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MIT 18 01 - Answers to Odd-Numbered Problems

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A-0 Answers to 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 .25 25; 22; t + 10 7 6; -30 9 v(t) = { Ofor t > .2 1110%; l2$% 29 Slope -2; 15 f 5 9 31 v(t) = 8 for O<t<T 8t for 0 5 t T -2 for T<t<5T lt) = { lOT -2t for T5 t _( ST 47 %v;;V 49 input * input -+ A input * input -+ A B * B -+ C input +I+ A input +A --+ output input +A --+ B B + C --+ output A * A -+ B A + B --+ output 61 3t+ 5,3t + 1,6t -2,6t -1,-3t -1,9t -4; slopes 3,3,6,6,-3,9 Section 1.2 Calculus Without Limits (page 14) 12 + 5 + 3 = 10;f = 1,3,8,11;10 3 f = 3,4,6,7,7,6; max f at v = 0 or at break from v = 1to -1 5 1.1,-2,s; f (6) = 6.6, -11,4; f (7) = 7.7, -l3,9 7 f (t) = 2t for t 5 5,10 + 3(t -5) for t 2 5; f (10) = 25 9 7, 28, 8t + 4; multiply slopes 11f (8) = 8.8, -15,14; = 1.1,-2,5 13 f (z)= 3052.50 + .28(x -20,350); then 11,158.50 is f (49,300) 15 19+% 17 Credit subtracts 1,000, deduction only subtracts 15% of 1000 19 All vj = 2;vj = (-l)j-';vj = ($)j 21 L's have area 1,3,5,7 23 fj = j; sum j2+ j; sum + 25 (1012 -9g2)/2 = 7 27 Vj = 2j 29 f31 = 5 31 aj = -fj 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 41 vj = 2(-l)j, sum is fj -1 45 v = 1000,t = lO/V 47 M, N 51 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 5Velocityatt=lis3 7Areaf=t+t2,slopeoffis1+2t 9 F; F; F; T 112; 2t 13 12 + 10t2; 2 + lot2 15 Time 2, height 1,stays above from t = $ to 17 f(6) = 18 21 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) 1lor, (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) 13 Speed $ 15 Area 2 17 Area 0Answers to Odd-Numbered Problems A-1 19 4 from speed, 4 from angle 21 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=sinit 37~=cos~,~=sin~,speed~,u~,=cos~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 ir28 9 d = 1,distance around hexagon < distance around circle 11T; T; F; F 13cos(2t+t) = cos2tcost -sin2tsint = 4cos3t -3cost 15icos(s-t)+~cos(s+t);~cos(s-t)-icos(s+t) 17cos8=secB=~tlat8=nr 19Usecos(t-s-t)=cos(t-s)cost+sin(t-s)sint 238=~+rnultipleof2n 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 2x+Ax+ 1;2x+ 1 -4 11&d=&+3-137;9;corner 15A=1, B=-1 17F;F;T;F 19 b = B; mand M; mor undefined 21 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 2fi AX 36 2 f 5 = 4u32 Section 2.2 Powers and Polynomials (page 56) 15 3x2 -1= 0 at x = fi and A 17 8 ft/sec; -8 ft/sec; 0 19 Decreases for -1 < x <fiz+h)-x 23 1 5 10 10 5 1adds to (l+l)'(x = h= 1) 253x2;2hisdifferenceofx's 27% =2x+Ax+3x2+3xAx+(Ax)2 +2x+3x2=sumofseparatederivatives 1 4 1297~~;7(x+l)~ 31~x4pl~~anycubic 33x+~x2+$x3+fx4+C 35~x,120x6 37 F; F; F; T; T 39 = .12 so 4= i(.12); sixcents 41 4= 1C-* = -3AX AX + A Adz 43E=X 1 10. lXn+l.2x+3 45ttofit 47i5x ,n+l ,dividebyn+l=O Section 2.3 The Slope and the Tangent Line (page 63)A-2 Answers to Odd-Numbered Problems 17 (-3,19) and (8, E) 19 c = 4, y = 3 -x tangent at x = 1 21 (1+ h)3; 3h + 3h2 + h3; 3 + 3h + h2; 3 23 Tangents parallel, same normal 25 y = 2ax -a2,Q = (0, -a2) ; distance a2 + i; angle of incidence = angle of reflection fi' 27~=2p;focushasy=$=p 29y-&=x+L-x=-2-4-4 31 y -= -12a(x-a);y= a2+ $;a= $ 33 ($)(1000) = 10 at x= 10 hours 55 a= 2 4157 1.01004512; 1+ 10(.001) = 1.01 39 (2 + AX)^ -(8 +6Ax) = AX)' + AX)^ 41 xl = i;x2 = -40 43T=8sec;f(T)=96meters 45a>tmeters/sec2 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; 2s 7 cos2B w 1-8' + f B4; f B4 is small 9siniBmiB 11:;4 13PS=sinh;areaOPR=isinh<curvedareaih 15 cosx=l- d-+L-... 17 &(cos(x+ h) -cos(x -h)) = ;(-sinxsinh) -+ -sinx2.1 4.3.2.1 193/=cosx-sinx=Oatx=q+ns 2l(tanh)/h=sinh/hcosh<~-+l -1.2,2,no 25y=2cosx+sinx;y"=-y 27y=-~cos3x;y=~sin3x23Slope~cos~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-~~sin~+4xcosx+2sinx92x-1-~112~sinxcosx+~x-1/2sin2x+~(sinx)-1/2cos~ 134x3cosx-x4sinx+cos4x-4xcos3x sinx 15~~~~0sx+2x~inx17019-~(~-5)~~/~+~(5-~)-~/~(=0?) 21 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' sint (l+t)' A=~(&+~cos~+%) A'=2(~ost-tsi~t+'-~~~~ lint 29 lot 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; itan2 t; ![(I + t)3/2 -11 59T;F;F;T;F 41degree2n-l/degree2n 43v(t)=cost-tsint(t<$);v(t)=-:(t>:) 45 y = 9+ 9,ha 2 = 0 at x = 0 (no crash) and at x = -L (no dive). Then 2 = ?($ + 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; …


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MIT 18 01 - Answers to Odd-Numbered Problems

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