A- 0 Answers t o Odd-Numbered Problems

CHAPTER 1 INTRODUCTION TO CALCULUS

Section 1.1 Velocity and Distance
(page 6)
2for 0 < t < 10 0 for 0 < t < T
1v = 30,0, -30;v = -10,20 3 v(t) = 1for 10 < t < 20 v(t) = for T < t < 2T
-3for 20 < t < 30 0 for 2T < t < 3T
20for t < .2 20t for t 5 .2
5 25; 22; t + 10 7 6; -30 9 v(t) = {
Ofor t > .2 1110%; l2$%
29 Slope -2; 15 f 5 9 3 1 v(t) =
8 for O < t < T 8t for 0 5 t T
-2 for T < t < 5 T lt) = { lOT -2t for T 5 t _( ST
47 %v;;V
49 input * input -+ A input * input -+ A B * B -+ C input +I + A
input +A --+ output input +A --+ B B + C --+ output A * A -+ B
A + B --+ output
6 1 3t + 5,3t + 1,6t -2,6t - 1,-3t - 1,9t -4; slopes 3,3,6,6,-3,9
Section 1.2 Calculus Without Limits
(page 14)
12 + 5 + 3 = 10;f = 1,3,8,11;10 3 f = 3, 4, 6, 7, 7, 6; max f at v = 0 or at break from v = 1to -1
5 1.1,-2,s; f (6) = 6.6, -11,4; f (7) = 7.7, -l 3, 9 7 f (t) = 2t for t 5 5,10 + 3(t - 5) for t 2 5; f (10) = 25
9 7, 28, 8t + 4; multiply slopes 11f (8) = 8.8, -15,14; = 1.1,-2,5
13 f (z)= 3052.50 + .28(x -20,350); then 11,158.50 is f (49,300) 1 5 19+%
1 7 Credit subtracts 1,000, deduction only subtracts 15% of 1000 1 9 All vj = 2;vj = (-l)j-' ;vj = ($)j
2 1 L's have area 1,3,5,7 23 f j = j ; sum j2+ j ; sum + 25 (1012 -9g2)/2 = 7 27 V j = 2 j 29 f31 = 5
31 a j = -f j 35 0; 1; .1 35 v = 2,6,18,54; 2 3j-I 37 = 1,.7177, .6956, .6934 -+ln 2 = .6931 in Chapter 6
39 V, = -(i)j 4 1 vj = 2(-l)j, sum is f j - 1 45 v = 1000,t = lO/V
47 M, N 5 1 4 < 2 . 9 < 92 < 29; (i)2< 2( i ) < @< 2lI9
Section 1.3 The Velocity at an Instant
(page 21)
16, 6, ya , - 12, 0, 13 34, 3. 1, 3+h, 2. 9 5 Ve l o c i t y a t t =l i s 3 7 Ar e a f =t +t 2 , s l o p e o f f i s 1 +2 t
9 F; F; F; T 112; 2t 1 3 12 + 10t2; 2 + lot2 1 5 Time 2, height 1, stays above from t = $ to
1 7 f(6) = 18 2 1 v(t) = -2t then 2t 23 Average to t = 5 is 2; v(5) = 7 25 4v(4t) 27 v,,, = t, v(t) = 2t
Section 1.4 Circular Motion
(page 28)
1l or , (0, -11, (- 1,O) 3 (4 cos t, 4 sin t ) ;4 and 4t; 4 cos t and -4 sin t
5 3t; (cos 3t, sin 3t); -3 sin 3t and 3 cos 3t 7 z = cost; J2/2; -&/2 9 2x13; 1; 2a
11Clockwise starting at ( 1, O) 1 3 Speed $ 1 5 Area 2 1 7 Area 0
Answers to Odd-Numbered Problems A-1
19 4 from speed, 4 from angle 2 1 from radius times 4 from angle gives 1in velocity
23 Slope i ; average (1 - $)/(r/6) = = .256 25 Clockwise with radius 1from (1,0), speed 3
27 Clockwise with radius 5 from (0,5), speed 10 29 Counterclockwise with radius 1from (cos 1,sin I), speed 1
31Left and right from (1,O) to (-1,0), u = -sin t 33 Up and down between 2 and -2; start 2 sin 8, u = 2 cos(t+8)
36Upanddownfrom(O, -2)to(0, 2);u=si ni t 3 7 ~ = c o s ~ , ~ = s i n ~ , s p e e d ~ , u ~ , = c o s ~ 360
Section 1.5 A Review of Trigonometry
(page 33)
1Connect corner to midpoint of opposite side, producing 30' angle 3 n 7 $ -r area i r 28
9 d = 1,distance around hexagon < distance around circle 11T; T; F; F
13cos(2t+t) = cos2tcost -sin2tsint = 4cos3t -3cost
1 5 i c o s ( s - t ) + ~ c o s ( s + t ) ; ~ c o s ( s - t ) - i c o s ( s + t ) 1 7 c o s 8 = s e c B = ~ t l a t 8 = n r
19Us ecos ( t - s - t ) =cos ( t - s ) cos t +s i n( t - s ) s i nt 2 3 8 = ~ + r n u l t i p l e o f 2 n
25 8 = f + multiple of n 27 No 8
29 4 = f
31 lOPl= a, 1OQ1= b
CHAPTER 2 DERIVATIVES
Section 2.1 The Derivative of a Function
(page 49)
1(b) and (c) 3 12+ 3h; 13 + 3h;3; 3 6 f(x) + 1 7 -6 9 2 x +Ax + 1; 2x+ 1
-4
1 1 &d = &+ 3 - 137;9;corner 1 5 A= 1 , B = - 1 1 7 F ; F ; T ; F
19 b = B; mand M; mor undefined 2 1 Average x2 + xl + 2x1
25 i ; no limit (one-sided limits 1,-1); 1; 1if t # 0, -1 if t = 0
27 ft(3); f (4) - f (3)
29 2x4(4x3) = BX7 31 = l=2 33 X = - L .
,, f1(2) doesn't exist d~ 2u 2 f i AX 36 2 f 5 = 4 u 3 2
Section 2.2 Powers and Polynomials
(page 56)
1 5 3x2 - 1= 0 at x =
fi
and A 17 8 ft/sec; - 8 ft/sec; 0 19 Decreases for -1 < x <
fi
z+h)-x
23 1 5 10 10 5 1adds to (l+l)'(x = h = 1)
253x2;2hisdifferenceofx's 2 7 % =2x+Ax+3x2+3xAx+(Ax)2 +2x+3x2=sumofseparatederivatives
1 4 1
2 9 7 ~ ~ ; 7 ( x + l ) ~ 3 1 ~ x 4 p l ~ ~ a n y c u b i c 3 3 x + ~ x 2 + $ x 3 + f x 4 + C 3 5 ~ x , 1 2 0 x 6
37 F; F; F; T; T 39 = .12 so 4= i(.12); sixcents 4 1 4= 1C- * =
-3 AX AX + A A d z
4 3 E = X 1 10. l X n + l .
2x+3 4 5 t t o f i t 4 7 i 5 x , n+l , d i v i d e b y n +l =O
Section 2.3 The Slope and the Tangent Line
(page 63)
A-2 Answers to Odd-Numbered Problems
17 (-3,19) and (8, E)
1 9 c = 4, y = 3 -x tangent at x = 1
21 (1+ h)3; 3h + 3h2 + h3; 3 + 3h + h2; 3 23 Tangents parallel, same normal
25 y = 2ax -a2, Q = (0, -a2) ; distance a2 + i ; angle of incidence = angle of reflection
fi' 2 7 ~ = 2 p ; f o c u s h a s y = $ = p 2 9 y - & = x + L - x = - 2 -4 - 4
31 y - = -1 2a ( x- a ) ; y= a 2 + $ ; a = $ 33 ($)(1000) = 10 at x = 10 hours 55 a = 2
41
57 1.01004512; 1+ 10(.001) = 1.01 39 (2 + AX)^ - (8 +6Ax) = AX)' + AX)^ 4 1 xl = i;x2 = -
40
43T=8s e c ; f ( T) =96me t e r s 45a >t me t e r s / s e c 2
Section 2.4 The Derivative of the Sine and Cosine
(page 70)
1(a) and (b)
3 0; 1; 5; $
5 sin(x + 2s); (sin h)/ h -t 1; 2 s 7 cos2B w 1-8' + f B4; f B4 is small
9 s i n i Bmi B 11:;4 13PS=s i nh; ar e aOPR=i s i nh<c ur ve dar e ai h
1 5 c o s x = l - d - + L - . . .
1 7 &(cos(x+ h) -cos(x - h)) = ;(-sinxsinh) -+ -si nx
2.1 4.3.2.1
1 9 3 / = c o s x - s i n x = Oa t x = q + n s 2 l ( t a n h ) / h = s i n h / h c o s h < ~ - + l
-1.
2 , 2 , n o 25y=2c os x+s i nx; y" =- y 2 7 y = - ~ c o s 3 x ; y = ~ s i n 3 x 2 3 S l o p e ~ c o s ~ x = ~ , 0 ,
1.
29 In degrees (sin h)/ h -+2x1360 = .01745 31 2 sin x cos x + 2 cos x(- sin x) = 0
Section 2.5 The Product and Quotient and Power Rules
(page 77)
122 5&-* 5 (2 -2)(x -3) + (2 - 1)(x-3) + (x - 1)(x-2)
7 - ~ ~ s i n ~ + 4 x c o s x + 2 s i n x9 2 x - 1 - ~ 1 1 2 ~ s i n x c o s x + ~ x - 1 / 2 s i n 2 x + ~ ( s i n x ) - 1 / 2 c o s ~
134x3cosx-x4sinx+cos4x-4xcos3x sinx 1 5 ~ ~ ~ ~ 0 s x + 2 x ~ i n x 1 7 0 1 9 - ~ ( ~ - 5 ) ~ ~ / ~ + ~ ( 5 - ~ ) - ~ / ~ ( = 0 ? )
2 1 3(sin x cos X) ~ ( COS ~ x -sin2 x) + 2 cos 22 23 u'vwz + v'utuz + w'uvz +z'uvw 25 -csc2 x -sec2 x
27 v = t;ytt, vt = cost-t sint-t' si nt
(l+t)' A = ~ ( & + ~ c o s ~ + % ) A ' = 2 ( ~ o s t - t s i ~ t + ' - ~ ~ ~ ~ lint
29 l ot for t < 10, &for t > 10
31 (l +t )' p 2t3+6t'
.(t+l)'-iTi)
(l+t)?
53 unv + 2u1v' + uu"; ut"v + 3u"v1 + 3u1v" + v"' 35 isin2 t; i tan2 t; ![(I + t)3/2 - 11
5 9 T ; F ; F ; T ; F 41degr ee2n- l / degr ee2n 4 3 v ( t ) = c o s t - t s i n t ( t < $ ) ; v ( t ) = - : ( t > : )
45 y = 9+ 9,h a 2 = 0 at x = 0 (no crash) and at x = -L (no dive). Then 2 = ?($ + f ) and
6 ~ ' h 2s
$#= r ( Z + 1).
Section 2.6 Limits (page 84)
after 5; 1.1111, y,all n; a,1, after 38; a- 1!, L = 0, after N = 10; E,oo, no N; i , ~ , 4, $, all n;
-i Ei , e = 2.718..., after N = 12. 3 (c) and (d)
5 Outside any interval around zero there are only a finite number of a's 9 111
7 $
1 3 1 1 5 sin 1 1 7 No limit 1 9 $ 21 Zero if f (x) is continuous at a 23 2
25.001,.0001,.005,.1 27l f ( x) - LI ; & 2 9 0 ; X=1 0 0 534; 03; 7; 7 353; nol i mi t ; O; l
if lrl < 1; no limit if lrl 2 1 39 .0001; after N = 7 (or 8?) 37
4 1 $
43 9;8;;an -8 = $(a,-1 -8) -+ 0
45 a, -L 5 b, - L 5 c, -L so Ib, - LI < E if la, -LI < E and Ic, -LI < E
Answers to Odd-Numbered Problems
Section 2.7 Continuous Functions
(page 89)
I c = s i n l ; n o c 3Anyc; c=O 5 c = Oo r 1; noc 7 c = l ; n o c 9 no c; no c
11 c = 1.
64, ' = 64
1 3 c =- l ; c =- 1 1 5 c = l ; c = l 1 7 c =- l ; c =- 1
1 9 c =2 , 1 , 0 , - 1 , ~ ~ ~ ; s a me c 21f ( x) =Oe xc e pt a t x=l 2 3 d x 25-ff 2 7 A
29One;two;two 31No;yes;no 3 3 x f ( x ) , ( f ( ~ ) ) ~ , ~ , f ( ~ ) , 2 ( f ( ~ ) - ~ ) , f ( ~ ) + 2 + 3 5 F ; F ; F ; T
37 Step; f (x) = sin $ with f (0) = 0 39 Yes; no; no; yes (f4(0) = 1)
41 g( i ) = f (1) - f (i) = f (0) - f (i) = -g(O); zero is an intermediate value between g(0) and g(;)
43 f(x) - x is 2 0 at x =O and 5 0 at x = 1
CHAPTER 3 APPLICATIONS OF THE DERIVATIVE
Section 3.1 Linear Approximation
(page 95)
I Y = ~ 3y =I +~( x - : ) 5 ~ = 2 ~ ( ~ - 2 4 726+6. 25. . 001 9 1
11 1 - I(-.02) = 1.02 13 Error .000301 vs. i (.0001)6 1 5 .0001- $lo-' vs. i(.0001)(2)
1 7 Error .59 vs. ?(.01)(90) 19 = A 2- = a a t x = O
2 1 $ ~ ~ = r f i = & a t u = 0 , c + ~ = c + $ l+u 2SdV=3(10)~(. 1)
25 A = 47rr2, dA = 87rr dr 27 V = 7rr2h, dV = 27rrh dr (plus 7rr2 dh) 29 1 + i x 31 32nd root
Section 3.2 Maximum and Minimum Problems
. (page 103)
1 x = -2: absmin 3 x = -1: relmax, x=0: abs mi n, x=4: absmax
5 x = -1: abs max, x = 0 , l : abs min, x = : re1 rnax 7 x = -3: abs min, x = 0 : re1 max, x = 1: re1 min
9 x = 1, 9 : abs min, x = 5 : abs rnax 11 x = : re1 max, x = 1 : re1 min, x = 0 : stationary (not rnin or max)
x = 0,1,2, . . : abs min, x = i, 4, 4, . . : abs rnax
151x/
1 : all min, x = -3 abs max, x = 2 re1 rnax
x = 0 : re1 min, x = $ : abs max, x = 4 : abs min
x = 0 : abs min, x = 7r : stationary (not min or rnax), x = 27r : abs rnax
19= 0 : re1 min, tan B = -? (sin B = 2 and cosB = - % abs max, sin B = - $ and COSB = % abs min),
8 = 27r : re1 rnax
h = $(62" or 158 cm); cube 25 A; 2 6 gallons/mile, miles/gallon at v = fi
(b) B = = 67.5' 29 x = compare Example 7; f = 4
6'
R z - C s . dR dC
R(x)-C(x); Ox ds; pr ~f i t 33x=+; r er o 3 5 x = 2
2( b 4
2
V=x( 6- 9) ( 12- 2x) ; xw1. 6 3 9 A=nr 2 +x 2 , x =f ( 4 - 2 a r ) ; r , , , i , =~
ma x a r e a 2 5 0 0 v s ~ = 3 1 8 5 4 3 x = 2 , y = 3 45P(x)=12-x;thinrectangleupyaxis
H
h = F , r = z 3 V = = ~f c ~n e v o l u me
r = , * ; best cylinder has no height, area 27rR2 from top and bottom (?)
