 2y)4, � 10(3x � 2y)4

5. fs � � , ft �

7. � (xy � 1)exy, � x2exy�z

�y

�z

�x

3

2s

3t

2s2

�z

�y

�z

�x

1

�2

33,093

�2

kg � m2

sec3

kg � m2

sec3

2.22

1.61.8

1.2

1.4

1

0.8

0.615,000 25,000 35,000 45,000


9. fx � , fy � �

11. fx � , fy �

13. � ln v, �

15. fx � , fy �

17. fx(�2, 1) � �22, fy(�2, 1) � 26

19. fx(0, 0) � 1, fy(0, 0) � 1

21. fxx � 60x2y3, fxy � 2(30x3y2 � 1), fyx � 2(30x3y2 �1), fyy � 30x4y

23. fxx � 2y(2x2y � 1) , fxy � 2x(x2y � 1) , fyx �2x(x2y � 1) , fyy � x4

25. fss � , fst � �

fts � � , ftt �

27. Daily output will increase by approximately 10 units.

29. (a) The marginal productivity of capital isapproximately 27 units. The marginal productiv-ity of labor is approximately 64 units.

(b) Additional labor employment

31. F(L, r) �

(a) F(3.17, 0.085) � 60,727.24k, � �

19,156.86k, � � ��2,857,752.58k4kL

r5

�F

�r

k

r4

�F

�L

kL

r4

s2

�(s2 � t2)3

st

�(s2 � t2)3

st

�(s2 � t2)3

t2

�(s2 � t2)3

ex2yex2yex2yex2y

2[y � (x � 2y) ln (x � 2y)]

y3(x � 2y)

1

y2(x � 2y)

u

v

�z

�v

�z

�u

� 5x

(y � x)2

5y

(y � x)2

2e2�x

y3

e2�x

y2(b) F(1.2L, 0.8r) � � 2.93F(L, r),

(1.2L, 0.8r) � 2.44 (L, r)

(1.2L, 0.8r) � 3.66 (L, r)

33. The monthly demand for bicycles decreases byapproximately 3.

35. The volume is increased by 72 cm3.

37. Substitute

39. Neither

41. Substitute

43. Yes

45. No

47. (a) An increase in x will decrease the demand D(x, y)for the first brand of mower. An increase in y willincrease the demand D(x, y) for the first brand ofmower.

(b) 0, � 0

(c) b 0, c � 0

49. P(x, y, u, v) � ,

Px � ,

Py � ,

Pu � , Pv �� 100xyu

(xy � uv)2

�100xyv

(xy � uv)2

�100uvx

(xy � uv)2

(xy � uv)100x � 100x2y

(xy � uv)2

�100uvy

(xy � uv)2

(xy � uv)100y � 100xy2

(xy � uv)2

100xy

xy � uv

�D

�y

�D

�x

�F

�L

�F

�L

�F

�r

�F

�r

k(1.2L)

(0.8r4)4


All of these partials measure the rate of change ofpercentage of total blood flow WRT the quantities x,y, u, and v.

51. (a) 0; For a fixed level of capital investment,

the effect on output of the addition of one workerhour is greater when the work force is small thanwhen it is large.

(b) 0; For a fixed work force, the effect on

output of the addition of $1,000 in capital invest-ment is greater when the capital investment issmall than when it is large.

53. (a) Q(37, 71) � 304,691, Q(38, 71) � 317,310,Q(37, 72) � 309,031

(b) Qx(37, 71) � 12,534 units, Q(38, 71) � Q(37,71) � 12,615 units

(c) Qy(37, 71) � 4,344 units, Q(37, 71) � Q(37, 71) �4,340 units

Chapter 7, Section 3Relative Relative Saddlemaximum minimum point

1. (0, 0) None None

3. None None (0, 0)

5. None None None

7. (�2, �1) (1, 1) (�2, 1); (1, �1)

9. None

11. (0, 1); (0, �1) (0, 0) (1, 0); (�1, 0)

13. None (0, 0)

15. (0, 3) None None

�4

3,

4

3�

�2, 7

2��4, 19

2 �

�2Q

�K2

�2Q

�L2

17. None None

19. (e, 1); (e, �1) None None

21. Jordan shirts (x): $2.70, O’Neal shirts (y): $2.50

23. The base of the box is a 2 ft by 2 ft square. Theheight is 8 ft.

25. Whole milk (x), 20 gallons; skim milk (y) 20 gallons

27. x � , y �

29. x � y � z �

31. x � 200; y � 300

33. S

35. y �

37. y � 3

39. (a)

(b) y � 0.51x � 0.41(c) Sixth year sales; approximately 3.5 billiondollars.

