3
., Problem Set 35 19. Let '.,.'1 =;:.:: 2 ::. in':: 1/;:'::). x -.03: -.02 UJi. .01 .02 .03: X=0 20. eX = y x = Iny 21. (sin -x{ cos (~ - x)][ see? (~ - x)] = -sin x sin x csc 2 x = -sin 2 x csc 2 x = -1 In x 22. y = log x = -- 2 In 2 23. y2 = 4x 2 + 16 represents a hyperbola. Let ',,·'1=..f'::4::<2+1E,) and '.,.'2= -V1. \V /1\ Vertices: (0,4), (0, -4) 24. 7! 4! 3! = 7.5 = 35 ways 1 25. mLD = -x 2 1 mLB = -y 2 1 mLCED = mLB + mLD = -(x + y) 2 72 -11 PROBLEM SET 35 1. L = w SA = 2Lw + 2wH + 2LH 100 = 2L2 + 2LH + 2LH 100 = 2L2 + 4LH 4LH = 100 - 2L2 H = 25L- 1 - .!.L 2 v = LwH = L(L) (25C 1 - ±L ) = (25L - ~L3 ) cnr' 2. S = S e kt t 0 60 = 50e k 6 k - = e 5 k = In 1.2 S = 50e(ln 1.2)(6) 6 = 50(1.2)6 = 149.2992 fathoms/s 3. f 3 sin x dx = 3 f sin x dx = -3 eos x + C 4. f 2t- l12 dt = '2 f rll2 dt = 2(2t Il2 ) + C = 4t 1/2 + C f 1 113 1 f V3 -_ -2 1 (-43U4l3) + C 5. "2 u du ="2 u du -- = ~U4/3 +C 8 6. f 3x dx = 3 f x dx = 3( ±x 2) + C = i x 2 + C 7. (2x 2 ) dy + y(4x) + 2y dy = -sin x dx dx dy (2x2 + 2y) = -sin x - 4xy dx dy dx = -sin x - 4xy 2x 2 + 2y Calculus, Second Edition Ii ~

1 2 ::. 1/;:'::). x - s3.amazonaws.com · 8. u2 + v2 = 2uv du dv dv du 2u-+ 2v-= 2u-+ 2v-dtdt dt du dv dv - (2u - 2v) = 2u - - 2v - dt dt dt dv 2(u - v)-du = dt dt 2(u - v)-du--dvdt

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Page 1: 1 2 ::. 1/;:'::). x - s3.amazonaws.com · 8. u2 + v2 = 2uv du dv dv du 2u-+ 2v-= 2u-+ 2v-dtdt dt du dv dv - (2u - 2v) = 2u - - 2v - dt dt dt dv 2(u - v)-du = dt dt 2(u - v)-du--dvdt

.,Problem Set 35

19. Let '.,.'1 =;:.:: 2 ::. in':: 1/;:'::).

x-.03:-.02UJi..01.02.03:

X=0

20. eX = y

x = Iny

21. (sin -x{ cos (~ - x)][ see? (~ - x)]

= -sin x sin x csc2 x = -sin2 x csc2 x = -1

In x22. y = log x = --

2 In 2

23. y2 = 4x2 + 16 represents a hyperbola.

Let ',,·'1=..f'::4::<2+1E,) and '.,.'2= -V1.

\V/1\

Vertices: (0,4), (0, -4)

24. 7!4! 3!

= 7 . 5 = 35 ways

125. mLD = -x2

1mLB = -y2

1mLCED = mLB + mLD = -(x + y)2

72

-11

PROBLEM SET 351. L = w

SA = 2Lw + 2wH + 2LH

100 = 2L2 + 2LH + 2LH

100 = 2L2 + 4LH

4LH = 100 - 2L2

H = 25L-1 - .!.L2

v = LwH

= L(L) (25C1- ±L )

= (25L - ~L3 ) cnr'

2. S = S ektt 0

60 = 50ek

6 k- = e5

k = In 1.2

S = 50e(ln 1.2)(6)6

= 50(1.2)6 = 149.2992 fathoms/s

3. f 3 sin x dx = 3 f sin x dx = -3 eos x + C

4. f 2t-l12 dt = '2 f rll2 dt = 2(2tIl2) + C

= 4t1/2 + C

f 1 113 1 f V3 -_ -21(-43U4l3) + C5. "2u du ="2 u du --

= ~U4/3 + C8

6. f 3x dx = 3 f x dx = 3( ±x 2 ) + C = ix 2 + C

7.