r = 2, h = 4 53 25 and 0 55 8 and -00
dFG-2 + Jq2 + (S - x)2. * = A -
8-2
9 d~
&- = 0 when sin a = sinc
y = x2 =
6 1 (1-1) ( - )
63 m = 1 gives nearest line 65 m = $ 67 equal; x = $
kx2
71 'Rue (use sign change of f")
Radius R, swim 2 R cos 0, run 2 RB, time + ; max when sin 0 = A, min all run
A-4 Answers t o Odd-Numbered Problems
Section 3.3 Second Derivatives: Bending and Acceleration
(page 110)
3 y = - l - x2; no . . . 5 False 7Tr ue 9Tr ue( f 1has 8zer os , f "has 7)
11 x=3i s mi n: f M( 3 ) =2 1 3 x = On o t ma x o r mi n ; x = ~ i s mi n :f M( ; ) =81
1 5 x = a is max: f " ( y ) = -a;x = is min: ft1(?) = fi
17 Concave down for x > $ (inflection point)
1 9 ~ = 3 i s ma x : f " ( 3 ) = - 4 ; z = 2 , 4 a r e mi n b u t f " = O 2 1 f ( Ax ) =f ( - Az ) 2 3 l + x - $
8 25 1-$ 27 1- ;x - Lx2 29 Error f " ( x ) ~ x 31 Error OAx + &f " ' ( x ) ( ~ z ) ~
37 & = 1. 0101~; = .909m 39 Inflection 4 1 18 vs. 17 43 Concave up; below
Section 3.4 Graphs (page 119)
1 120; 150; 9 3 Odd; x = 0, y = x 5 Even; x = 1,x = -1, y = 0 7 Even; y = 1 9 Even
11 Even; x = l , x = -1, y = 0 13 x =O, x =- l , y =O 1 5 x = 1,y = 1 1 7 Odd 1 9 3
21 x + & 23 d G 25 Of the same degree 27 Have degree P < degree Q; none
29 x = 1and y = 32 + C if f is a polynomial; but f (x) = (x - 1)'13+ 32 has no asymptote x = 1
3 1 ( ~ - 3 ) ~ 3 9 x = f i , x = - &y = x 4 1 ~ = 1 0 0 s i n ~ 4 5 ~ = 3 , d = l O ; c = 4 , d = 2 0
47 X* = JS= 2.236 49 t j = x - 2; Y = X ; y = 2~ 5 1 xmax = -281,Zmin = 6.339; xinfl = 4.724
53 xmin = -393, xmaX = 1.53, xmin = 3.33; Zinfl = .896,2.604
55 xmin = -.7398, xmaX = .8l35; xins = .O4738;x~~,,,, = k2.38 57 8 digits
Section 3.5 Parabolas, Ellipses, and Hyperbolas
(page 128)
1dyldx = 0 at 2 3 V = (1,-4), F = (1,-3.75) 5 V = ( O, O) , F = (0,-1) 7 F = (1,l)
9 V=( O, f 3 ) ; F=( o , f f i ) 11V = ( O , f l ) ; F = ( ~ , f f i ) 1 3 Twolines, a = b = c = O; V= F = ( 0 , 0 )
11
1 5 t ~ = 5 x ~ - 4 x 1 7 Y + P = J x 2 + ( Y - p ) 2 - - + 4 p y = x 2 ; ~ = ( ~ , ~ ) , Y = - ~ ; ( f ~ , 1 2 )
19 x =a y 2 with a > 0 ; y = W ; y = - a x 2 +a x wi t h a >0
z2+ Y1
,
= 1.
,
( x- 3) ' + ( ~ - 1 ) ~
21 $ + y 2 = 1 ; ~ + ( y - 1 ) 2 = 1 2 3 % ,,
= 1;x2 + y2 = 25
-
25 Circle, hyperbola, ellipse, parabola 27 *= -2; y = -$x +5
32
29 b*2 = 1
dz 49 40 , 2 ( ~ 5)
2 333x12+y12=2 ~ 5 ~ ~ - $ ~ ~ = 1 . ~ - ~ = l ; ~ ~ - ~ ~ = 5 ~ 1 ~ i r ~ l ~ ; ( 3 , 1 ) ; 2 ; X = y , Y = ~ ' 9 9
37 2 - & = 1 39 # -4y + 4, 2x2 + 122 + 18; -14, (-3,2), right-left 25
41 ~ = ( k $ , ~ ) ; y = k : 43 ( ~ + y + 1 ) ~ = 0
45 (a2 - 1)x2+ 2abxy + (b2 - 1)y2 + 2acx + 2bcy + c2 = 0; 4(a2 + b2 - 1); if a2 + b2 < 1then B2 -4AC < 0
Section 3.6 Iterations xn+l = F( x n )
(page 136)
1-.366;oo 3 1 ; l 5: ; f oo 7-2; -2
9 attracts, 9repels; $ attracts, 0 repels; 1attracts, 0 repels; 1 attracts; $ attracts, 0 repels;
f \ / Z repel
11Negative 13 .900 1 5 .679 17 la1 < 1 19 Unstable IFII> 1 21 x* = k;la1 < 1
Answers to Odd-Numbered Problems
23 $2000; $2000
25 XO, 6/ 00, X O, ~ / x o , . .
27 F' = - A x - 3 / 2 2 = -: at . *
29 F1 = 1 - 2cx = 1 - 4c at x* = 2;O < c < ) succeeds
31 F1 = 1 - 9c(x - 2)8 = 1 - 9c at x* = 3; 0 < c < succeeds
xa -2. sin 2 %-
SQ xn+l = Xn - &;
= xn- C 0 8 X m
3 5 ~ * =4 i f x O > 2. 5; ~* = 1 i f ~ o < 2.5
37 m = 1 + c at x* = 0, m = 1 - c at x* = 1 (converges if 0 < c < 2)
39 0 43 F' = 1 at x* = 0
Section 3.7 Newton's Method and Chaos (page 145)
1 b:+;Y =
25 r is not afraction 27= f x : + ) + S;Z = A 29 162 - 80z2 + 1282 - 64z4; 4; 2
31 lxol < 1 33 A x = 1, one-step convergence for quadratics 55 = *; x2 = 1.86
37 1.75 < x* < 2.5; 1.75 < x* < 2.125 39 8; 3 < x* < 4 4 1 Increases by 1; doubles for Newton
45 xl = xo + cot xo = xo + r gives x2 = xl + cot xl = X I + r 49 a = 2, Y' s approach ;
Section 3.8 The Mean Value Theorem and 1'H6pita19s Rule
(page 152)
I c = fi S No c 5 c = 1 7 Corner at ) 9 Cusp at 0
11 sec2 x - tan2 x = constant 13 6 15 -2 17 -1 l 9 n 21 -) 23Not %
1 -sin x
25 -1 27 1; TT~;;;; has no limit
29 f l (c) = $$;c = \/j
31 0 = x* - xn+1 + -#$(x* - xn)' gives M m 33 f l (0); v; singularity 35 # -+ 37 1
CHAPTER 4 DERIVATIVES BY THE CHAIN RULE
Section 4.1 The Chain Rule
(page 158)
1s = y3,y = x2 - 3,s' = 6x(x2 - 3)2 3 2 = cosy, y = x3,z' = -3x2sinx3
5 ~ = ~ , ~ = s i n x , z ' = c o s x / 2 ~ ~ 7 z =t a ny +( 1 / t a nx ) , y =l / x , d=( ~) s e c 2 ( ~) - ( t a nx ) - 2 s e c 2 x
9z =cosy, y=x2+x+1, d=- ( 2x+1) si n( x2+x+1) 1117cos17x 13si n(cosx)si nx
15x2cosx+2xsi nx 1 7 ( ~ o s ~ ~ ) ~ ( x + l ) - ' / ~ 1 9 ) ( 1 + s i n ~ ) - ~ ~ ~ ( c o s z ) 2 l c o s ( & - ) ( ~ ~ )
2 3 8 ~ ' = 2 ( ~ ~ ) ~ ( 2 ~ ~ ) ( 2 x ) 2 5 2 ( ~ + 1 ) + c o s ( x + r ) = 2 ~ + 2 - ~ o s x
27 (x2 + +I 2 + 1; sin U from 0 t o sin 1; U(sin x ) is 1 and 0 with period 27r; R from 0 t o x; R(sin x ) is half-waves.
29 g(x) = x + 2, h( x ) = x2 + 2; k ( x ) = 3
31 f t ( f ( x ) ) f l ( x ) ; no; ( - l / ( l / ~ ) ~ ) ( - l / x ~ ) = 1 and f ( f ( x ) ) = x
33 ? ( ) x + 8) + 8; i x + 14; &
35 f ( g( x) ) = x, g ( f ( y ) ) = y
37 f ( g( x) ) = d f (4) = 1 - $ 8 f ( f (4) = x = g(g(x)), g ( f ( g( x) ) ) = = f ( g ( f (4))
39 f ( y) = y - 1, g(x) = 1
48 2 cos(x2 + 1) - 4x2 sin(x2 + 1); - (x2 - 1)-'I2; - (cos &)/4x + (sin f i / 4 x 3 I 2
45f ' ( u( t ) ) u1( t ) 47( c os 2u( x ) - s i n2u( x ) ) g 4 9 2 x u ( x ) + x 2 ~ 511/4d=4=
53 df / dt 55 f ' (g(z))g1(x) = = 122" 57 3600; 4; 18 59 3; 5
A-6 Answers to Odd-Numbered Problems
Section 4.2 Implicit Differentiation and Related Rates
(page 163)
I -xn-l / yn-l 3 2 4 5 2 = 1 7 ( y2-2xy) / ( x2-2xy) or 1 1
F' ( v) g & o r ~ i ; l
11 First 2 = -E , second 2 = j 13 Faster, faster 15 222' = 2yyt -+ 2' = E y' = y' sin6
2 1 $ =- g . * = 17 sec2 0 = I S ~ O O ~ ; ~ O O J ~ 3 , d t - 2f i ; oot he nO
23 V=Tr 2h. dh = -- I dV -- -- in/sec
25 A = i absi n 9 , % = 7 27 1.6 m/sec; 9 m/sec; 12.8 m/sec
9 dt 4r dt
L C O s 2 ~ &. g u =&
29 - g
3 1 d"- a&.&-
dt '
,, y" - &jcos3 O~ i n B( y ' ) ~
dt - 2 dt 1 d t - 10
Section 4.3 Inverse Functions and Their Derivatives
(page 170)
( x unrestricted -,no inverse)
11 y = ,
1
13 2 < f - ' ( x) < 3 15 f goes up and down
f ( x ) g( x )and & 19 m# 0;m20; Iml > 1
2 1 $ = 5x4, 2 = iy-4/5
2
- 1
25 & = -=1_ & -
= 3x2. dz = $ ( I + y) - 2/ 3
' dY
27 y ; i y 2 + C
7 - d x ( 3- 1l 2 d~ -
39 2/& 4 1 l / 6cos 9
f ( g( x ) )= -1/3x3; g- l ( y)
Decreasing; $ = &
g(g-' (x)) : ; =
< 0 45 F; T ; F 47 g ( x )= xm, f ( y ) = yn, x
= x
= (2'1" 1 ' I r n
g ( z ) = ~ ~ , f ( y ) = y + 6 , x = ( z - 6 ) ~ / ~5 1 g ( x ) = 1 0 x , f ( y ) = l o g y , x = l o g ( l ~ Y) = y
y = x3, y'' = 62, d2x/ d$ = -$ yV5I3;m/ sec2,sec / m2 55 p =
fl
- 1;0 < y 5 1
,ax = G = 3
gY
413 GI=
2y
113
59 y2/100
9
Section 4.4 Inverses of Trigonometric Functions
(page 175)
CHAPTER 5 INTEGRALS
Section 5.1 The Idea of the Integral
(page 181)
8 11, 3, 7, 15, 127 3 - 1 - 1 - 1 = 1 - 8 1 5 f j - f O = 2 7 3 ~ f o r x ~ 7 ~ 7 x - 4 f o r x ~ 1
s2m,&,&G g 11 1 1 Lo we r b y 2 13Up, down;rect angl e 15 , / X- &; A~; ~; $
17 6; 18; triangle 19 18 rectangles 21 62 - $x2 - 10;6 -x 23 25 x2;x2; i x 3
Answers to Odd-Numbered Problems A-7
Section 5.2 Antiderivatives (page 186)
i x6+$x6*P ' 3 32f i ; 2 5Qx413(1+21/3);q(i+21/3) 7 - 2 ~ 0 s x - ~ c o s 2 ~ ; ~ - 2 c o s i - ~ c o s 2
9xsi nx+cosx; si n1+cos1- 1 11i s i n2x; i s i n21 1 3 f = C; O 15f ( b) - f ( a) ; f 7- f 2
5 , 36 ,oo
23 f(x) = 2&
25 5, below -1; +, q 1 7 8 + * 19:(1+&);:(3+fi);2 2 l 5 = m *
27 Increase - decrease; increase - decrease - increase
29 Area under B - area under D; time when B = D; time when B - D is largest 33 T; F; F; T; F
Section 5.3 Summation Versus Integration
(page 194)
n n
7 x akxk; x sin - 9 5.18738; 7.48547 11 2(a; + 6;) 13 2" - 1; if - 1 5 F; T
1 7 $ + C; f p - fs - fl + fo
1 9 fl = 1; n2 + (2n + 1) = (n + 1)2
21 a + b + c = 1,2a + 4b + 8c = 5,3a + 9b + 27c = 14; sum of squares 23 S4oO = 80200; E400 = .0025 = i
25 Sloo,l/3 w 350, Eloo,l/3 w .00587; Sloo,3 = 25502500, Eloas = .0201 27 vl and v2 have the same sign
Section 5.4 Indefinite Integrals and Substitutions
(page 200)
1 $ ( 2 + x ) ~ / ~ + C ~ ( x +l ) " +' / ( n +l ) +C( n # - I ) 5 & ( ~ ~ + 1 ) ~ + C 7- +c os 4z +C
9 -!cos42x+C l l s i n - l t + c 1 3 $(1+t2)312-(1+t2)112+C 1 5 2 f i + x + C
17 s e c x + ~ 1 9 - COSX+C 21 ax3 + $x3/2 23 -$(I - 2~)3/ 2 25 y = 6
27 ?x2 29 asinx + bcosx 31 &x' /~ 33 F; F; F; F 35 f ( x - 1);2f(:)
57 x - tan-' x 39 I ?du 41 4.9t2 + Clt + C2
43 f (t + 3); f (t) + 3t; 3 f (t); $f (3t)
Section 5.5 The Definite Integral
(page 205)
1 C = - f (2)
S C = f (3)
5 f (t) is wrong 7 C = 0 9 C = f(-a) - f(-b)
1 5 u = s e c x ; ~ ~ ~ ~ d u = ~ ( s a m e a s 1 3 ) 1 7 u = ) , x = ~ , d x = = $ ; ~ , ' ~ ~ ~
19 s= $(++I)' + + ( I + I ) ~ ; s = ;(o) + +( ++q4
21 s = + i3 + (;)3 + 23]; s = ?[03 + (+)3 + i3 + (;)3]
1 17 4
23 S = z[(E) + (q)4 + (%)( + 2' 1
25 Last rectangle minus first rectangle
27 S = .07 since 7 intervals have points where W = 1. The integral of W (x) exists and equals zero.
29 M is increasing so Problem 25 gives S - s = Ax(1- 0); area from graph up to y = 1 is $ 1 + A ' + . . =
A
4 2
+( I + + & +.-.) = = i; area under graph is i.