3

2

1

010 2 3 4 5

x

y

1

4x �

3

2

�5

8, �

1

8�

3�V0

�2

2�2

2

��3

2, 1�


41. (a) Let x denote the number of hours after the pollsopen and y the corresponding percentage ofregistered voters that have already cast theirballots. Then

(b) y � 3.05x � 6.10(c) When the polls close at 8:00 P.M., x � 12 and so

y � 3.05(12) � 6.1 � 42.7, which means that ap-proximately 42.7% of the registered voters can beexpected to vote.

43. � 2x � 4y; � 2y � 4x. Thus (0, 0) is a critical

point. Since � 2 � 0, the second derivative test

tells us there is a minimum in the x direction. Likewise,

�2f

�2x

�f

�y

�f

�x

40

30

20

10

020 4 6 8 10 12

x

y

x y xy x2

2 12 24 4

4 19 76 16

6 24 144 36

8 30 240 64

10 37 370 100

x y xy x2

� 30 � 122 � 845 � 220

x 2 4 6 8 10

y 12 19 24 30 37

� 2 � 0 implies a minimum in the y direction.

However, along the curve determined by y � x, wehave f � �2x2, which has a relative maximum at (0, 0).

45. fx �

fy �

Critical points:

47. fx � 8x3 � 22xy � 36x, fy � 4y3 � 11x2

Critical point: (0, 0)


1.

3. f(1, 1) � f(�1, �1) � 2

5. f(0, 2) � f(0, �2) � �4

7. (max);

f(0, 1) � �3 (min)

9. f(8, 7) � �18

11.

13.

15. ,

�56

�14 (min)f ��

4

�14, �

8

�14, �

12

�14� �

f� 4

�14,

8

�14,

12

�14� �56

�14 (max)

f�8, 4, 8

3� �256

3 (max)

f��2, ��2� � f ��2, �2� � e�2 (min)f��2, �2� � f ��2, � �2 � � e2 (max)

f ��3

2, �

1

2� � f ��3

2, �

1

2� �3

2

f �1

2,

1

2� �1

4

��7e, e�, ��7e, e�

y(x � 14y) ln y � (x2 � xy � 7y2)

xy(ln y)2

x2 � 7y2

x2 ln y

�2f

�2y

� � � �


17. 40 meters by 80 meters

19. 11,644 cubic inches, when x � 18, y � 36

21. r � 1.51 inches; h � 3.02 inches

23. Development (x), $2,000; promotion (y), $6,000

25. Output will be increased by 31.75.

27. H � 2R

29. smax � 4L

31. x � 8.93 cm, y � 10.04 cm

33. x � y � z �

35. 11.5 ft � 15.4 ft � 7.2 ft (front length) � (side length) � (height)

37. � � 306.12, which gives the approximate change per$1000. Since the difference is only $100, the profit isincreased by approximately 0.1($306.12) � $30.61.

39. (a) x � 35 units, y � 42 units(b) � � 14.33 is the approximate change in the maxi-

mum utility resulting from a one-unit increase inthe budget.

41. Increases by � �

43. Then, Q(x, y) � production; C(x, y) � px � qy � k. And Cx � p, Cy � q, therefore, Qx � �p; Qy � �q;

.Qx

p�

Qy

q

��

a��

��

b��1

3�V0

45. x � 0, y � �2. The critical point (0, �2) is an inflec-tion point, not a relative extremum.

47.

49.

51. f(2.1623, 1.5811) � 1.6723

53. f(0.9729, �0.1635) � 2.9522


1.

