Problem Set 80

22. y = x + sin x + arctan (X2) + e" csc (2x)cos x

dy =dxcos x(l + cos x) - (x + sin x)(-sin x)

cos2 x

+ 2 ~ + eX[-2 csc (2x) cot (2x)](x) + 1

+ eX csc (2x)

l+cosx sinx(x + sin x)+

cos x cos? X

dydx

+ ~ + e' csc (2x) [-2 cot (2x) + 1]x4 + 1

dy = see x + 1 + see x tan x (x + sin x)dx

+ ~ + eX csc (2x) [-2 cot (2x) + 1]X4 + 1

dy = see x + 1 + x see x tan x + tarr' xdx

2x [ ]+ -4-- + eX csc (2x) -2 cot (2x) + 1x + 1

~ 2 2x= see x + x see x tan x + see x + -4--dx x +1

+ eX csc (2x) [-2 cot (2x) + 1]

d 2x!x = see x (1 + xtanx + secx) + X4 + 1

+ e' csc (2x) [-2 cot (2x) + 1]

hex) = f(g(x) = f(sin x) = sin2 xh'(x) = 2 sin x cos xh'(x) = sin (2x)

23.

24. lim f(a + h) - f(a) = rea)h....•O h

I· f(x) - f(a)imX •...• a X - a

The correct choice is B.

25. (a)

~

J

\J(b) f(x) = 2x3 - 3x2 - 12x + 20

rex) = 6x2- 6x - 12

o = 6(x2 - X - 2)o = (x - 2)(x + 1)x = -1,2

f(-I) = 27, f(2) = 0

(-1,27), (2, 0)

184

PROBLEM SET 801. aCt) = 2t

vet) = t2 + C

v(3) = 32 + C

1O=9+C

C = 1

vet) = t2 + 1

17 = t2 + 1

t2 = 16

t = -4,4

2. y

4

dy

----.j!'---!L.....o_ X

w = S depth . weight density . volume

= S: (4 - y)(2000)(1)(3) dy

= 6000 S: (4 - y) dy

= 6000[ 4y - ±y2 I= 6000(8 - 2) = 36,000 joules

3. (a) aCt) = -9.8

vet) = -9.8t + C

v(/) = -9.81 + 50x(t) = -4.9t2 + SOt + C

x(/) = -4.9t2 + SOt + 160

(b) vet) = -9.8t + 50

o = -9.8t + 50

t "" 5.1020 s

(c) x(t) = -4.9t2 + SOt + 160

o = -4.9t2 + SOt + 160

-50 ± ~2500 - 4(-4.9)(160)t =

-9.8

t "" 12.7626 s (t '" -2.5585 s)

Calculus, Second Edition

I 12 I( 2)-112(b) V = -7rX . - 48 - x (-2x)3 2

2+ -7rx(48 - x2)lI23

-.!.7rx3(48 - x2t1l23

2+ -7rx(48 - x2)1123

.!.7rx(48 - x2)-1I2 [_x2 + 2(48 - x2)]3

.!.7rx(48 - x2t1l2 (-3x2 + 96)3I _-7rx(48 - x2) 1/2 (-3x2 + 96)3

4. (a)

(c)

Problem Set 80

s. . 3x5 - 2x3 + 1 3lim = -

x~~ 2x5 - 1 2

Horizontal asymptote: y = ~2

x + 1 16. y = -- = 1 +-

x xx2 + y2 = (4f3?y2 = 48 _ x2

y= ~x2

1-7rr2h3

1 2 f-7rX ,,48 - X23

1· x + 1 11m --- =X~~ X

V=Horizontal asymptote: y = 1

Vertical asymptote: x = 0

Zero: x = -1v= y

V' -

6

5

4

3

2

-•• - - - - - - - 1

1 1 1 1 ~ I 1 1 1 I. X-5 -4 -3 -2 -1 1 2 345

V' --3'

-4

V' - -24x + 6x27. y = 3x2 _ 27

6x(x - 4)3(x + 3)(x - 3)