31 f (x) = 3 +
v(x)dx; f (x) = I; v(x)dx
33 T;F;T;F;T;F;T
A-8 Answers to Odd-Numbered Problems
Section 5.6 Properties of the Integral and Average Value
(page 212)
1 ~ = ~ ~ ~ ~ x ~ d x = ~ e ~ u a l s c ~ a t c = f ( ~ ) ~ ~ ~ ~ ~ = ~ J ~ c o s ~ x d x = ~ e ~ u a l s c o s ~ c a t c = ~ a n d $
2 d z
6 i r = / 1 2 = ~ e q u a l s $ a t c = f i 7J:v(x)dx gFalse, takev(x)<O
11The; 3 J',v(x)dx + $ .
J: v (x)dx = i J,S v(x)dx
1 3 False; when v(x) = z2 the function x2 - i is even
15 False; take v(x) = 1; faetor ? is missing 1 7
= A Ja v(x)dx
19 0 and ?
b-a
21 v(x) = Cx2; v(x) = C. This is 'constant elasticity" in economics (Section 2.2) 23 V +0; + 1
25 i J i ( a - x) dx= a + 1i f a > 2;;s; la- xldx= ? area = $ - a + 1i f a < 2; distance = absolute value
27 Small interval where y = sin B has probability $;the average y is J :
= 2A
29 Area under cos 0 is 1. Rectangle 0 < 0 5 5 , O 5 y 5 1has area 5. Chance of falling across a crack is $= 1.
%dt = -220- g s i n % = Vave 31 $,&,..., $;10.5 33 5 J, ' ~~ocos
35 Any V(X) = veve,(x) odd(^); (X +
= (3x2 + 1)+ (x3 + 3%);;)i=
- &
31 16 per class; $;E(X) = 64 = 22.9 39 F; F; T; T 8
Section 5.7 The Fundamental Theorem and Its Applications
(page 219)
1cos2 x S O S ( X ~ ) ~ ( ~ X ) = ~ X ~ ~ v ( x + I ) - V ( X ) g e m - 2. J: sin2 t dt
ll/;v(u)du 1 3 0 152si nx2 17u(x)v(x) 19t h-' (si nx)cosx=xcosx
21 F; F; F; T 23 Taking derivatives v(x) = (xcos x)' = cos x -xsin x
25 Taking derivatives -v(-x) (- 1) = v(x) so v is even 27 F; T; T; F
29 Jr v(t)dt = J; v(t)dt - v(t)dt = +- & (in revised printing)
31 V = s3; A = 3s2; half of hollow cube; AV rr 3s2dS; 3s' (which is A)
33 dH/dr = 2?r2r3 35 Wedge has length r rr height of triangle; $r2d0 = $
1 . do . ~ 4 4 do = t a . e + = ~
c o s 8 ~ 2 e o s 2 8 ~ 02cos28 T O 2
39 x = y2;J; y2dy = = t ;vertical strips have length 2 -fi
41 Length &a; Jo
1
ada = 43 The differences of the sums f j = vl +v2+-. *+vj are f j - fj-1 = vj width 3;
Section 5.8 Numerical Integration
(page 226)
1? A X ( U ~ -vn)
3 1,-5625, ,3025; 0, -0625, -2025 5 L8 W .1427, T8 W .2052, S8 U .2OOO
- l a# $ 9 For y=x2, error +(AX)' from i - s, yl ' -2Ax - 7 p = 2 : for y = z 2 , f . ~ ~ + I - ( i ) ~ + f
2
13 8 intervals give %[:& + = < .001 15 fl'(c) is yl(c) 1 7 00;.683, .749, .772 + 2
19 A+ B + C = l , ? B + C = & , ~ B + c = $;Simpson
1
21 y = 1and x on [0,1]: L, = 1 and i - &,R, = 1 and + k,so only ?L, + $R, gives 1and 5
23 Tlo N 500,000,000; Tloow 50,000,000; 25, 000~
25 a = 4, b = 2, c = 1; 1,'(4x2 + 22 + 1)dx = y; Simpson fits parabola 27 c = &
Answers to Odd-Numbered Problems
CHAPTER 6 EXPONENTIALS AND LOGARITHMS
Section 6.1 An Overview
(page 234)
15; -5; -1.1.. 3.2
5 1; -10; 80; 1; 4; -1 7nl ogbx gmaa 3 , 10 13 lo5
5' 2 '
1 5 0 ; I S F = 1 0 7 ~ O~ 8 . 3 + l ~ g l o 41 7 A=7 , b =2 . 5 1 9 A= 4 , k = 1 . 5
21 A; -&;log2
23 y - 1= cx; y - 10 = c(x - 1) 25 (.l-h- l)/(-h) = (loh - l )/ (-h)
27 3/' = c2bX; x0 = -l/c# 29 Logarithm
Section 6.2 The Exponential eZ
(page 241)
149e7" 3 8e8" 5 3% in 3 7 ($)" in $ 9- (i+e:)2 1 1 2 13xex l5(e~+e-z)2 4
1 7 esin x
cos x + ex cos ex 1 9 .1246, .0135, .0014 are close t o ; i ; ~ 21 1.1
e ' e
2 3 Y( h ) = l + &; Y( l ) = ( l + &) ' O= 2 . 5 9 2 5 ( l + ~ ) " < e < e x < e 3 x / 2 < e 2 x < 1 0 x < z x
3s 72 z3 e-z3
2 7 %+ ? 2 9 x + & + & 31 %+2ex 33%- - 2
35 2exl2 + $ 37 e-" drops faster at x = 0 (slope -1); meet at x = 1; e-"'/e-" < e-g/e-3 < &for x > 3
39 y -ea = ea(x -a); need -ea = -aea or a = 1
> 0
d~ 43 $(e-x y) = e-" *-e-"y = 0 so e-x y = Constant or y = Cex
2z ninz
2 A
45 !L]i= I-'
47 &]L1 = g= ,,, , 49 -e-"IF = 1 5 1 el+"]: = e2 -e 53 = 0
= 55 J F d x = -e-u +C; J ( eu) 2edx = +eZU+C 57 yy' = 1gives iy2x +C or y = 4-
59 = (n - X ) X " - ~ / ~ " < 0 for x > n; F(2x) < -+0 6 1 m 117;( : ) 6 m 116; 7 digits
Section 6.3 Growth and Decay in Science and Economics
(page 250)
47 (1.02)(1.03) +5.06%; 5% by Problem 27 49 20,000 e(20-T)(.05) = 34,400 (it grows for 20 -T ears)
- 1)
.005
51 s = -cyoect/(ect - 1)= -(.01) ( 1 0 0 0 ) e . ~ ~ / ( e . ~ ~
53 yo = m(1
- e-.005(48)
1
55 e4c = 1-20 so c =
57 24e36.5 =? 59 TO-00; constant; to + oo
6 1 = 60cY; = 60(-Y +5); still Y, = 5
41 3/ = xx(lnx + 1)= 0 at %,in = :; y" = ! ] + 1)2 + xx[(ln x
A-10 Answers to Odd-Numbered Problems
Section 6.4 Logarithms (page 258)
1 $ 3 - 1 5 l nx 7 ~ 0 8 5 =
x(ln x)a s i n s x 9 11 $ l n t + C I 3 i n$
1 5 i l n 5 17- l n( l n2) 191n( s i nx) +C 21- $l n( cos 3x) +C 2 3 $ ( l n ~ ) ~ + C
27 in y = $ ln(x2 + 1); 2 =
29 * = esin
cos x
dFE dx
3 1 2 = exee' 33 l n y = e x l n x ; ~ = y e x ( l n x + ~ ) 5 5 l n y = - 1 s o y = : , z = O 3 7 0
39 -1 4 1 sec x 4 7 . l ; .095; .095310179 49 -.01; -.01005; -.010050335
5 1 lYHSpital: 1 53 1 5 5 3 - 2 in 2 57 Rectangular area i + . . + < $: $ = I nn
In b
59Ma xi muma t e 6 1 0 6310gl oe or & 6 5 1 - x ; l + x l n 2
( t +2) a -+ y = 1 - 1 never equals 1
67 Raction is y = 1 when l n( T + 2) - In 2 = 1 or T = 2e - 2 69 y' = -2-
t+2
7 1 l np = xl n2; LD 2"l n2; ED p = eZLn2, p' = In2 esln2
75 24 = 42; yl n x = xl n y -+ '"2 = decreases after x = e, and the only integers before e are 1 and 2.
y ' s
Section 6.5 Separable Equations Including the Logistic Equation
(page 266)
I 7et - 5 3 ($x2 + 1)lI3 5 x 7 e l - ~ ~ ~ t 9 ( ?+&) a 11 y, =O; t = 1
YO
1 5 z = l +e - t , y is in 1 3 1 7 ct = l n3, ct = l n9
19 b = c = 13 . y, = 13 . lo6; at y = & (10) gives ln = ct + In c_'::,b so t = 1900 + = 2091
2 1 # dips down and up (avalley) 23 sc = 1 = sbr so s = $, r =
25 Y = l+e-NY(N-l) ; ~=!d!!$l-+o
27 Dividing cy by y + K > 1 slows down y'
29 dR = CK
dy ( y + ~ ) f > 09 * -+
3 1 = 6; multiply e ~ l K = e- ct l Key~l K ( EL ) by K and take the Kt h power t o reach (19)
33 f / = ( 3 - y ) 2 ; & = t + $ ; y = 2 a t t = 2 3
35 A e t + D = A e t + B + ~ t + t - + ~ = - l , B = - l ; y o = A + B g i v e s A = l
37 y + 1 from yo > 0, y -+ -oo from yo < 0; y -+ 1 from yo > 0, y -+ -1 from yo < 0
39 $ Cyiydy =
dt -+ ln(sin y) = t + C = t + In i. Then sin y = i e t stops at 1 when t = In 2
Section 6.6 Powers Instead of Exponentials
(page 276)
a 3 a 3
1 l - x + y - % + . . . 3 l f x + ~ f ~ + - 5 1050.62; 1050.95; 1051.25
7 1+n( $) + w(+)2 + 1- I + 4 9 square of ( I + i)"; set N = 2n
11 Increases; l n ( l + $) - & > 0 1 3 y(3) = 8 1 5 y(t) = 4(3') 1 7 y(t) = t
19 y(t) = $(3t - 1) 2 1 s ( 2 ) if o # 1; st if a = 1 23 yo = 6 25 yo = 3
b
27- 2, - 10, - 26+- 00; - 5- =- ?- +- 12 9 2 , 2 9 P = = 3 1 10.38% 33 100( 1. 1) ~~ = $673
100 000 1 12
35 & = 965 37 Y ( 1 . l z 0 - 1) = 57,275 39 y, = 1500 4 1 2; ( g ) 5 2 = 2 69. ye
43 1.0142'~ = 1.184 -+ Visa charges 18.4%
Section 6.7 Hyperbolic Functions
(page 280)
1 ex, e-x eax-eeax
2 4
= $ sinh 22 7 sinh nx 9 3 sinh(3x + 1) 11 - eoah = - t anhx sech x
1 3 4 cosh x sinh x 1 5 ~ ( s e c h 4 G ) ~ 1 7 6 sinh5 x cosh x
19cosh( l nx) = i(x+;) = l a t x = 1
2 1 139 '3 5 1 -B 5 I -3 12, -5 12
23 O , O Y ~ Y ~ Y ~
25 sinh(2x + 1) 27 $ cosh3 x 29 ln(1 + cosh x) 3 1 ex
A-11 Answers to Odd-Numbered Problems
33 J y d z =J s i n h t(sinh t d t ) ; A = i s i nh t c o s h t - J y d x ; ~ l = ~ ; A = o at t = ~ s o A= i t .
4 1 eY = x + d m ,y = In[x+ d-]
47 4 ln 1% 1
49 sinh-' x (see 41) 5 1 -sech-'z
53$1n3; oo 5 5 y ( x ) = ~ c o s h c x ; $ c o s h c ~ - $
57 5/' = y -3 3 . L(
Y
= 1 -y3 is satisfied b y y = isech2:
9 2 2Y
CHAPTER 7 TECHNIQUES OF INTEGRATION
Section 7.1 Integration by Parts
(page 287)
$ ( x 2 +1 ) t a n - ' x - %+C 21x3s i nx+3x2c os x- 6xs i nx- 6c os x+C
ex(x3-3x2 + 6x -6) + C 25 x t an x + ln(cos x ) + C 27 -1 29 -:e-2 + 3 1 -2
3l n10- 6+2t anV' 3 35 u = x n, v =e x 37 u = x n, v =s i nx 39 u = ( l nx ) " , v =x
u = xs i nx, v = ex +/ e x s i n x d x in 9 and - $xcos xexdx. Then u = - xcosx, v = ex + ~ e x c o s x d x
in 10 and -J x sin x exdx (move t o left side): ( x sin x - xcos x + cos x) . Also t ry u = xex, v = -cos x.
$ $ u s i n u d u = $( s i nu- uc os u) = $( s i nx2- x2cos x2) ; odd
3. step function; 3ex. step function 49 0; x6( x) ] -$6( x) dx = -1; v ( x ) d( z ) ] -I v( x) 6( x) dx
~ ( 4 = Jxl f (+x
u( x ) = 51,"v( x) dx;+(: - $);f for x 5 i,~ ( Z X - x2 - 4 ) for x 2 i;:for X Ii, &for x > i.
u=x 2 , v =- c o s x +- x 2 c o ~ x +( 2 x ) s i nx - J 2 s i nx dx 57Compar e23
1
uw']A-Jo' u'wl -u1w]A+ So u'w' = [uwl- ul w];
No mistake: ex cosh x -ex sin hx = 1 is part of the constant C
Section 7.2 Trigonometric Integrals
(page 293)
1 J ( 1 - ~ o s ~ x ) s i n x d x = - ~ o s x + ~ ~ o s ~ x + C 3 i s i n 2 x + C
5 $ ( 1 - u 2 ) 2 u 2 ( - d t l ) = - $ c 0 s 3 x + ~ c 0 s 5 x - ~ c 0 s 7 x + ~ 7 $ ( s i n ~ ) ~ / ~ + ~
3 2 7) 9 i J s i n 3 2 x d x = &( - c o s 2 x + $ c o s 3 2 x ) + ~ 1 1 3 L( 5 2 +s i n 6 x + C
15 x + C 17 cos5 x sin x + $ cos4 x dx; use equation ( 5 )
19 $:I2 dx =
$:I2 c0sn-2 dx = . . . = &. . .
i$:I2 n n n-2 d~
21 I = -sinn-' x cos x + ( n- 1) J xcos2 x dx = -sinn-' x cos x + ( n- 1)J x dx - ( n- 1) I .
So nI = -sinn-' x cos x + ( n- 1)$ x dx.