3. �1

5. 4 ln 2 � ln 16

7. 0

9. ln 3

11. 6 cubic units

7

6

dy

dx�

y

x2 � (1 � xy2)exy2

� ln (x � y) �x

x � y

2x2yexy2

�1

x�

x

x � y

�P

�K� �

�C

�K;

�P

�L� �

�C

�L ; � �

�P

�K

�C

�K

�

�P

�L

�C

�L

(0, –2)

(1, 0)

(C = 0)

(C = –1)

(C = 1)

(C = 2)

y

x

y = x5 + x – 2


13. ln 2 � ln 3 cubic units

15. cubic units

17.

19. 0.921

21. 0.53775

23. 0.2011

25. 0.4627

27. 235.75 units

29. $2,489,650

31. 630 ft

33. 0.3335

35. $118,370

37. 1.8908 cubic units

39. 0.228

Chapter 7, Review

1. (a) fx � 6x2y � 3y2 � , fy � 2x3 � 6xy �

(b) fx � 5y2(xy2 � 1)4, fy � 10xy(xy2 � 1)4

(c) fx � y(xy � 1)exy, fy � x(xy � 1)exy

(d) fx � , fy �

(e) fx � ,

fy �x

y(x � 3y)�

1

y�

3

x � 3y

3y

x(x � 3y)�

1

x�

1

x � 3y

�(x2 � 4xy � y2)

(2x � y)2

2(x2 � xy � y2)

(2x � y)2

1

x

y

x2

1

6

e2 � 1

2e2

2. (a) fxx � 2, fyy � 6y � 4x, fxy � fyx � �4y(b) fxx � 2(2x2 � 1)e , fyy � 2(2y2 � 1)e ,

fxy � fyx � 4xye

(c) fxx � 0, fyy � � , fxy � fyx �

3. Daily output will increase by approximately 16 units.

4. � 0

5. (a)

(b)

6. (a) (b) �

7. The level of unskilled labor should be decreased byapproximately two workers.

1

2

2

3

f = 2

f = 1

f = 0

y

x

2

1.5

1

�.5

�1

�1.5

.51 2 3�3 �2 �1

f = –2

f = 2

(0, 2)

y

x

�2Q

�K�L

1

y

x

y2

x2�y2

x2�y2x2�y2


8. (a) Relative maximum at (�2, 3); relative minimumat (0, 9); saddle points at (0, 3) and (�2, 9)

(b) Relative minimum at ; saddle point at

(c) Relative maximum at (2, �1); saddle points at

, (0, 0), and .

(d) Saddle points at and

9. Relative maximum: f(1, ) � f(1, � ) � 12; relative minimum: f(�2, 0) � 3

10. Area � f(x, y) � xy; perimeter � g(x, y) � 2x � 2y �

C; fx � y, gx � 2, fy � x, gy � 2; � � means

y � x, so the rectangle is a square.



13. Maximum profit will increase by approximately$234.57.

14. D � 0, fxx � 0, relative minimum at (2, 3), D � 0 forsome (x, y) where x � 0 and y � 0

15. (a) �4(b) 0.5466

(c)

(d) (e2 � e�2)

(e) 2 ln 5(f) 3.4366

16. 81

1

4

4

3

y

2�

x

2

�3�3

��2

3,

5

6��2

3, �

5

6��10

3, 0��40

9, �

25

3 �

�1

2, 1�

��23

2, 5�

17. 2

18. 18 cubic units

19. (e�2 � e�3) cubic units

20. 8.1667

21. e�8 � e�6 � e�2 � 1 � 0.8625

22. e�15/4 � e�5/2 � e�5/4 � 1 � 0.6549

23. x � y � z �

24. x � , y � , z �

25.

26. y � x � 1

3.5

3

2.5

1

1.5

2

0.50.5 1 1.5 2 2.5 3 3.5 4 4.5

y

x

4

9

�10; at (0, ��10, 0)

30

7

90

7

60

7

20

3

3

2


27. (a)

(b) y � 11.54x � 44.45(c) Approximately $102,150

28. Average temperature � 2°; Arnold will move.

29. (a) Dick should wait at x � 0.4243 miles; Mary should wait at y � 2.2361 miles. (2.06 miles from F )

(b) Tom, Dick, and Mary will win by 0.2080 hours (13 minutes).


30. �

31. Q(x, y) � xayb

Qx � axa�1yb; Qy � bxayb�1

xQx � yQy � x(axa�1yb) � y(bxayb�1)

� (a � b)xayb � (a � b)Q

If a � b � 1, then xQx � yQy � Q.