(2x2) dy + y(4x) + 2y dy = -sin xdx dx

dy (2x2 + 2y) = -sin x - 4xydx

dydx = -sin x - 4xy

2x2 + 2y

Calculus, Second Edition

Ii~

Page 2: 1 2 ::. 1/;:'::). x - s3.amazonaws.com · 8. u2 + v2 = 2uv du dv dv du 2u-+ 2v-= 2u-+ 2v-dtdt dt du dv dv - (2u - 2v) = 2u - - 2v - dt dt dt dv 2(u - v)-du = dt dt 2(u - v)-du--dvdt

8. u2 + v2 = 2uv

du dv dv du2u- + 2v- = 2u- + 2v-

dt dt dt dt

du dv dv- (2u - 2v) = 2u - - 2 v -dt dt dt

dv2(u - v)-du = dtdt 2(u - v)

du dv- --dt dt

9. y2 _ x2 = 1

dy2y - - 2x = 0dx

dy x- =-dx y

dyl _ Q _ 0dx (0,1) 1

y = 1

10. The graph of y = .J(x + 2)2 crosses the x-axis atx = 0 and touches the x-axis at x = -2. When x islarge and positive, y is large and positive. When x islarge and negative, y is large and negative.

The correct choice is A.

110/\ 0'(\", n 0'(\12. s(t) = tI/3 - 2tI12 + 3t-I

s'(t) = _!'r2l3 - r1l2 - 3t-23

s"(t) = _3..t-513 + _!'r3l2 + 6t-39 2

s"(2) = _3.. (2)-513 + !(2)-312 + 6(2)-39 2

s"(2) "" 0.8568

ex+Llx _ eX1im

Llx-tO Llx- !!.-ex = eX- dx13.

1. x2 + X - 61m =

x-t2 X - 2

= 1im (x + 3) = 5x-t2

1im --'-(x_+_3----')----'-.(x_-_2---=-)x-t2 X - 2

14.

Calculus, Second Edition

Problem Set 35

15. 1. x + 11m ---

x~oo X

1 +!= 1im x..

x-t~ 111 + 0

1

16. Y

-2I ! I I I •• X

(_00,00)

17. 82x-I = 4(23)2x- 1 = 22

26x-3 = 22

6x-3=2

5x = -

6

18. =lJ 2 0 1 k~ -2 2 -32 -2 3 [ill

k - 3 = 0

k = 3

19. Y

t Jf

x2 3

20. g(x) = -3 + 2f( x - ~)

= -3 + 2 sin (x - ~)Y

x---A-1

-2Jr -3 I {-7r .11"1 < I }

-5

73

Page 3: 1 2 ::. 1/;:'::). x - s3.amazonaws.com · 8. u2 + v2 = 2uv du dv dv du 2u-+ 2v-= 2u-+ 2v-dtdt dt du dv dv - (2u - 2v) = 2u - - 2v - dt dt dt dv 2(u - v)-du = dt dt 2(u - v)-du--dvdt

Problem Set 36

21. 2 sin2 x - 3 sin x + 1 = 0

(2 sin x - l)(sin x-I) = 0PROBLEM SET 36

1. Circumference of circle:

C = 2,.r = 20,. em1sinx =

2sin x = 1

x =,. 5,.6'6

Circumference of cone:

2,. - x---(20,.) = 10(2,. - x) cm2,.

x =,.2

I" .::...f..:....(x_+_h~)_-~f--,,(x....:..)22. f'(x) = imh--+O h

= lim [2(X + h)3 + 3(x + h) - 4 - 2x3h--+O h

+ -3xh+ 4J

= lim [2x3 + 6x2h + 6xh2 + 2h3 + 3x + 3hh--+O h

+ ~4 - 2x3

h- 3x + 4]

= lim 6x2h + 6xh2 + 2h3 + 3hh--+O h

= lim (6x2 + 6xh + 2h2 + 3) = 6x2 + 3h--+O

Radius of cone:

2m = 10(2,. - x)

r,. = 5(2,. - x)

10,. - 5xr =

12. y = -x3 - x

3

y' = x2 - 1

o = x2- 1

o = (x + l)(x - 1)

23. x2 - 2x + y2 + 4y = 4

(x - 1)2 - 1 + (y + 2)2 - 4 = 4

(x - I? + (y + 2)2 = 9

Critical numbers: x = -1,1

This represents a circle centered at (1, -2).

Y1=-2+(9-(X-l)~)Y~=-2-(9-(X-l)~) Local maximum: (-1, ~)

Local minimum: (1, -~ )

"""'-f ••

\U3. f(x) = x3 - 2.x2 + 6x + 3

2

f'(x) = 3x2 - 9x + 6

o = 3(x2 - 3x + 2)o = (x - 2)(x - 1)

Critical numbers: x = 1, 224. A=

A =

52 Local maximum: (1,~1)

Local minimum: (2, 5)

74 Calculus, Second Edition