230, +, 0, 0, 0, - ~ ~ - $ c o s ~ x , o 27- ; ( & 2 + T ) , O
200
C0s200x 29 + ( s i n2003 + si;2x), 0
31 -+ 33 1: = A sin2 x dx + A $ : = x sin x dx 55 Sum = zero = (l ef t + right) cos x, 0 2
37 p is even 39 p - q iseven 4 1 sec x + C 43 $ tan3 x + C 45 $ sec3 x + C
47 $ t a n 3 x - t a n x +x +C 49 l nI si nxl +C 51 &+c 53 A=&, - f i s i n( x +: )
55 4JZ 57 59 1-cosx si nx
6 1 p and q are 10 and 1
l +c o a x ~ s i n x+ C
A-12 Answers to Odd-Numbered Problems
Section 7.3 Trigonometric Substitutions
(page 299)
7 ~ = i t a n - ' z + ? + + ~ t a n 8 ; $ ~ 0 ~ ~ 8 d f I =
9 X = 5sec8; S5(sec28- l ) d8= d n - 5 s e c - I ; + C
I I X = S ~ C ~ ; J C O S ~ ~ ~ = ~ + C I ~ X = ~ ~ ~ ~ ; $ C O S B ~ O = - + C
-5
15 x = 3 sec 8; $ 'g"'?,dee = & + C = -
dm
9 @z i +c
1 7 x = sec8; Jsec3 8 dB = &sect9 t an8 + i l n( sec8 + t an8) + C = & x d G + ? l n( x+ d m ) + C
1 g X = t a n 8 ; $ c ~ s ~ d ~ = - L + C = -+C
sina 6 sin 0 x
2 1 $
= -8 + C = -cos-' x + C; with C = 5 this is sin-' x
= -ln(cos 8) + C = 1n4- + C which is iln(x2 + 1)+ C 23 $ t a n ~ ~ $ ~ e
25 x = a sin 8; $: L72 a2 cos2 8 d8 = = area of semicircle
27 sin-' x]f5= 5 - 2 = t
"12 cos8d0 = -14 2
29 Like Example 6: x = sin 8 with 8 = 5 when x = oo,8 = 5 when x = 2, Jnl3
,h
"12 3seca de = g "12
dx = $xn-'dx = $
3 1 x = 3 t an 8; $-r12 9seca e 3]-n/2 =
33 $ xnTcln-l
35 x=s e c e ; i ( e f +e- f ) = L( x+J=+ .+;=)= 2 ? ( x + d Z + x - - d G ) = x
37 x = cosh 8; $ dB = cosh-' x + c
39 x = cosh 8; $ sinh2 8 dB = i (si nh 8 cosh 8 -8) + C = $xd= - $ ln(x + d r l )+ C
4 1 x = tanh 8; $ dB = tanh-' x + C 43 (x -2)2 + 4 45 (x -3)2 -9
47 (x +
1 x 2
49u=x- 2, $- &= i t an-' : = i t a n - ( +j - ) +C; u=x- 3, $*= Ll nU- 5= '1 ~ - 6
u -9 6 ~ + 3 n ~ + c ;
u = x + 1 , $ + = L - ' + c x+1 u
dU
5 1 u = x + b; $ u'-ba+c
u ~ e ~ u = a s e c 8 i f b ~ > ~ , ~ = a t a n 8 i f b ~ < c , e ~ u a l s - ~ = ~ i f b ~ = c
53 cos 8 is negative (-d-) from 5 $: then F; to - + 4-dx = 7 = area of unit circle
55 Divide y by 4, multiply dx by 4, same $ y dx
57 No sin-' x for x > 1; the square root is imaginary. All correct with complex numbers.
Section 7.4 Partial Fractions
(page 304)
~ ~ A + + + M . A = - L
4 '
B = L
4 '
C = OD = - L
x+1 x- x'+l ' 2
1 7 Coefficients of y : 0 = -Ab + B; match constants 1= Ac; A = $, B =
1 9 A= l , t h e n B = Z a n d C = 1 ; ~ 5 + $ % =
ln(x - 1)+ ln(x2 + x + 1)= ln(x - l ) ( xZ+ x + 1)= ln(x3 - 1)
2 1 u = e ~ ; $ ~ = $ ~ - $ ~ = l n ( ~ ) + ~ = l n ( ~ ) + ~
~ ~ u = c o s ~ ; $ . & + C. We can reach = - $ J A - I $ k =& l n ( l - - u ) - ~ l n ( l + u ) = $ l n ~
1in ('-CO.B)1 = In 1-cose
~ - C O S ~ O -ln(csc8 - cot e) or a different way +I n =In- ~ + C OS e = -1,- sm e =
-ln(csc 8 + cot 8)
25 u = e x ; d u = e x d x = u d x ; $ ~ d u = $ ~ + $ ~ (1-U)U = - 2 l n ( l - e x ) +l n e x +C= - 2 l n ( l - e x ) + z + C
2
Answers toOdd-NumberedProblems A-13
2 7 x + 1 = u 2 , d x = 2 u d u ; $ ~ =J [ 2 - &] d u =2 ~ - 2 1 n ( l +u ) +C=
2,/2+1-21n(i+,/z+l)+c
. - s +&bydefinitionof derivative. At adoubleroot Q'(a) =0.
29Note Q(o)=0. Then
=
Section7.5 Improper Integrals
(page309)
1 - P
1 5 divergesforeveryp! 1 7 Less than $? 3 =
+$PO ,q=tan-' XI; - -$]I" = +2 19Less than $, ' , &
21Less than$PO e-'dx =$, greater than -+
23Less thani,'e2dx +e$re-('-')'dz =c2+e$ ' e - ~l d u=e2+'- Jsr
3 $ ; + 1 lessthan -
251,' -+
=2 27p! =ptimes (p-I)!; 1=1times01
31$ ;
-2
L d x =i f i
7~
$ ; +
=G : - 33w=3 p l ~--- tmV; a, =
1000e--~~dt-10, 000e-. ~~]r=$10,000 = 29u=x,dv=xe-"'dz :- x<] r
$=Jree--+ln2dx =C! I I 00- 1
- I n210 - m
$ ; 35
37$: I2 (seex-tanx)dx=[ln(secx+tanx)+ln(cosx)]:~' =[l n(l +sinx)];l2 =In2.
Theareasunder secx andtanxseparately areinfinite 39Onlyp=0
CHAPTER8 APPLICATIONS OF THEINTEGRAL
Section8.1 AreasandVolumesby Slices
(page318)
1x2-3=1givesx=f 2 ; ~!~[(1-
32
(x2-3)ldx=7
33 =x=9gives y=f3;$_S3[9-y2]dy=36
5x4-2x2=2x2givesx=f2(orx=0);$!2[2x2 - (x4-2x2)]dx=
7y=x2=-x2 +182gives x=0,9;$:I(-x2 +182)-x2]dx=243
9 y = c o s x = c o s 2 x w h e n c o s x = 1 0 r 0 , x = Oo r ~ o r ~ ~ ~ ~ ~ ~ ~ ( c o s x - c o s ~ x ) d x = 1 - ~
-1
11ex =e2z-1 gives x=1;$:[ex -e2'-']dz =(e- 1)- (y)
4
1 3 Intersections(O,O),(l, 3), (2,2);$,'[3x -xldx+~: [4-x-xldx=2
1 5 Inside,since 1-x2<J D ; $: l [ dn- (1-x2)ldx=5 - $
1 7 V=$: a ay2dx=$faab2(l- $)dx =9;around yaxisV=w; rotating
x=2,y=0 aroundyaxis givesacirclenot in thefirst football
v; $ : = - 2ax(8 $ ; a(8 1; 21
~ ( x ~ ) ~ d x =F; $,'27r(l- x4)xdx=
I9V 2x2sinxdx=27r2 x)dx= (sameconetippedover) -X ) ~ ~ X =
23J,' a.12dx-I,'
-
25r
25 ~ ( 3 ~ ) d x =y; 2rx(3- $)dx=7
271,'~ [ ( x ~ l ~ ) ~ =$;lo'2ax(x213-X ~ / ~ ) ~ X - ( ~ ~ l ~ ) ~ ] d x = (noticexysymmetry)
29x2=R2-y2,V=$R-h T ( R ~-y2)dy=r ( ~ h ~
R
- $)
3 1 j : a ( 2 d m) 2 d x =?a3
33J,'(2d=)'dy =2
371A(x)dxorinthiscase$o(y)dy
39Ellipse; J s t a n 8;$(I-x2)tan8; tan8
41Half of ar2h;rectangles 43 ~ ( 5 ~ -22)dx=42r 45J: a(4' - 12)dx=30a
A-14 Answers to Odd-Numbered Problems
59 2 r
6 1 1,' 2ry(2 - &)dy =
63 3re 65 Height 1; $ : 2 r z dz = ra2; cylinder
67 Length of hole is 2d- = 2, so b2 - a2 = 1 and volume is !f 69 F; T(?); F; T
Section 8.2 Length of a Plane Curve
(page 324)
Graphs are flat toward (1,O) then steep up to (1,l); limiting length is 2
~=\ / 36s i n23t +36c os 23t =6 2 3 J , ' a d y = &
1
J!, J - d y = J!, 3(e'+ e-Y)dy = $(ev - e-')]L1 = e - - e
Using x = cosh y this is dy = 1 cosh y dy = sinh y]kl = 2 sinh 1
Ellipse; two y's for the same z 29 Carpet length 2 # straight distance */Z
( dd2 = ( d ~ ) ~ + ( d ~ ) ~ + ( d ~ ) ~ ; ds = \/(%)l + (%)a + (%)2dt;
ds = \/sin2 t + cos2 t + l dt = h d t ; 2 a 4 ; curve = helix, shadow = circle
L = I,' t/TTZ?dz; Jt d G S d z = 1,' JGG 2du = 2L; stretch xy plane by 2 (y = x2 becomes : =
Section 8.3 Area of a Surface of Revolution
(page 327)
1 J" 2rfiJ-dx = 1: 2rdZ+!dx = 3 2 1,' 2 r ( 7 x ) md x = 14s-
1
5 J', - 2 a d = m d x = I-, 4rdz = 8 r 7 1: 2r x J1+(22)2dx = f (1 + 4x2)312]~ = f [173/2 - 11
9 $: 2rzz\/Zdx = 9 r f i 11 Figure shows radius s times angle I9 = arc 2r R
13 2rrAs = r ( R + Rt)(s - st ) = aRs - uR's' because Rts - Rs' = 0
15 Radius a, center at (0, b); +
= a2, surface area st" 2r(b + asin t )a dt = 47r2ab
17 J: 2rx J - dx = 1:
= r2 + 2 r (write 22 - z2 = 1 - (x - 112 and set x - 1 = sin 19)
19 $t12 2 r x d q d z (can be done)
21 Surface area = JF 2r: J x d x > JT 00 = 2 r l n x J r = a, but volume = JF ~ ( $ ) ~ d x = r
23 J : 2 r sin t d 2 sin2 t + cos2 t dt = J : 2 r sin tt/- dt = 2rt/Z--;du =
rut/= + 2 r sin-' 3 ] L 1 = 2 r + 9
Section 8.4 Probability and Calculus
(page 334)
1 p ( X < 4) = i, P(X = 4) = & , P(X > 4) = $
s ir p(x)dx is not 1; p(x) is negative for large x
5 1; e-'dz = -$;/ll.O1e-'dx (J (.01); 7 p(x) = $; F( z) = : for 0 5 x 5 r (F = 1 for x > A)
$ ;
Answers t o Odd-Numbered Problems
9 p = 1 . 1 + ; . 2 + . . . + 1 . 7 = 4 11$* 2xdx =
Iln(1 +x2)]F =+ m 7 7 o n(l+x3) m
~ x e - ~ " d z = [-xe-""]F + e-axdx = a
2dx
= Z tan-' x. JX e-"dx = 1- e-X. ae-aXdx = 1-e-ax 17 $= Le-x/10dZ = -e-~/10 w
1
J X
0 I"n(l+x3) rr 0 0 10 10 I10 = ;
Exponential better than Poisson: 60 years --+ .01e-.~" dx = 1-e-s6 = .45
y = 7;three areas = $ each because p -o to p is the same as p t o p +o and areas add to 1
-2p J xp(x)dx +p2 J p(x)dx =-2p .p +p2 e = -pa l
p = o . $ + 1 . $ + 2 . + 1 ; 0 2 = ( o - 1 ) 2 . $ + ( 1 - 1 ) 2 . $ + ( 2 - 1 ) 2 . $ = Z 3 '
A l ~ o x n ~ ~ , - p ~ = O . $ + 1 . 1 + 4 . ~ - 1 = Z 3 3
00 ~ e - * / ~ d x
-
-
2; 1-Jo 7 p =Jo -7
4 e-s72dx
= 1+[e-x/2]: =e-2
Standard deviation (yes - no poll) 5 1= = & Poll showed = %peaceful.
2 n
95% confidence interval is from %- & to + &, or 93% to 100% peaceful.
31 95% confidence of unfair if more than $= &=2% away from 50% heads.
2% of 2500 = 50. So unfair if more than 1300 or less than 1200.
33 55 is 1.50 below the mean, and the area up to p - 1.50 is about 8% so 24 students fail.
A grade of 57 is 1.30 below the mean and the area up t o p - 1.30 is about 10%.
35 .999; .999100 = (1- &)loo' = $ because (1- i)"4$.
Section 8.5 Masses and Moments
(page 340)
3 I F = ? s z = r 4 5 ~ = 3 . 5 7 z = + g 9 z = + g I I Z = L $ = ~ IS^=$,$$
15 Z = & = g
21 1 = $ x ~ ~ d x - 2 t $ x ~ d x + t ~ $ ~ d x ; ~ = - 2 ~ x ~ d x + 2 t $ ~ d x = 0 f o r t = ~
23 South Dakota 25 2n2a2b 27 M, =0, M, 75 29 $ 31 Moment
33 I =xmnrz;
Crn,rzwz; o
35 14nt$; 14d$;
37 $; solid ball, solid cylinder, hallow ball, hollow cylinder 39 No
a, 41 T =5.mby Problem 40 so T =a,m,4
Section 8.6 Force, Work, and Energy
(page 346)
12.4 ft lb; 2.424 ... ft lb 3 24000 lb/ft; 835 ft lb 5 lox ft lb; lox ft lb 7 25000 ft lb; 20000 ft lb
9 864,000 Nkm 115.6. lo7 Nkm I3 k = 10 lb/ft; W = 25 ft lb 1 5 $6Owh dh =48000cu, 12000w
17 i wAH2; ~ W A H ~ 19 9600w 21 (1- $- ) - 3/ 2 23 (800) (9800) kg 25 f force
CHAPTER 9 POLAR COORDINATES AND COMPLEX NUMBERS
Section 9.1 Polar Coordinates
(page 350)
I ~ O < Y < O O , - ; < B < ~ ; O < r < m , n < ~ < 2 n ; & < r < J S , 0 < 0 < 2 n ; 0 ~ r < m , - ~ < B <
19y=xt anB, r =xs ecB 2 1 B = ~ , a l l r ; r = s i n e ~ e o s e ; r = ~ ~ s B + ~ i n B
2 3 x 2 + y 2 = y 2 5 ~ = r s i n B c o s 8 , y = r s i n ~ 8 , ~ ~ + ~ ~ = ~
(e)2
1 2
+ ( Y - * ) ~ = 2 9 x = C O ~ @ sine Z' 1 x 2 +y 2 =x +y , ( x - Z)
cos @+sin0 9 Y = cos @+sin8
31 (x2 +y2)3 =24
A-16 Answers to Odd-Numbered Problems
Section 9.2 Polar Equations and Graphs
(page 355)
1Line y = 1 3 Circle x2 + y2 = 25 5 Ellipse 3x2 + 4y2 = 1-22 7 x, y, r symmetries
9 x symmetry only 11No symmetry 13 x, y, r symmetries!