0=1· -24x + 6x2 21m =

X~~ 3x2 - 27x = 0 -3x2 + 96 = 0

x2 = 32x = 4.J2em y = 4em

Horizontal asymptote: y = 2Vertical asymptote: x = 3, x = -3

Zero: x = 0,4

(4,0)

y

I----!J,------- I

1 * 1'-= •. x

Window: XIYJi n= -10, XIYJax= 10,YIYJin=-30, YIYJax=150

-3IIIII

"IlXifVlUfVl~=5.656B526 1'1'=13'1.0'1129

(d) V = .!.7r(4.fi)2~48 - (4.fi)231 128= -7r(32)(4) = -7r em 33 3

Calculus, Second Edition 185

Asymptote: y = x + 3

Vertical asymptote: x = 3

Zero: x = -1, 1

x2 - 19. y =

x - 3

8y= x+3+--

x - 3

18

16

14

12

10

8

6

4

(x + 1)(x - 1)=

x - 3v = f 7rr2dx

= 7r f>:=4 y2 dx>:=0

y

= 7r f04

X dx

= 7r[~x2 J:= 87runits''

15. yy = x3 y = X

...•"1IIII

3 4 5 6 7 8 9I

IIIII A = 2 f~(x - x3

) dx

[ J1 2 1 4= 2"2x -"4x 0

= 2(~- ±)1 't2= -um2

10. lim x lim 1 1-x--+o sin (45x) >:--+0 45 cos (45x) 45

2 r 2x11. ti x lim 2x2 =m-- 1m - 00

x--+~ In x x--+~ 1 x--+~

x

12. I, cos x-IIm----

x--+o 52 sin x

186

-sin xlim --->:--+0 52 cos x

o52

= 0

Calculus, Second Edition

Calculus, Second Edition 187

Problem Set 81

16. x2 + y2 = 25f

,ex21. (In x + 43X

) dx = x In x - x + -- + CIn 43dy

2x + 2y dx

dydx

= 0

x 22. f J x dx- 6 - 4X4

- ! f 4x dx- 4 ~(.)6)2_ (2x2)2

1 2x2

= "4 aresin -16 + C

=y

2 = x

Y2y = -x

2.,j~25---x-:'-2 =-x

4(25 - x2) = x2

5x2 = 100

x2 = 20

x = ±2j5(2-J5, --J5), (-2-J5,-J5)

f x dx23. 6 + 4x'

- ~1~ f .)6(4x) dx- 4.)6 (.)6)2+ (2x2)2

1 2x2= -arctan- + C4-16 -16

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

3 + r f(x) dx = 5r f(x) dx = 2

24. f(x) = sin (arctan x)

Range of arctan x: {y E IR I - ~ < y < %}Range of sin (arctan x): {y E IR I -1 < y < I}

25. lim f(x) = f(1)x~l-

lim (2x + 1) = a +x~l-

18. y = arctan (sin x) + x21n [sin xl + esecx

dy cos x 2 COS x In I' I (2x)- = + x -- + Sill Xdx sin 2 x + 1 sin x

+ see x tan x esecx

3 = a +a = 2

PROBLEM SET 81dy =dx

eosx 2 •2 + x eot x + 2x In ISID xl

SID x + 1+ seex tan x eSecx

1. 3Y = --x

4

y

19. LW = 100

L dW + W dL = 0dt dt

L(-0.8) + W(5) = 0

L(-0.8) + 100 (5) = 0L

-0.8L + 500el = 0

500 = 0.8LLL2 = 625

L = 25 em, W = 4 em

I I I I ''.<:: I •. x

W = f depth . weight density . volume

20. u = x3 + eX du = (3x2 + eX) dx

= f;:: (3 - y)(9000)(-x)(l0)dy

124= 90,000 -y(3 - y) dyo 3

= 120,000 f: (3y - y2) dy

= 120,ooo[ly2 _ .!.i]22 3 0

= 120,000 (6 - ~) = 400,000 joulesf 3x2 + e" f 1

3 dx = - du = In lul + Cx + eX u

= In Ix3 + eXI + C