J - s3.amazonaws.com fileProblem Set 80 22. y = x + sin x + arctan (X2) + e" csc (2x) cos x dy = dx...
Transcript of J - s3.amazonaws.com fileProblem Set 80 22. y = x + sin x + arctan (X2) + e" csc (2x) cos x dy = dx...
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