1 5 x2 + y2 = 6y + 82 -t (x -4)2 + (y -3)2 = 52, center (4'3)
17 (2,0), (0,O)
l g r = l - &
2
B=s". 4 , r = I + +, 8 = ".(o,o) 21 r = 2 , ~ f~
12'
f~
12' 4 '
=
1 2 '
5-
12
Section 9.3 Slope, Length, and Area for Polar Curves
(page 359)
1Area 3 Area 9 5 Area 7 Area - a 9 $: I3 7r/3 (2 2 Cos2 6 - =
11Area 87r 13 Only allow r2 > 0, then 4 j;l4 i cos - 28 d6 = 1 1 5 2 + q
17 8=O; left points r = +, 8 = f F , x = -I4 , Y = f 9
19 $]i4= 40,000; $[ r J F T F + c2 ln(r + J 7 7 7 ) 1 : 4 = 40,000.001
21t a n$=t a n8 23x=O, y=1i sonl i maconbut not ci rcl e 25iln(27r+J=)+7rd1+4?rZ
r 27 ?f 29 & (base) (height) FJ i ( r ~ 8 ) 31 ?& 33 2s(2 -&) 35 ! f 39 sec 19
Section 9.4 Complex Numbers
(page 364)
1Sum = 4, product = 5
5 Angles F,?f ,
7 Real axis; imaginary axis; + axis x 2 0; unit circle
g c d =5 +1 0 i ,c = u,, 112 cos 8, l ; -1,l 1 3 sum = O, product = -1
1 5 r4e4"
' r
le-'O Le-4'e
, r 4
1 7 Evenly spaced on circle around origin 19 eit, e-" 21et , e-t , e0 23cos7t,sin7t
2 9 t = - z , y = -ex/+ 3 1 F; T; at most 2; Re c < 0 33 be-", x = $ cos8, y = - $ sin 8; fLe-' e/2
J;
CHAPTER 10 INFINITE SERIES
Section 10.1 The Geometric Series
(page 373)
1Subtraction leaves G -XG= 1or G = &
3 L. 9.W. 3 4
5 2 - l + 3 . 2 x + 4 . 3 x 2 + . . - =
29 5' 11 9 99
7 .I42857 repeats because the next step divides 7 into 1 again
9 If q (prime, not 2 or 5) divides l oN - loM then it divides 10N-M - 1 11This decimal does not repeat
19 '"5
3
87 123
1 5 a 17 6 1-111 x 21 23 tan-'(tan x) = x
25 ( ~ + x + x ~ + x ~ - . . ) ( ~ - -x3. . . ) = 1 + x 2 + x 4 z + x 2
2 1
272(.1234 ...) i s 2- &. *=8; 1- . 0123 . . . i ~ l - - ~ ~ loo (1-&)1 - 81 - - ~ 2 9 5 s =1
3
31- l n( 1- . l ) =- l n. 9 3 3 i l n Y 35((n+1)! 3 7 y = L 1 - b ~
39 All products like a1b2 are missed; (1+ 1)(1+1)# 1+ 1 41 Take x = in (13): In 3 = 1.0986
43 In 3 seconds the ball goes 78 feet 45 tan z = $; (18) is slower with x = $
Answers t o Odd-Numbered Problems
Section 10.2 Convergence Tests: Positive Series
(page 380)
1 ? + f + is smaller than 1 + $ + .
1 2 n 1
8 ~ n = S n - S n - 1 = ~ , S = 1 ; ~ n = 4 , S = ~ ; ~ n = h * - ~ n ~ = ~ n ~ , ~ = h 2 n+l n 1
5 No decision on x b, 7 Diverges: &(I + + . *) 9 x - converges: 5 is larger
11 Converges: 5 is larger 1 5 Diverges: x is smaller 15 Diverges: & is smaller
1 7 Converges: x 8 is larger 19 Converges: C 5 is larger 2 1 L = 0 23 L = 0 25 L = 5
2 7 r o 0 t ( v ) ~ + L = $ 29s=l ( onl ysur vi vor ) 3 l I f y d e c r e a s e s , ~ ~ y ( i ) ~ ~ ~ y ( x ) d z ~ ~ ; - ' y ( i )
1
35 x: e-" imO e-ldx = 1; $ + 7 + + . = 55 Converges faster than fi
zC- I+ 1
37 Diverges because ST = ln(x2 + 1)Ir = oo 39 Diverges because Srxe-"dx = = oo
4 1 Converges (geometric) because i;(f)'dx < oo 43 (b) J' +' $ > (base 1) (height &)
45 After adding we have 1 + 5 + . . + & (close to ln 2n); thus originally close to ln 2n - In n = In % = ln 2
1000 &
47 Jloo 2 = 2 loo - looo - - .009
49 Comparison test: sin an < an; if an = m then sin a, = 0 but C an = oo
51an=n- 6/ 2 5 3 a n = 5 65Ratiasarel,~,l,i,...(nolimitL);(&)'l"= ' ; yes
5 7 Ro o t t e s t &- r L=O 5 9 Ro o t t e s t L=& 6 1 Di v e r g e n c e : Nt e r ms a d d t o ~ ~ + m
65 Diverge (compare i) 65 Root test L = Q 67 Beyond some point $ < 1 or an < b,
Section 10.3 Convergence Tests: All Series
(page 384)
1 Terms don't approach zero 3 Absolutely 5 Conditionally not absolutely 7 No convergence
9 Absolutely 11 No convergence 1 3 By comparison with C la, 1
1 5 Even sums + f + a + . diverge; an's are not decreasing 1 7 (b) If an > 0 then s, is too large so s - s, < 0
19 s = 1 - $; below by less than
2 1 Subtract 2($ + f i + . -) = i(fr + & + . . .) = from positive series to get alternating series
23 Text proves: If C lanl converges so does C a,
25 New series = ( 4) - f + (i) - is.. = i(1 - I +
- ..
2 -) 27 In 2 : add in 2 series to $ (In 2 series)
29 Terms alternate and decrease to zero; partial sums are 1 + 8 + + ;! - In n + 7
31 .5403? 53 Hint + comparison test 55 Partial sums a, - ao; sum -a0 if a, + 0
57 && = 3 but product is not 1 +
+ . - .
39 Write x to base 2, as in 1.0010 which keeps 1 + and deletes i, f , . .
4 1 + & + adds to = 6 and can't cancel +
43 a I-cos 1 = cot ? (trig identity) = tan (g - 1). 2 ' s = C 2 n = - log(1- e') by 10a in Section 10.1;
take imaginary part
Section 10.4 The Taylor Series for eZ, sin x, and cos x
(page 390)
+ . . . ; derivatives 2"; 1 + 2 + $ + . . 3 Derivatives in; 1 + i x + .
1 l + 2 x +
5 Derivatives 2"n!; 1 + 22 + 4x2 + . . 7 Derivatives -(n - l)!; -X - E?, -
-
2 3
g y = 2 - e ~ = l - x - I ) - . . . 11 y = x - $ + ... =sin 13 y=~e'=~+~~+d+.-
21 21
15 l + 2 ~ + ~ ~ ; 4 + 4 ( ~ - 1 ) + ( ~ - l ) ~ 1 7 - ( X- I ) ~ 19 i - ( ~ - i ) + ( ~ - l ) ~ - -
21 ( %- 1)- w+ - ... =l n( l + (x-1)) 25 e-'el-= =e-' (1- (x- 1)+ - a * . )
3
25 x+2z2+2x3 27 A - ~ + . 2 24 720 2 g X- d . + & 18 600 3 1 l + x 2 + $ 35 l + x - $
A-18 Answers t o Odd-Numbered Problems
x5 2~~
35ooslope; l + &( z - l ) 3 7 x - 3 - 4 - 5 3 9 ~ + % + ~ 4 1 l + x + $ 43 14- OX- x2
ei e +e-?O i e - -i B
45 cos 8 = ,sin8 = + 47 99th powers - 1, -i, e3"14, -i
49 e-"I3 and - 1; sum zero, product - 1 53 i;, it + 27ri 55 2ex
Section 10.5 Power Series
(page 395)
1 1 + 4 ~ + ( 4 x ) ~ + - . . ; r = !;x= f 3 e ( l - x + < -. . . ); r = co
5 l n e + l n ( l + i ) = 1+ 5 - i ( 5 ) 2 + - . . ; r = e ; x = -e
7 1 < 1 or ( - 1 , )
9 l x - a1 < 1; -l n(1- ( x- a))
l-(1-Lx?..)
1 1 1 + ~ + $ + . . . ; a d d t o l a t x = 0 13 al , a3, . . . ar eal l zer o 1 5 - + L 2
1 7 f ('1 (c) = cos c < 1; alternating terms might not decrease (as required)
xn+l n+ l
1 9 f = & , l R n l I w ; R n = ~ ; ( 1 - ~ ) 4 = 1 - ;
n+ 1
21 f("+')(x) = *,, lRnI 5 -(A) -' 0 when x = 4 and 1 - c > i
23 R2 = f (x) - f (a) - f t ( a) ( x - a) - i f U( a) ( x -
so Rz = R; = R" 2 - - 0 at x = a, R: ' = f"';
Generalized Mean Value Theorem in 3.8 gives a < c < c2 < cl < x
25 1 + i x 2 + ;(x2)' 27 (-l)n; (-l )n(n + 1)
29 (a) one friend k times, the other n - k times, 0 5 k 5 n; 21 33 (16 - 1)'14 EI 1.968
35 (1 + I ) = ( ) ( ) + ( I ) 1.1105 37 1 + $ + 5ZI-r 24 ' = 5 41 x + x2 + $x3 + $x4
43 x2 - 5x4 + &x6 45 1 + + + 2 47.2727 49 -' 6 - 3 = -' 2 5 1 r = 1, r = 5 - 1
CHAPTER 11 VECTORS AND MATRICES
Section 11.1 Vectors and Dot Products
(page 405)
1(0, 0, 0);(5, 5, 5);3;-3;cose = -1 3 %- j -k; -i -7j +8k; 6; -3; cosB = -I 2
5 (v2, -vi); ( ~ 2 , -vl, 0), (v3,0, -v1) 7 (0,0);(0,0,0)
9 Cosine of 8; projection of w on v
11 F;T;F 13 Zero; sum = 10 o'clock vector; sum = 8 o'clock vector times
15 45' 1 7 Circle xZ + J = 4; (x - 1)2 + # = 4; vertical line x = 2; half-line x 2 0
1 9 ~ = - 3 i + 2 j , w= 2 i - j ; i = 4 v - w 2 1 d = - 6 ; C= i - 2 j + k
23cos8 = -&cos8 = -&;cos8=
2 5 A. ( A+ B) = l + A . B = l + B . A = B - ( A + B ) ; equilateral,600
27 a = A . I, b = A . J 29 (cos t, sin t) and (- sin t, cos t) ; (cos 2t, sin 2t) and (-2 sin 2t, 2 cos 2t)
31C=A+B,D=A-B;C.D=A.A+B.A-A-B-B-B=r2-r2 = O
S S U+ V- W= ( 2 , 5 , 8 ) , U- V+ W = ( 0, - 1, - 2) , - U+V+W= (4, 3, 6)
35 c and JFTF; b/a and J a 2 + b2 + c2
~ ~ M ~ = ~ A + c , M ~ = A + ~ B , M ~ = B + $ c ; M ~ + M ~ + M ~ = ~ ( A + B + c ) = o
39 8 5 3 3; 2 & j 5 x + y 41 Cancel a2c2 and b2d2; then b2c2 + a2d2 2 2abcd because (be - ad)2 2 0
43F; T; T; F 45al l 2fi ; cosB = - +
Section 11.2 Planes and Projections
(page 414)
1( 0, 0, 0) and( 2, - l , O) ; N=( l , 2, 3) 3( 0, 5, 6) and( 0, 6, 7) ; N=( 1, 0, 0)
5 (1,1,1) and (1,2,2); N = (1,1,-1) 7 x + y = 3 9 x + 2y + z = 2
Answers to Odd-Numbered Problems A-19
11 Parallel if N V = 0; perpendicular if V = multiple of N
13 i + j + k (vector be tween ~oi nt s) is not perpendicular to N; V . N is not zero; plane through first three
is x + y + z = 1; x + y - z = 3 succeeds; right side must be zero
1 5 a x + b y + c z = O ; a ( x - x o ) + b ( y - yo) +c( z- zo) =O 17cosB= $,$,*
19 &A has length $ 21 P = $A has length $ 1 ~ 1 23 P = -A has length IAl 25 P = 0
27 Projection on A = (1,2,2) has length g; force down is 4; mass moves in the direction of F
29 IPlmin = & = distance from plane to origin 31 Distances 2 and 2 both reached at ($, $, - $)
6 6
3 3 i + j + k ; t = -$;(!,-5,-;);-&
35 Same N = (2, -2, l); for example Q = (0,0,1); then Q + $N = (2, -$, v) is on second plane; $ 1 ~ 1 =
37 3i + 4j; (3t,4t) is on the line if 3(3t) + 4(4t) = 10 or t = g; P = (g, g), IPI = 2
~ 9 2 x + 2 ( ~ - f x ) ( - f ) = 0 s o x = ~ = ~ ; 3 x + 4 ~ = 1 0 g i v e s y = ~
41 Use equations (8) and (9) with N = (a, b) and Q = (xl , yl ) 43 t = A'B B onto A
45 aVL = ?LI - ?LIII; aVF = $LII + $LrII
4 7 V. LI =2 - l ; V. LI I =- 3- l , V. LI I I = - 3 - 2 ; t h u s ~ . 2 i = 1 , ~ - ( i - &j ) = - 4 , and^= $ i + U e 2 J
Section 11.3 Cross Products and Determinants
(page 423)
10 33i - 2j - 3k 5- 2i +3j - 5k 7 2 7 i +1 2 j - 1 7 k
9 A perpendicular to B; A, B, C mutually perpendicular 11 I A x B I = a, A x B = j - k 1 3 A x B = 0
15 [ A x BIZ = (a: + ag)(b: + bg) - (albl + a2b2)2 = (alb2 - a2b1)2; A x B = (alb2 - a2bl)k
1 7 F ; T; F ; T 19N=( 2, 1, O) or 2i +j 2 1 x - y + z = 2 s o N = i - j + k
23[(1,2,1)-(2,1,1)]x[(1,1,2)-(2,1,1)]=N=i+j+k;x+y+z=4
25 (1,1,1) x (a, b, c) = N = (c - b)i + (a - c) j + (b - a)k; points on a line if a = b = c (many planes)
27 N = i + j, plane x + y = constant 29 N = k, plane z = constant
31 1 1 0 = x - y + z = O
I : : I
33 i - 3j; -i + 3j; -3i - j 35 -1,4, -9
