Problem Set 65

24. (a)

l = b2_ (~)\2

y = ~b2 _ (;)\2,(b) r ~b2 - (;)\2 dx

x> 0

25. From problem 24 in Problem Set 63, these curvesintersect when x = t. In the same problem,Yl = (1 - x2)112 and Y

2= (_x2 + 2x)1I2. In the

fourth quadrant,Y

1= -(1 - x2)112

y1' = x(1 - x2)-112

Y2

= _(_x2 + 2x)1I2

y2' = (x - 1)(_x2 + 2x)-1/2

1m1 = y/ll12 = .J3

1m2 = y2'1112 = -.J3

m1

- m2tan 8 =

1 + m1m2

8 = ~-,[ ~: t18 = tan" -f3 = 60°

PROBLEM SET 65

dx = -100 mdt s

1. x = 2000m

cot8 =x

1000x = 1000 cot 8

dx = -1000 csc? 8 d8dt dt

. (-100) = _1000(1000.J5)2 d81000 dt

~ = 5 d810 dt

d8 1 rad= ---

dt 50 s

142

2. aCt) = -9.8

vet) = -9.8t

h(t) = -4.9t2 + 500

h(3) = 455.9 m

v(3) = -29.4 mts

a(3) = -9.8 mts2

3. aCt) = -9.8

vet) = -9.8t + 30

h(t) = -4.9t2 + 30t + 100

o = -9.8t + 30

t '" 3.0612 s

h(3.0612) '" 145.9184 m

4. Y = cos x

x = cosy

. dy= -sm y-dx

dydx = -cscy

dy 1=

dx --/1 - x2

~rl(x) = 1dx --/1 - x2

(f-l),(0.2) '" -1.0206

5. . xarcsm -3

y =

=-J55

6. xy = arctan-

2

dydx

Calculus, Second Edition

Problem Set 65

7. f(x) = 2x3 - 3x2 - I2x + 7

rex) = 6x2 - 6x - 12

o = 6(x - 2)(x + 1)

11. W = r 2x dx

= [x2]~

= 4 - 1

Critical numbers: x = -2, -1,2,3

f(-2) = 3; fe-I) = 14; f(2) = -13; f(3) = -2

= 3joules

12. f(x) = ax2 + b

rex) = 2ax

r(2) = g'(2)

4a = 4 + a

g(x) = x2 + ax

g'(x) = 2x + aMaximum: 14 Minimum: -13

8. Critical numbers: x = -1, 1,3

fe-I) = 2 f(1) = 0 f(3) = 2

Maximum of 2 at x = -1 and x = 3.

Minimum of 0 at x = 1.

4a = -

3

With f(l) = 5, a + b = 5

4- + b = 53

9. y

b = 113

13. f cos (2x) [1 + sin (2x)]-I12 dx

= ~ f 2 cos (2x) [1 + sin (2x)]-112dx

= ~1 + sin (lx) + C

I I I I I I •• X-3 -2 -1 I 1 2

Maximum: 6 Minimum: 1

110. (a) A = -bh

2

= .!..(2x)(3 - 1-x2 )

2 12

1= 3x - _x312 15. y

f 2 1 f 214. xe" +Jr dx ="2 2xex +Jr dx

1 x2+Jr + C- -e- 2

(b) A' = 3 - .!..x2

4

y=x

Y= 5

1o = 3 - _x2

4I ¥ I I I •• x1_x2 = 3

4

x2 = 12x = 2.../3

A = 3(2-!3) - 1-(24-!3)max 12

= 6.../3 - 2.../3= 4.J3 units2

A = f~(x1l2 - x) dx

= [~x3/2 _ .!..X2]13 2 0

=2 13 2

1= - unit2

6

Calculus, Second Edition 143

Problem Set 65

16. (a) 3x2- k2 = _k2x2 + 3

(3 + k2)x2 = 3 + k2

x2 = 1

x = ±1

y

--IHHHI\-~- x

A(k) = r==~1[(_k2x2 + 3)