32. (a) �

�

sw � st � 0 only when t � 0

(b) 28.9 m(c) Writing exercise; responses will vary.

W � B

k�

W � B

ke�kgt/w�s

�t

�t(W � B)

kWe�kgt/wt

k�

1

k2g(2W � B)(e�kgt/w � 1)

�s

�W

D

2V

160

140

120

80

100

442 4 6 8 10 12

y

x


Appendix: Algebra Review, Section A1

1. 1 x � 5

3. x � �5

5.

7.

9. 4

11. 5

13. �3 � x � 3

15. �6 � x � �2

17. x � �7 or x � 3

19. 125

21. 4

23. 4

25.

27.

29.

31. 2

33.

35. n � 10

37. n � 1

39. n � 4

41. n �

43. x4(x � 4)

45. �25(x � 7) � 25(7 � x)

47. 2(x � 1)2(x � 2)2(7x � 2)

13

5

1

4

1

4

1

2

1

2

–2 0 x

2 x

49. x�1/2(6x � 1)

51. 2(x � 3)

53.

55. 34

57. 0

59.

61.

63.

65. (a) Surface area is approximately 5.212 � 108 km2;mass of the atmosphere is 5.212 � 1018 kg.

(b) 127,400 years

Appendix: Algebra Review, Section A2

1. (x � 2)(x � 1)

3. (x � 3)(x � 4)

5. (x � 1)2

7. (4x � 5)(4x � 5)

9. (x � 1)(x2 � x � 1)

11. x5(x � 1)(x � 1)

13. 2x(x � 5)(x � 1)

15. x � 4; x � �2

17. x � �5

19. x � 4; x � �4

21. x � � ; x � �1

23. x � �3

2

1

2

�8

j�1 (�1) j�1j

�6

j�1 2xj

�6

j�1 1

j

2(x � 3)3(5 � x)

(1 � x)3


25. x � 1; x � �5

27. x � 1; x � �2

29. x � �1

31. x � � ; x � �1

33. No real solutions

35. x � �

37. x � 3; y � 2

39. x � 4, y � 2

41. x � �7, y � �5 and x � 1, y � �1

Appendix: Algebra Review, Review Problems

1. �2 � x 3

2. x � 4

3.

4.

5.

6.

7. 3

8. 3

9. 2 � x � 4

10. x �2 or x � 1

11. 243

12.

13. 4

1

16

2 7 x

1 x

–1 5 x

–3 2 x

3

2

1

2

14.

15.

16. 243

17. 73

18.

19.

20.

21. n �

22. n �

23. n � �1

24. n � �5

25. 21

26. 54

27. 95

28. � 0.3348

29.

30. 3j2

31. x2(x � 3)(x � 3)

32. (x � 3)(x2 � 3x � 12)

33. x4(x6 � 4)(x3 � 2)(x2 � 2)

34. �2(x � 3)2(5x2 � 16x � 1)

35. x(x � 1)2

36. 4x(x � 2)

37. (x � 5)(x � 3)

�5

j�1

1 � �7

k�2 (�1)k

k

1

4

7

18

9

�3 4

3

2

16�2

16 4�2

1

343


38. (2x � 1)(x � 3)

39. (2x � 3)2

40. (4x � 3)(3x � 1)

41. (x � 1)(x � 1)(x � 3)

42. (x � 1)2(x � 2)(x � 2)

43. x � �4; x � 1

44. x � � ; x � 2

45. x � �7

46. x � �8; x � 8

47. x � �1; x � 2

48. x � 3

2

1

2

49. x � � ; x �

50. x � � ; x �

51. No real solutions

52. x � 0.362; x � �0.790

53. x � �2; x �

54. x � �1.129; x � 4.871

55. x � �2, y � 1

56. x � 3, y � 1

57. x � 1, y � 2 and x � 15, y � �26

58. x � 2.093, y � 3.234 and x � 0.573, y � �0.567

1

3

5

8

2

3

1

2

3

7

Answers to Odd-Numbered Problems and Review Problems calculus... · 2010-08-24 · 628 Answers to Odd-Numbered Problems and Review Problems 31. V S; V increases by a factor of . 33.

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