39 +c1
b2 b3
- c2
b l b2
41 area2 = ( i ~ b ) ~ + ( ? u c ) ~ + ( $ 6 ~ ) ~ = (21A 1 x B1)2 when A = ai - bj , B = ai -ck
43 A = $(2 1 - (-1)l) = i; fourth corner can be (3,3)
45 ali + a j and bli + b j ; lad2 - a2b1 I; A x B =
+ (alb2 - azbl)k
47 A x B; from Eq. (6), (A x B) x i = -(asbl - alb3)k + (a1b2 - a2bl)j; (A . i )B - ( B . i)A =
al(bli + b j + b3k) - bl(ali + a j + a&)
4 9 N= ( Q- P ) x ( R- P ) = i + j + k ; a r e a $ & ; X + ~ + Z = ~
Section 11.4 Matrices and Linear Equations
(page 433)
A-20 Answers to Odd-Numbered Problems
15 ad - bc = -2 so A-l =
[ :;-:; ]
17 Are parallel; multiple; the same; infinite
19 Multiples of each other; in the same direction as the columns; infinite
21 dl = .34, d2 = 4.91 23 .96x + .02y = .58, .O4x + .98y = 4.92; D = .94,x = .5, y = 5
25 a = 1gives any x = -y; a = -1 gives any x = y
-:] ; D 27 D= - 2 , ~ - l = -1 - = -8, (2A)-' = +A-'; D=
-2 '
(Aw1)-' = original A;
L .I
D = -2 (not +2), (-A)-' = -A-'; D = 1,I-' = I
39 Line 4 + t, errors -1,2, -1 41 dl -2d2 + ds = 0 43 A-' can't multiply 0 and produce u
Section 11.5 Linear Algebra
0 -1
5 det A = 0, add 3 equations -,0 = 1 7 5 a + l b + O c = d , A V 1 =
9 b x c; a . b x c = 0; determinant is zero 11 6, 2, 0; product of diagonal entries
-2 4 0 2 -1
15 Zero; same plane; Dis zero
17 d = (1,-1.0); u = ( 10, 0) or (7,3,I) 19 AB = 4: 2: , det A S = 12 = (det A) times (det B) ] [
18 12 0
2 3 -3
I 1 A + C = [ 1 4 z ] , de t ( A+C) is not det A + det C
0 -1
2 s P = l
2)(3)-(0)(6)
6 6 = 1 , q = -(4)(3)+(0)(0)= -2 25 ( ~ - l ) - lis always A
33 New second equation 32 = 0 doesn't contain y; exchange with third equation; there is a solution
35 Pivots 1,2,4, D = 8; pivots 1,- l , 2, D= -2 37 al;! = 1,a21 = 0,
aijbjk = row 2, column k in AB
CHAPTER 12 MOTION ALONG A CURVE
Section 12.1 The Position Vector
(page 452)
= 1~ ( 1 )i + 3j; speed m;
3 2 =
= %;tangent t o circle is perpendicular t o " =
Y
5 v = e t i - e - t j = i - j ; y - 1 = - ( x - l ) ; x y = 1
Answers to Odd-Numbered Problems
7 R = (1,2,4) + (4,3,O)t;R = (1,2,4) + (8,6,O)t;R = (5,5,4) + (8,6,O)t
9 R = ( 2+t , 3, 4- t ) ; R= ( 2+ $, 3, 4- $);the same line
Line; y = 2+2t , z = 2+3t ; y= 2+4t , z = 2+6t
Li n e ; t / m= 7 ; ( 6 , 3 , 2 ) ; l i n e s e g me n t 15$; l ; $ I I x = t , y = mt + b
v = i - &j,IvI = ~W, T = v/lvl;v = (cost -tsint)i + (sint +tcost)j; Ivl = d m ;
R = -sint i + cost j + any &;same R plus any wt
v = (1-sin t)i + (1-cos t)j; Ivl= 4 3 -2 sin t -2 cos t, Ivlmin = d r f i ,IVI~.. = d c f i ;
a = -cost i +si nt j , l al = 1; center is on x = t, y = t
Leaves at (9, $);v = (-&,&);a = (9, $) + v(t - P)
R = cos l i + sin i j + l k
fi fi \/z
v = sec2t i +sect t ant j ; Ivl = s e c 2 t m ; a = 2sec2t t ant i + (sec3t+secttan2t) j;
curve is y2 -x2 = 1; hyperbola has asymptote y = x
If T = v then lvl = 1; line R = ti or helix in Problem 27
- (240)
0 5 t 5 3 (3 -2t, 1) 15 t 5 q
(x(t)3
- (1,2t - 1) 3 5 t 5 1 (0,4 -2t) q 5 t 5 2
~ ( t ) = 4 c o s i , ~ ( t ) = 4 s i n i 3 7 F ; F ; T; T; F 3 9 f = t a n e b u t t # t a n t
v and w; v and w and u; v and w, v and w and u; not zero
u = (8,3,2); projection perpendicular to v = (1,2,2) is (6, -1, -2) which has length
x = G(t), y = F(t); y = x2I3;t = 1and t = -1 give the same x so they would give the same y; y = G(F-I(%))
Section 12.2 Plane Motion: Projectiles and Cycloids
(page 457)
1(a) T = 16/gsec, R = 128&lg ft, Y = 32/g ft
(b) ; , (c) 0
3 z= 1.2 or 33.5
5 y = x - i x 2 = ~ a t ~ = 2 ; ~ = z t a n x - ~ 2(v, cosa)2= 0 at x = R 7 x = v o e
9 vo M 11.2, tan a M 4.32 11vo = a= am/sec; larger 1 3 +j/2t~ = 40 meters
15 Multiply R and H by 4; dR = 2vi cos 2ada/g, dH = v; sin a cos a da/g
19 T= ~l-cOse)i+sinei
17 t = set; y = 12 - %r, -2.1 m; + 2,lm
,/-
21 Top of circle 25 ca(1- cos 8), casin 8; 8 = r,$ 27 After 8 = r :x = r a + vot and y = 2a - i gt 2 29 2; 3
31 v ; 5 9 a 3 33 x=cos8+8si n8, y=si n8- 8cos8 35 ( a =4) 6 r
57 y = 2sin 8 -sin 28 = 2 sin 8(1 -cos 8); x2 + y2 = 4(1- cos 8)2; r = 2(1- cos 8)
Section 12.3 Curvature and Normal Vector
(page 463)
1-&-5 0 (line)
7 &$&
3 $
9 (- sin t2, cos t2); (- cos t2, -sin t2)
11(cost,sint);(-sint,-cost) 1 3 ( - ~ s i n t , ~ c o s t , ~ ) ; ~ v ~ = 5 , n = &; ~ l o n g e r ; t anB=$
1
16,N = i 1 9 (0,O); (-3,0) with $ = 4; (-1,2) with != 2 f i
l52\/za,/l-cos8
1 7 n = 9
2 l R a d i u s ~ , c e n t e r ( 1 , f ~ ~ f o r n ~ 1 2 3 U- V' 2 5 l ( s i n t i - c o s t j + k ) 2 7 ;
29 N in the plane, B = k, r = 0 33 a = 0 T + 5w2N 55 a = -&T + &N 31 e5
\/z
3 7 a =* ~ +- Ja 39 IF2+ 2(F1)' -FF"I/(F2+ F " ) ~ / ~
\l&N
A-22 Answers t o Odd-Numbered Problems
Section 12.4 Polar Coordinates and Planetary Motion
(page 468)
9r $$+22g=O=Ld( r 2$) 1 1 ~ = . 0 0 0 4 r a d i a n s / s e c ; h = r 2 ~ = 4 0 , 0 0 0 r dt
47r2 150 1017
kg 1 3 mR x a;torque 15T ~ / ~ ( G M / ~ ~ ) ' / ~ 1 7 4n2a3/T2G 19 ( 3 6 5 ~ ) 2 ~ 2 4 ) 1 ( ( 3 ~ 0 ~ 2 ( 6 6 6 7 ~ 1 0 0 1 1
23Us ePr obl eml 5 2 5 a + c = &, a - c = - ,&, solve for C, D
27 Kepler measures area from focus (sun) 29 Line; x = 1
10
33 r = 20 -2t, 0 = z,v= -2ur + (20 -2 t ) g u s ; a = (2t -20) ( %) ~u,-4(%)us; So lvldt
CHAPTER 13 PARTIAL DERIVATIVES
Section 13.1 Surfaces and Level Curves
(page 475)
3 x derivatives ca,-1, -2, -4e-4 (flattest) 5 Straight lines 7 Logarithm curves
9 Parabolas 11No: f = (x + y)" or (ax + by)" or any function of ax + by 1 3 f (x, y) = 1 - x2 - y2
1 5 Saddle 1 7 Ellipses 4x2 + y2 = c2 19Ellipses 5x2 + y2 = c2 + 4cx + x2
2 1 Straight lines not reaching (1,2) 23 Center ( 1, l ) ; f = x2 + y2 - 1 25 Four, three, planes, spheres
27 Less than 1, equal t o 1, greater than 1 29 Parallel lines, hyperbolas, parabolas
3 1 $ : 482 - 3x2 = 0, x = 16 hours 33 Plane; planes; 4 left and 3 right (3 pairs)
Section 13.2 Partial Derivatives
(page 479)
7 -22 . -2
1 3+ 2xy2; -1 + 2yz2 3 3x2y2 -2x;2x3y - eY
5 a;(z%)2 (z2+Y2); (z2+$)2
z&2 i 7% l1Z+ z2:y2 132, 3, 4 156( x+i y) , 6z( x+i y) , - 6( x+zy)
2z2- 2 . 2 2-z2
17( f =! ) f zZ= , s Y , f z y = y ; f y y = yr s
19-a2 cos ax cos by, ab sin ax sin by, -b2 cos ax cos by
2 1 Omit line x = y; all positive numbers; fz = -2(x - Y ) - ~ , fy = 2(x -y)-3
23 Omit s = t; all numbers; 2,A,H,&$
2 5 x > O, t > Oa n d x = O, t > 1 andx=-1,-2,...,t=e,e2,...;fz = ( l nt ) ~' " ~- ' , ft = ( ~ n x ) t ~ ~ ~ - '
27 y, x; f = G(x) + H( y) 29 = Y v ( x y ) = yv(zy)
3 1 fzzz = 6 9 , fyyy = ex3, fzzy = f zyz = fuzz = 18x9, fyyz = fyzy = fxyy = 18x2y
3 3 g(y) =
35 g(y) = ~ e ' y / ~ + ~ e - ' y / ~
37 ft = -2 f , fzz = fyy = -e-2t sin x sin y; e-13' sin 2x sin 3y
39 sin(x + t ) moves left 4 1 sin(x - ct), cos(x + ct), ez-"
43 (B- A) hy (C*) = ( B - A) [fy (b, C*) - fy (a, C*) ] = ( B - A) ( b -a) fyz (c*, C*); continuous fxy and fyx
45 y converges t o b; inside and stay inside; d, = J(x, -a) 2 + (y, - b) 2 -+ zero; d, < E for n > N
47 E, less than 6
49 f (a, b);
1
51 f (0,O) = 1; f (0,O) = 1; not defined for x < 0
or ( x- l ) ( y- 2)
Answers to Odd-Numbered Problems
Section 13.3 Tangent Planes and Linear Approximations
(page 488)
9 Tangent plane 2 4 2 -a)-2xo(x -xo) -2yo(y -yo) = 0; (0,0,O) satisfies this equation because
zi -xg -yi = 0 on the surface; cos 9 = ,m= dL,
N-k
= (surface is the 45' cone)
zg+Y: +r,l
11dz = 3dx -2dy for both; dz = 0 for both; Az = 0 for 3% -2y, Az = .00029 for x ~ / ~ ' ; tangent plane
13 z = z o + Fzt; planeB(x-4) + 12( y- 2) +8( ~- 3) =O; normalline x = 4 +6 t , y = 2+12t , z = 3+8t
15 Tangent plane 4(x -2) + 2(y -1) + 4(z -2) = 0; normal line x = 2 + 4t, y = 1+ 2t, z = 2 + 4t; (0,0,0)
at t = - 1
2
1 7 dw = yodx + xody; product rule; Aw -dw = (x -xo)(y-yo)
19 d I = 4000dR + .08dP; d P = $100;I = (.78)(4100)= $319.80
21 Increase = - = &,decrease = - = &;dA = Adz - Sdy; 3 23 A@ M - Y ~ ~ + ~ ~ Y
Y ,/z'+y'
25 Q increases; Q8 = - y , Q t = +,pa = -.2Q8 = El3' Pt = -.2Qt = $; Q=50- Z$l(s- .4) - $( t - 10)
P8 = -Qu =
2 7 s =l , t =l Og i v e s Q=4 0 :
s Qa +Q=Q8+40 ;Q8=-2O, Qt=-;, p8 =20, Pt = $
Pt = -Qt = sQt + 1= Qt + 1
2 9 s - 2 = x- 2+2( y- 1) and z - 3 = 4 ( ~ - 2 ) - 2 ( ~ - 1 ) ; ~ = 1, y= ; , Z = O
2 , ~ 1 1 31 AX = -$, Ay = A - = 5,yl = -$; line X + ~ = O
33 3 a 2 ~ x -Ay = -a -a3 gives Ay = -Ax = f&&;lemon starts at (I/&, -I/&)
-Ax + 3a2Ay = a + a3
35 If x3 = y then y3 = x9. Then x9 = x only if x = 0 or 1or -1 (or complex number)
37 AX = -xo + 1,Ay = -yo + 2, ( XI , yl) = (1,2) = solution
x1
39 G = H = 2xn: 1
4 l J = [: : Y ] , AX= - l +e- xn, Ay= -1- (..-l + e - ~ n ) e - ~ n
43 ( ~ 1 , = ~ 1 ) (0, :), (-:, :)I (;to)
Section 13.4 Directional Derivatives and Gradients
(page 495)
1grad f = 2xi -2yj, Du f = f i x -y, Du f (P) = fi
3 grad f = ex cos y i -ex sin y j, Du f = -ex sin y, Duf ( P) = -1
t i f = ~ ~ ~ + ( ~ - 3 ) ~ , g r a d f = j i + y j , & f = r , & f ( P ) = L fi 7gr adf = *i +, &j
9 grad f = 6xi + 4yj = 6i + 8j = steepest direction at P; level direction -8i + 6j is perpendicular; 10, 0
l l T; F( gradf i savect or) ; F; T
1 3 ~ = ( * \/W a2+b' ' b - ) , ~ u f = d =
(&,s), f 15 grad f = (ex-Y, -ex-Y) = (e-', -e-') at P ; u = = h e - '
17gradf=Oatmaximum;levelcurveisonepoint I gN=( - I l l , - l ) , U=( - 1, 1, 2) , L=( l , l , O)
21 Direction -U = (-2,0, -4)
23 -U = ( d m ,
- xP- Yl
l - z l - ~ l )
25 f = (x + 2y) and (x + 2y)2; i + 2j;straight lines x + 2y = constant (perpendicular to i + 2j)
27 grad f = f(A,3); grad g = f(2& &), f = f(3- %) + C,g = f( 2 h x + &) + C
. .