- (3x2 - k2)] dx

= [- k32

x3 + 3x - x

3 + k2

XI~~I= (_k3

2

+3-1+e)

_ (~2 _ 3 + 1 _ k2 )

= (~k2 + 2) - ( _~k2 - 2)

(b) A(k) = .±k2 + 4 = 73

'±e = 33

k = 32

17.dA = 5 units2

k = 15-dt s

A = .±k2 + 43

dA ~k dk- =dt 3 dt

(5) = ~(15) dk3 dt

dk 1 unit- =dt 8 s

144

18. r f(x) dx + s: f(x) dx = f f(x) dx

r f(x) dx + (-2) = 10

r f(x) dx = 12

19. f" ~ dx = [In x]~ = 1 - 0 = 1J1 X

20. y = 2 sinx

dy = 2 cosxdx

d2y = -2 sinxdx2

d3y = -2 cosxdx3

d4y = 2 sinxdx4

d4yl = 2(1) = 2dx4nfl

21. y = (sin x)1f2 + e2x cos x + (In x)(csc x)

y' = ~(sin x)-1I2 cos x + e2x(-sin x)2+ 2 cos x e2x + (In x)(-csc x cot x)

+ (csc x{~)y'= cosx 2 •JSiDX + e X(2 cos x - sm x)2 sin x

+ csc x (~ - In x cot x )

22. y

-1I

IIIII

Concave up: (-2, -1), (2, 00)

Concave down: (-00, -2), (-1,2)

23.eX _ e2

lim ---x~2 X - 2

Calculus, Second Edition

(XI) x2 1 124. A. f- =-:;:---x2 xI xI x2

B. f[~) = In ~ = Inxi - Inx2x

2x2

C. f[ ::) = [::)' * x,' - x,'

D. f[~) = sin ~ :;: sin XI - sin x2x2

x2

The correct choice is B.

25. y=2x+by2 = 4x2 + 4bx + b2 y2 = 4x

The curves intersect when

4x = 4x2 + 4bx + b2

o = 4x2 - 4x + 4bx + b2

o = 4x2 + (-4 + 4b)x + b2

4 - 4b8~ _

~16 - 32b + 16b2 - 4(4)(b2)+ ~--------~~- 8

X =

For 2 distinct roots, or points of intersection:

16 - 32b + 16b2 - 16b2 > 032b < 16

1b < -

2

PROBLEM SET 66

1. W = r (x2 - 3x) dx

[1 3 J5

= "3x3 - "2x2 I

_ 125 75 (1 3)- ---- ---3 2 3 2

16= - joules3

2. Acceleration due to gravity is always -9.8 m/s2•

3. aCt) = -9.8vet) = -9.8t + 20

h(t) = -4.9t2 + 20t + 160

h(2) = 180.4 mv(2) = 0.4 m/s

a(2) = -9.8 m/s2

Calculus, Second Edition 145

Problem Set 66

4. u = nx2 du = 2nx dx

S~xcos(nx2)dx

1 i1C

2n 0 cos U du

12n[sin u]~ = 0

5. u = cos (5x) du = -5 sin (5x) dx

s: [sin (5x)]eCOS (5x) dx

1 S-I 1 J'-- eU du = - eU du5 I 5 -I

![eu]1 = !'(e _ !)5 -I 5 e

6.

~1

~

y = csc X

X = csc y

dy= -csc Y cot Y dx

dydx = -sin y tan y

dy - (!)( 1 1dx--x ~

dydx

-1

= x~x2 - 1

-1(f-I),(X) = x.Jx2 _ 1

7. . XY = arcsm -

3

dy =dx

1.)9 - x2

1 1fl=dY!

dx 3/2 - ~9 - ~ =

8. y = arctan (sin x)

dy _dx

cos xsin2 x + 1

2-J39