19. Centerline: y = 35 Amplitude: 40

n C = 2n = 4 5n 7nPeriod: "2 Phase: - or-

n 8 8-2

y = 35 + 40 sin [ 4( e - 5;)] or

y = 35 - 40 sin [4( e _ 7;)]

20. y

I .., I I I I •• X

Answers may vary: (1,0), (0, -1)

!= 11

slope "" 0 - (-1)1 - 0

21. The argument of a logarithm must be positive, so-x > 0, or x < O.

Domain: {x E IR I x < O}Range: IR

22. sin2 x + cos2X =

sin243° + cos243° = 1

23. (a) tan (2A) = sin (2A)cos (2A) =

2 sin A cos Acos? A 2 tan A

I-tan2A

2 sin A cos Acos? A - sin? A

= cos 2 A sin 2 A =cos2 A - cos2 A

(b) tan (2A) = 2 tan A 2(2)1 - tan2 A = 1 - (2)2

4 4- 1 - 4 = - -

3

24. 1 1 1V = (X2)1/3 = X 213

3r--=z _ ( -2) 113 _ -213-i x : - x - x

X-213

With x > 0 the quantities are equal: C

25. By drawing a diagonal and "triangulating" thequadrilateral, the quadrilateral is split into twotriangles each having a sum of interior angles of1800

• Therefore, x + y + z + t = 3600

Calculus, Second Edition

Problem Set 17

PROBLEM SET 1740M + 60B1. Average rate = • • _ mph

2.~

~I

w

lw = 400

I = 400w

(400 )P = 2(1 + w) = 2 ---;- + w

= SOOw-1+ 2w

3. 1. 3x3 - 2x + 4im

x-+~ 1 - 2x3

3 - 0 + 00-2

4. x3 - 6xlim 2

x-+-~ 5x + x

- 00

3 2 4--2 +-

lim x x3

x-+~ 1- - 2x3

32

limx~-oo

6x--

x

~ + 1x

-00 - 0

o + 1

5. 1. x2 a2 li (x + a)(x - a)1m = 1m -'----'-'-----'-x-+a X - a x-+a X - a

lim (x + a) = a + a = 2ax-+a

6. x alim 2 2

x-+a X + a

7. lim f(x) = 00x-+-l

s. lim f(x) = -00

x-+l-

9. lim f(x) = 00

x-+l+

a - aa2 + a2

o2a2 o

10. The limx~l f(x) is undefined because the limitfrom the left does not equal the limit from the right.

11. Increasing: (_00, -1)

Decreasing: (-1,1), (1, 00)

33

Problem Set 17

12. sin2 x + 2 cos x - 2 = 0

(1 - cos2 x) + 2 cos x - 2 = 0

cos? X - 2 cos x + 1 = 0

(cosx - 1)(cosx - 1) = 0

cos x = 1

x = 0

13. 21n x = In (x - 1) + In (x - 2)

In x2 = In (x2 - 3x + 2)

x2 = x2- 3x + 2

3x = 2

2x = "3

Since this gives a negative argument there isno solution.

14. 42x = 16' - x

42x = (42)' - x

42x = 42 - 2x

2x=2-2x

4x = 2

1x =2

15. y = e'

In y = In (e")

x = lny

16. y = -2 + 3 sin (4x)

Amplitude: 3 Period:27r 7r- =

4 2

Centerline: y = -2

17. For the key trigonometric identities, seesection 12.A in the textbook.

cos(14) = cos(A + A)

cos (14) = cos A cos A - sin A sin A

cos (14) = cos2 A - sin2 A

cos (14) = cos? A - (1 - cos2 A)

cos (14) = 2 cos? A-I

2 cos2 A = 1 + cos (14)

1 1= 2" + 2" cos (2A)

34

18. 2 sin (7r _ x) _1_ - 12 sec -x

= 2 cos x (_1_) - 1 = 2 cos x (cos x) -sec x

= 2 cos2 x-I = cos (2x)

19. X Y1.006 1.0312.005: 1.0269.00Lt 1.0223.003 1.0176.002 1.0125:t&i. 1.0069ERROR

X=0/(0.003) = 1.0176

/(0.002) = 1.0125

/(0.001) = 1.0069

lim f(x) = 1x->o+

20. Find the slope perpendicular to the given line.

2y - x + 3 = 6

2y = x + 3

1 3y = -x +-2 2

1- slope = -2

Find a perpendicular line through (1, -I).

y + 1 = -2(x - 1)

y = -2x + 1

Find the intersection of 2 lines.

1 3-x + - = -2x +2 25 I-x =2 2

1x =

5

Intersection: (-~,~)

Find the distance between 2 points.

d= ~(~+lr+(-~-lr= c;r + (~r= ) 1:~=

6-155

Calculus, Second Edition

Problem Set 18

21. 42p-5 = 73p+2PROBLEM SET 18

(2p - 5) In 4 = (3p + 2) In 7

2p In 4 - 5 In 4 = 3p In 7 + 2 In 7

2p In 4 - 3p In 7 = 2 In 7 + 5 In 4

p(2 In 4 - 3 In 7) = In 49 + In 1024

In 50,176p =

In 16 - In 343

11. RA = "6 1R = -

B 3

(~ + ~)T = 1

1-T = 12

T = 2hr

22. PH = 20,000

H = 20,000P

y 2.20,000 I H

sq. meters

p

F=2H+P

= 2eO~00) + P= 40,000p-l + P

I---=r= 1.1 I I I •. X

Answers may vary: (1,2), (-0.5,0)

2 -2-0 --=1.3slope » 1 _ (-0.5) - 1.5 3. (f + g)(x) = x2 + 1 + ~

(f + g)(5) = 52 + 1 + ~ = 28

4. (fg)(x) = (x2 + 1) ~(fg)(5) = (52 + 1) -!5=1 = 52

23. lim f(x + h) - f(x)h~O h

1. 2(x + h)2 - 2x21m ---'---""""'------

h~O h 2 + 15. ( f )eX) = ~x-I

52 + 1 _ 13(f}5) = -!5=1 -I. 2(x2 + 2xh + h2) - 2x21m ~~------~~~

h~O h

. 2x2 + 4xh + 2h2 - 2x2=hm----------

h~O h 6. x-I> 0x > 1

{x E IR I x > I}

7. (f 0 g)(x) = f(~) = (~)2 + 1

= x-I + 1 = x

= lim h(4x + 2h)h~O h

= lim (4x + 2h) = 4x + 2(0) = 4xh~O

24. x + y + z = 1800

y + a + b = 1800

(straight angle)

(sum of interior angles)(f 0 g)(3) = 3

8. x-I ~ 0x ~ 1The quantities are equal: C

Domain: {x E IR I x ~ I}Range: {y E IR I y ~ I}

25. (x + y)2 = x2 + 2xy + y2

= (x2 + y2) + 2xy

= (20) + 2(8)

= 36

9. g(x) = 2x - n

x --- I 9 I --- (2x - n)

If (x + y)2 = 36 then x + y = ±6. Since x and yare both positive x + y = 6.

((x) = cos x

(2x - n) --- I ( I --- cos (2x - n)

Calculus, Second Edition 35