29 9 = constant along ray in direction u = 7;grad 9 = wi = *;u-grad t9 = 0
x +Y
31 U = (fx, fy, fi+ f i ) = (-1, -2,s); -U = (-1, -2,5); tangent at the point (2,1,6)
33 grad f toward 21 +j at P,j at Q, -2i +j at R; (2, ?) and (21,2); largest upper left, smallest lower right;
z,,, > 9; z goes from 2 to 8 and back to 6
A-24 Answers to Odd-Numbered Problems
35 f = iJ(x - 112 + (y - 112; ( 3 ,
= (3
~ J Z ) 2J i '
37 Figure C now shows level curves; lgrad f 1 is varying; f could be xy
39 x2 + xy; ex-'; no function has 3= y and % = -x because then f,, # f,,
4 1 v = (1,2t); T = v / &S F ; % = v . (2t, 2t2) = 2t + 4t3; $ = (2t + 4t3)/J-
43 v = (2,3); T = -&; 3 = v . (2xo + 4t, -2yo -6t) = 4xo -Byo - lot; $ =
45 v = (et,2e2',-e-');T = G;grad f = (; , $, $) = ( ~- ' , e- ~' , e' ) , % = 1+2 -
0
1, = -2-
Ivl
47 v = (-2 sin 2t, 2 cos 2t), T = (- sin 2t, cos 2t); grad f = (y, x), 2 = -2 sin2 2t + 2cos2 2t, % = i s ;
zero slope because f = 1on this path
4 9 2 - 1 = 2 ( x - 4 ) + 3 ( y - 5 ) ; f = l +2 ( x - 4 ) +3 ( y - 5 ) 51 grad f . T = O ; T
Section 13.5 The Chain Rule
(page 503)
1f, = cfx = c COS(X + ey) 3 f, = 7fx = 7ex+7'
5 3g2*& ax dt + 3 2 % 2
7 Moves left at speed 2
9 2= 1 (wave moves at speed 1)
11sf(x + iy) = f t t ( x+ i y), -@-f(x+iy) = i 2f t t ( x+i y)
so f i x + f,, = 0; (x + ~ y ) ~ = (x2 - #) + i(2xy)
1 3 % = 2 ~ ( 1 ) + 2 ~ ( 2 t ) = 2 t + 4 t ~ 1 5 $ = y $ + x $ = - 1 1 7 * = l d . + 1 * = 1 dt x+ydt s+ydt
19 V = STr2h dV
27rrh dr
7rr2 dh = 3GT
dt=--3 d t + ~ d t
90
90a+90a 903+90
= Ji m ~ h ; dD
2 1 % = d z ( 6 0 ) + d7(45)
7 T = d- 60
(60) + J-
45 (45) cl 74 mph
23 $ = U I % + U ~ % + U ~ % 25 g = l wi t h x a n d y f i x e d ; % = 6
27 ft = fxt + f , W ft t = fxtt + fx + 2fytt + 2f, = (fxxt+ fYX(2t ))t + fx + 2(fx,t + f,,(2t))t + 2f,
29
= gg + gg = ~ C O S B + U s i n B , ~is fixed
x3
= h..
a?
3 , a(:)-- r - .,-2& ax = L - 2 - 3,2 2 +~ 1 ) 3 ax r ~ 3 - $
1
2 )(2), = I; first answer is also J&= eosr
35 f r = f ~cosB+f , si nB, fro = -fxsin~+f,cosB+fx,(-rsinBcos~)+fx,(-rsin2~+rcos28)+f,,(~~~~~~i~~)
-
37 yes (with y constant): 2 = yex', 2 = 2 - &
39 ft = fxxt + f,yt; ftt = fxxx; + 2fX,xtyt + fyyy?
9 ( 2 ) y = a; (E).= gg = 4 1 ( % ) , = % + % 2 = a - 3 b . $6
4 3 1 4 5 f = y 2 s o f x = ~ , f , = 2 y = 2 r s i n ~ ; f = r 2 s o f r = 2 r = 2 J ~ , ~ f e = ~
47 gu = fxxu + f,yu = f x + f,;gu = fxxu + f,yu = fx - f,; guu = f,& + fx,y, + f,,x, + f,,yu
-
- f i x + 2 f ~ y+ fyy; ~ U U= ~ X X X ~ + fXyyu - fyXxu-fyyyu= fix - 2fxy + fyy Add gUu+ guu
49 False
Section 13.6 Maxima, Minima, and Saddle Points
(page 512)
1(0,O) is a minimum 3 (3,O) is a saddle point 5 No stationary points 7 (0,0) is a maximum
9 (0,0,2) is a minimum 11All points on the line x = y are minima 13 (0,0) is a saddle point
1 5 (0,O) is a saddle point; (2,O) is a minimum; (0, -2) is a maximum; (2, -2) is a saddle point
1 7 Maximum of area (12 - 3y)y is 12
2(x + y) + 2(x + 2y -5) + 2(x + 3y -4) = 0 x = 2;
19
gives
y = - 1
min because ExxEyy= (6)(28) > E: , = 1z2
2(x + y) + 4(x + 2y - 5) + 6(x + 3y - 4) = 0
2 1 Minimum at (0, i ) ; ( 0, l ) ; ( 0, l )
Answers to Odd-Numbered Problems
23 %=0whent ant =&; f m, =2at ( i , $) , f mi n=- 2at (-+,-'$)
1
25 (ax + by),, = W; (x2 + y2)min =
2 7 0 < c < f
29 The vectors head-to-tail form a 60-6@6O triangle. The outer angle is 120' 31 2 + &; 1 + fi; 1 +
35 Steiner point where the arcs meet 39 Best point for p = oo is equidistant from corners
41 grad f = (& ?+ y+ y,\/Z ?+
+ 7); angles are 90-135-135
43 Third derivatives all 6; f = 5x3 + *x2 + $29 + 5 y3
1 2 3 3
45 (&)n(s)m ln(1- ~ y ) ] ~ , ~ = n!(n - I)! for rn = n > 0, other derivatives zero; f = -xy -
2
- 3 - . .
47 All derivatives are e2 at (1,l); f N e2[l + (x - 1) + (y - 1) + i ( x - 1)2 + (x - l)(y - 1) + ?(y - I ) ~ ]
4 9 x = l , y = - 1 : f, = 2, f, = -2, f,, = 2, fx, = 0, f, = 2; series must recover x2 + y2
51 Line x - 2y = constant; x + y = constant
5 3 ~ f . , + z y f x , + f f , , ] ~ , ~ ; f x , > Oa n d f x z f u v > f ~ a t ( ~ , ~ ) ; f x = f v = O 5 5 Ax =- l , Ay =- 1
57 f = x2(12 - 42) has fmax = 16 at (2,4); line has slope -4, y = 5 has slope = -4
59 If the fence were not perpendicular, a point to the left or right would be closer
Section 13.7 Constraints and Lagrange Multipliers
(page 519)
2k kkl
3 A = -4, Xmin = 2, Ymin = 2
1 f = x2+ (k- 2 ~ ) ~ ;
= 22- 4(k- 22) = 0; (-g-, g) , -g
5 X = : (x, y) = ( ~k2' / ~, 0) or (0,f21/6), fmin = 2lI3; X = ' 3 (x, Y) = (*I, f 1 ) s f m a ~ = 2
7 X = i, (x, y) = (2, -3); tangent line is 22 - 3y = 13
9 (1 - c ) ~ + (-a- c)'+ (2 - a - b - c)'+ (2- b - c ) ~ is minimized at a = - $, b = t , c = Q
1
11 (1, -1) and (-1,l); X = -5
13 f is not a minimum when C crosses to lower level curve; stationary point when C is tangent to level curve
15 Substituting = = = 0 and L = fmin leaves = X
1 7 x2 is never negative; (0,O); 1 = A(-3y2) but y = 0; g = 0 has a cusp at (0,0)
19 2x=X1+X2,4y=X1,2s=X1 - X2 , x +y +z =0 , x - z = 1 gives X1 =0,X2 = 1, fmin= ? at (;,o,-?)
21(1, 0, 0);(0, 1, 0);(Xl , X2, 0);x=y=O 2 3 %a n d d ; X= O
25 (1,0,0), (0,1,0), (0,0,1); at these points f = 4 and -2 (min) and 5(max)
27 By increasing k, more points are available so fmax goes up. Then X =
2 0
29 (0,O); X = 0; fmin stays at 0
31 5 = X1 + X2, 6 = X1 + As, X2 2 0, As 5 0; subtraction 5 - 6 = X2 - X3 or -1 2 0 (impossible);
x = 2004, y = -2000 gives 52 + 6y = - 1980
33 22 = 4X1 + X2, 2y = 4X1 + As, X2 2 0, X3 2 0,4x + 4y = 40; max area 100 at (10,0)(0,10); min 25 at (5,5)
CHAPTER 14 MULTIPLE INTEGRALS
Section 14.1 Double Integrals
(page 526)
A-26 Answers to Odd-Numbered Problems
& L .-L
37 + q L ! = , C L f (n,
~ ) i s e x a c t f o r f = 1 , x , y , x y 39Vol ume8. 5 41Vol umesl n2, 2l n( l +&)
n
&dy $: = xydx dy J: $: 4 3
45 Wi t h long rectangles
1 1
Ins = l n2;J0 loxydy dx = $' o " - l dx = In2
yi AA = A A = 1 but $$ y d A =
Section 14.2 Change to Better Coordinates
(page 534)
Wdu dv
1 $:;i4 $:
1 dl-"'
5 R is symmetric across the y axis; So So u du dv = 5 divided by area gives (a,U) = ( 4/ 3r , 4 / 3 4
'+-dy dx; xy region R* becomes R in the x*y* plane; dx dy = dx'dy* when region moves
7 2So
I *
Sl+x
g J =
COSO* -r*sinO*
= r*;$:7i4so1r*dr*dO*
11 I y = $$Rx2dx dy = $:Y/~$: r2cos2O r dr do = 5 - i;Is = 5 + i;I. =
13 (0, 0), (1, 2), (1, 3), ( 0, l ) ; area of parallelogram is 1
15 x = u, y = u + 3v + uv; then ( u, v ) = ( 1, 0) , ( 1, I ) , ( 0 , l ) give corners ( x , y) = ( 1,O) , ( 1, 5) , ( 0, 3)
3
17 Corners ( 0, 0) , (2, 1), (3, 3), (1, 2); sides y = i x , y = 22 - 3, y = i x + 5 , y = 22
I 9 Corners ( 1, I ) , (e2, e) , (e3, e3) , (e, e2) ; sides x = y2, = x2/ e3, x = y2/e3, y = x2
1
r dr dB = 2
3 S = quarter-circle with u > O and v 2 0;So So
2 1 Corners (0,0
/
, (1, 0), (1, 2), ( 0, l ) ; sides y = 0, x = 1, y = 1 + x2, x = 0
1 1
3e3u+3"dU dy $: $ :
e 2 ~ + ~ ze2u+v
1 3 J = = 3, area soZdu dv = 3; J = eu+2v 2eu+2v -
3e3u+3v, - =
1 0
1 25 corners ( x , y) = ( 0, 0) , ( 1, 0) , ( 1, / ( 1 ) ) , ( 0, f ( 0 ) ) ; ( $ 9 1) gives x = $ 9 Y = f ( $ ) i J = v u f(,)
27 ~2 = 2 $:I4 $:Isine e-r' r dr do = - 1
" 8 2n 1
29f = // r2dr dB/ // r dr dB = So ,a
3
cos3 B dd/xa2 = 97r
3 1 /, sor2r dr dB = 5
33 Along the right side; along the bottom; at the bottom right corner
1 1
35 $$ xy dx dy = So So (ucos a - v si na) ( usi n a + v cos a) du dv = f (cos2 a -sin2 a )
37 $:" $ ' r2r2r dr do = Y(S6-46)
39 x = cos a -sin a , y = sin a + cos a goes t o u = 1, v = 1
Section 14.3 Triple Integrals
(page 540)
21 Corner of cube at
1 1
( &,z, sides 5;
3&
z); area
4
23 Horizontal slices are circles of area r r 2 = a( 4 -z ) ; volume = lor ( 4 -z ) dz = 8r ; centroid
4
has z = 0 , g = 0 , z = sozl r(4--z)dz/ 8r= 5
AnswerstoOdd-NumberedProblems
f dxdz,&
$ givesaeros;
= dydz, f $ : = I: --I,"f
25I =
J : ~ ( ~ ~ + z2)dxdydz= y;J/Ix2dv= t ;3$JJ(x $! , $: , 27 - T ) 2 d ~ =
29 J: dxdydz= 6 $1Tkape~oidalruleissecond-order; correctfor 1,x,y,z,xy,xz,yz,xyz
Section 14.4 Cylindrical and Spherical Coordinates
(page 547)
1(r,8,2) = (D,0,0);(P,498) = (Dl:, 0) 3(r,#,a)= (0,anyangle,Dl;(P,4,8)= (D,0, anyangle)
5(x,y,z)=(2,-2,2fi);(r,8,~)=(2@,-f,2@) 7(x, y, z)=(O, O, -l);(r, d, z)=(O, anyangle, -1)
94 = tan-'(:) 1145' coneinunitsphere: y(1-A) 1 3conewithout top: 2
15 hemisphere: 1 7 $ 19Hemisphereof radiusr ::r4 21r ( R2-z2);4 r t - d n
23$a3t ana(see8.1.39)
27 = p-DcOsC - nearlide
-
Q hypotenuse = COs
31Wedges arenot exactlysimilar;theerror ishigherorder + proof iscorrect
33Proportionalto1+ i(\/02 + (D-h)2-@TP)
a cos8 -rsin# 0
35J= b = abc;straight edgesatright angles 37 sin8 rcos8 0 = r
C 0 0 1
3g e.n
3 '3
41p3; pa;force= 0 insidehollowsphere
CHAPTER 15 VECTOR CALCULUS
Section 15.1 Vector Fields
(page 554)
l f ( z , y ) = x + 2 y 3f ( x, y) =si n( x+y) 5f ( x, y) =l n( x2+#) =21nr
7 F= xyi+ Gj, f(x,Y) = 9
9 = O sof cannot depend onx;streamlines arevertical (y= constant)
1 1 F = 3 i + j I S F = i + 2 y j 1 5 F = 2 x i - 2 y j 17F=ex- vi - ex- Yj
Y ' l g z = - l ; y = - x + C 2 1 $ =- E- x 2 +y 2 =C 233 x2+ y2 = C 25parallel = *-= 7;
2 7 ~ = Fi+yj
2 9 F = -:fG(xi+yj)- ((x-1)2+Ya)3/2 ~ M G ((x- 1) i+ Yj)
t l ~ = $ ~ i - q ~ j J J ~ z = = = - ~ . ~ K = 2 9 d ~ * = 2
35
= gE= g:; 5 = g f ; f(r)= C givescircles
37T;F(noequipotentials); T;F(notmultipleof xi+ yj+ zk)
39FandF+ i and 2Fhave thesamestreamlines(differentvelocities) andequipotentials(differentpotentials).
But if f isgiven,Fmust be grad f.
Section 15.2 Line Integrals
(page 562)
l $ ; d Wd t = &; j , ' 2 d t = 2 3c t 2\ / Zdt +~: 1. ( 2- t ) dt = $+;
5JtU(-3 sint)dt= 0(gradientfield);J: " -9 sin2t dt= -97r = - area
7No, xyjisnot a gradient field;takelinex= t,y= t from (0,O) to(1'1) and $ t2dt# ?
g No , f o r a ~ Lc l e ( 2 7 r r ) ~ # 0 ~ + 0 ~l l f = x + ~ # ; f ( O, l ) - f ( 1 , 0 ) = - i
13f = +xay2;f(0, l )-f( 1, O) = 0 15 f = r = dm; f(0, l )- f( l , ~ ) = 0
17Gradient forn = 2; after calculation - = ~3
ax rn
19x=acos t , z= asint, ds=adt , M=$, ( a+asi nt ) adt = 2ra2
A-28 Answers to Odd-Numbered Problems
2 1 x = a cos t ,y = a sin t , ds = a dt , M = a3 cos2 t dt = nu3, (3,$) = ( 0, 0) by symmet ry
2i+2tj
4+4t +;F = 3 x i + 4 j = 6 t i + 4 j , d s = 2 d m d t , ~ . ~ d s =
23T=\r, =d-
( 6t i +4j ) - ( - $=$) 2~mdt =
2Ot dt ; F .d R = (6t i + 4). ( 2 dti + 2t dt j ) = 20t dt ; work = J1
2
20t dt = 30
25 ~f
= t hen M = cay + 6 , N = ax + c , constants a, b, c
27 F = 4xj (work = 4 f rom (1,O) u p t o ( 1, l ) )
29 f = [ X -2ylIt:ij = -1 3 1 f = [ x y 2 ] ~ ~ : ~ ~ = 1
-' - ( t i $: 3 3 Not conservative;
1
t j ) . ( dt i + dt j ) = $ 0 dt = 0; (t 2i -t j ) ( dt i + 2t dt j ) = so-t2dt =
3
35 = ax, = 22 + 6 , so a = 2,b is arbitrary 37 = 2yebx = w-f = -y2e-"
BY ay ax 9
a M = ~ = ~ . f = r = J ~ = 1 x i + y j 1
3 9 ~ ax , f-
Section 15.3 Green's Theorem
(page 571)
1 $:"(a cos t ) a cos t dt = r a2; Nz - My = 1, $$ dx dy = area r a2
0
3 J, ' xdx+J1 x ~ x = O , N ~ - M ~ = O , J $ O ~ X ~ ~ = O
27r
= 4 ' 5 $ x 2 y d x = $: 7r( a~ost ) 2( asi nt ) ( - asi ntdt ) = -$so ( ~ i n 2 t ) ~ d t-d.
27r a
N, - M y = - x 2, $$( - x 2) dx dy = SO So - r2cos2@ r dr d0 = -$-
7 J x dy - y dx = $' ( c os 2 t + sin2 t ) dt = r;$ / ( I + 1) dx dy = 2 (area) = s;$ x2dy - x y dx = $ + 1;
Jl
( 22+ x) dx dy = $
9 4 $i n( 3 cos4 t sin2 t + 3 sin4 t cos2 t ) dt = i stff 3 cos2 t sin2 t dt = $2 (see Answer 5 )
11 $ F d R = 0 around any loop; F = :i + Fj and $ F d R = $:"[- sin t cos t + sin t cos t ] dt = 0;
= z gives $$o dx dy
ay
2n
1 3 x = cos 2t , y = sin 2t , t f rom 0 t o 2 r ; So -2 sin2 2t dt = - 2s = -2 (area);
$:7r -2dt = -47r = -2 times Example 7
1 5 J ~ d y - ~ d x = ~ ~ " 2 s i n t c o s t d t = 0 ; ~ $ ( ~ , + ~ ~ ) d x d ~ = $ $ 0 d x d ~ = 0
2lr
1 7 M = ~ , N = ~ , $ ~ d y - ~ d x = $ , ( c o s 2 t + s i n 2 t ) d t = 2 r ; $ $ ( ~ x + ~ y ) d x d y = $ $ ( ~ - $ . + ~ - $ ) d x d y =
$$ k d x dy = $$ dr dB = 2 s
19 $ ~ d -yNd x = / - x2y dx = - x2( 1 1: - x) dx = A;$' o o $ I - Y x2dx dy = &
2 1 J$( M, + Ny ) d x dy = $$ di v F dx d y = 0 between t he circles
23 Work: $ a dx + b dy = $$(%- E ) d x dy; Flux: same integral
25 g = tan-' (:) = 0 is undefined at (0, 0) 27 Test My = N, : x2dx + y2dy is exact = d( 5x3+ 5y3)
2 9 d i v F =2 y - 2 y =O; g =x y 2 3 1 d i v F = 2 x + 2 y ; n o g 3 3 d i v F =O; g =e x s i n y
35 div F = 0; g = $
37 N, - My = -22, -6xy, 0, 2x - 2y, 0 , -2ex+Y; i n 3 1 and 3 3 f = 5( x3 + y3) and f= ex cos y
39 F = ( 3x2 -3y2)i -6 x y j ; d i v F = 0 4 1 f = x4 -6x2y2+ y4; g = 4x3y -4xy3
4 3 F = ez cos y i - e x s i ny j ; g = e x s i ny
f 45 N = f ( x ) ,$ Md x + Nd y = I,' +
f ( 0) dy= f (1) - f (0); $$( N, - My ) d x dy =
$$ g d x dy = I,' g d x (Fundamental Theorem of Calculus)
Answers to Odd-Numbered Problems
Sect ion 15.4 Surface Integrals
(page 581)
2rr 2
1 N = -2xi - 2yj + k; dS = dl + 4x2 + 4 9 dx dy; lo /, d w r dr dB = :(17~/' - 1)
3 ~ = - i + j + k ; d ~ = f i d x d y ; area fir
.
-21-
+k ; d S= d2d
2~ 1 / f i rdrd8
5 N= d & 0 0 J--+fi)
7 N = -7j + k; dS = 5 f i dx dy; area 5 4 ~
9 N= ( y 2 - x 2 ) i - 2 x y j + k ; d S = ~ l + ( y 2 - + 4x2y2dx dy = dl + (y2 + ~ 2 ) ~ d x dy;
JtCJ,' d m r dr d0 =
N = 2i + 2j + k; dS = 3dx dy; 3(area of triangle with 2% + 2y 5 1) =
A = -sinu(cosv i +s i nvj ) +cos uk; B = - ( 3+cosu) si nvi + (3+cosu)cosv j;
N = -(3 + cosu)(cosucosv i +cosusinv j + sinu k); dS = (3 + cosu)du dv
$ J ( - M~ - N% + P)dx dy = JJ(-2x2 - 2 3 + z)dx dy = -r2(r dr d0) = -87r
F . N= - z +y+z =Oonpl a ne
N = - i - j + k , F = ( v + u ) i - u j , J ~ F . N d S = I I - v d u d v = ~
2rr 2rr
JJ dS = so Jo (3 + cos u)du dv = 127r2
31 Yes 33 No
A = i + f'cos0 j + f' sin0 k ; B = -f sin8 j + f cos8 k ; N = ff' i - f cos8 j - f sin0 k; dS = INldz dB =
f ( x ) d m dx dB
l d i v F = l , J J J d ~ = Y 3 d i v F = 2 ~ + 2 y + 2 z , ~ / $ d i v ~ d V = 0 5 d i v F = 3 , ~ ~ 3 d ~ = ~ = ~
2~ ~ / 2
7 F N = pa, JJp=a p2dS = 47ra4
9 div F = 22, lo I.
J: 2pcos 4(p2 sin 4 dp d# dB) = i r u 4
11 J: J : J:(2x + 1)dz dy dz = a' + a3; -2a2 + 2u2 + 0 + a' + 0 + a3
I ~ ~ ~ v F = $ , J I J $ ~ v = o ; F . ~ = x , J J x ~ s = o 1 5 d i v F = l ; J I I i d V = ~ ; ~ $ J 1 d ~ = ~
R div R
1 7 div (7) = 7 + R grad$ = 3 - $R gradp
19 Two spheres, n radial out, n radial in, n = k on top, n = -k on bottom, n = on side;
@T7
n = -i, -j, -k, i + 2j + 3k on 4 faces; n = k on top, n = l ( ' i + fj -
fi
k) on cone
21 V = cylinder, / div F dV = /I(% + +)dx dy (a integral = 1); IJ F - ndS =
Mdy - Ndx, z integral = 1 on side, F - n = 0 top and bottom; Green's flux theorem.
23 div F = -:yM = -47rG; at the center; F = 2R inside, F = 2(:)3R outside
25di vu, = : , q= y , / J E - n d ~ = $ I l d ~ = 4 a 2 7 F ( di vF=O) ; F; T( F. n< 1); F
29 Plane circle; top half of sphere; div F = 0
Section 15.6 Stokes' Theorem and the Curl of F
(page 595)
l c u r l F = i + j + k Sc u r l F=O 5 c u r l F = O 7 f = + ( x + y + ~ ) ~
9 curl xmi = 0; xnj has zero curl if n = 0 11 curl F = 2yi; n = j on circle so $$ F - ndS = 0
1 3 c u r l ~ = 2 i + 2 j , n = i , ~ ~ c u r l ~ ~ n d ~ = ~ ~ 2 d ~ = 2 7 r
A-30 Answers to Odd-Numbered Problems
15 Both integrals equal F dR; Divergence Theorem, V = region between S and T, always div curl F = 0
17 Always div curl F = 0 19 f = xz + y 21 f = e2-' 23 F = yk
25 curl F = ( a s k - a2bs)i + (alb3 - a3bl)j + (a2bl - alba)k 27 curl F = 2wk; curl F .
= 2 w / 4
29 F = x(a3z + a2y)i + y(alx + a3z)j + z(al x + a2y)k
2% r / 2
31 curl F = -2k, JJ -2k . RdS = Jo Jo
-2 cos 4(sin 4 d4 dB) = -2r; J y dx - x dy =
J:"(- sin2 t - cos2 t)dt = - 2r
2% %/2
33 curl F = 2a, 2 / / (al x + a2y + a3z)dS = 0 + 0 + 2a3 Jo Jo
cos 4 sin 4 d4 dB =
35 curl F = -i,n = *,JIB' . n d S = - ~ r r ~
h A
3 7 p = d - I ' = stream function; zero divergence
39 div F = div ( V + W) = div V so y = div V so V = $j (has zero curl). Then W = F -V = xyi - $j
41 curl (curl F) = curl (-2yk) = -21; grad (div F) = grad 22 = 2i; Fx2 + F,, + Fzz = 4i
aB
43 curl E = -= = a s i nt so E = ?(a x R) si nt
CHAPTER 16 MATHEMATICS AFTER CALCULUS
Section 16.1 Linear Algebra
(page 602)
1 All vectors c 3 Only x = 0 5 Plane of vectors with xl + x2 + x3 = 0
7 + = [ ~ ] , A ( X ~ + ~ O ) = [ : I + [:]
9 A(xp + xo) = b + 0 = b; another solution
1 0 1
13 C C ~ = [ 0 1 2 ] ; CTC = [ : : ] ; (2 by 3) (2 by 3) is impossible
1 2 5
15 Any two are independent 17 C and F have independent columns
23 det ( F - XI) = det [ 2 ; X 2!X] = ( 2 - X ) 2 - 1 = 3 - 4 X + X 2 = ~ i f X = l o r X = 3 ;
l - X 1
1 - X 1 = (1 - - 3(1- A) + 2 = X3 - 3X2 = 0 if X = 3 or X = 0 (repeated)
1 l - X ' I
Answers to Odd-Numbered Problems
3 1 H
33 Fif b 2 0; T;T; F(eAtis not a vector); T
= [ -2
2
-:]
Section 16.2 Differential Equations
(page 610)
13Best -Best = 8est gives B = 4 :y = 4est 3 y = 3 - 2 t + t 2 5 Ae t + 4 e s t = 7 a t t = Oi f A= 3
7 Add y = Ae-' because y' + y = 0; choose A = -1 so -e-' + 3 -2t + t2 = 2 at t = 0
e" - 1 tekt
g y = *;,= t; by19~bpi t al lim- = lim- = t
k+O k k+O 1
11Substitute y = Aet + Btet + C cost + D sin t in equation: B = 1,C = i,D = -i,any A
13Particular solution y = Atet + Bet; y' = Atet + (A + B)et = c(Atet + Bet) + tet
-1
g i v e s A=c A+l , A+B=c B, A= &, B==
15X2eXt+ 6XeXt+ 5eXt= 0 gives X2 + 6X + 5 = 0, (A + 5)(X + 1)= 0, X = -1 or -5
(both negative so decay); y = Ae-' + Be-5t
1 7 (A2 + 2X + 3)eXt= 0, X = -1 f\/=Z has imaginary part and negative real part;
+ ~~( - 1 - f i ~i ) t ; y = ~ ~ ( - l + f i i P y = Ce-' cos f i t + De-' sin f i t
19d = 0 no damping; d = 1underdamping; d = 2 critical damping; d = 3 overdamping
21 X = -:z t is repeated when b2 = 4c and X = -i;(tX2 + 2X)ext + b(tX + 1)e" + ctext = 0
when X2 + bX + c = 0 and 2X + b = 0
23 - most -bsint -asi nt + bcost + acost + bsint = cost if a = 0, b = 1,y = sint
25 y = Acos3t + Bcos5t;y" + 9y = -25Bcos5t + 9Bcos5t = cos5t gives B = G;
yo = 0 gives A =
1
27 y = A(cos wt -cos wet), y" = -Aw2 cos wt + Aw: cos wot, y" + wiy = cos wt gives A(-w2 + wg) = 1;
breaks down when w2 = w i
2 9 y = BeSt ; 25B+3B=1, B= $ 3 1 y = ~ + ~ t = $ + i t
ss y"-25y = e5t;y" + y = sin t; y" = 1+ t; right side solves homogeneous equation so particular
solution needs extra factor t
35 et ,e-" ee", e-it 37 y = e-2t + 2te-"; y(27r) = (1+ 4 ~ ) e - ~ " r~ 0
39 y = (4e-" -r2e-4tlr)/(4 -r2) -+ 1as r -+ 0 43 h 5 2; h 5 2.8
Section 16.3 Discrete Mathematics
(page 615)
1Two then two then last one; go around hexagon 3 Six (each deletes one edge)
5 Connected: there is a path between any two nodes; connecting each new node requires an edge
1 3 Edge lengths 1,2,4
15No;1,3,4onleftconnectonlyto2,3onright;1,3onrightconnectonlyto2onleft 1 7 4
19Yes 2l F( mayl oop) ; T 2516
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Gilbert Strang
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