Calculus Solutions 25

Calculus Solutions 25 - A-24 Answers to Odd-Numbered...

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A-24 Answers to Odd-Numbered Problems 35 f = i J(x - 112 + (y - 112; (3, = (3 ~JZ) 2Ji' 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,, # 41 v = (1,2t); T = v/&SF; % = 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 = (;, $, $) = (~-',e-~',e'),% = 1+2- 0 1, = -2- Ivl 47 v = (-2 sin 2t, 2 cos 2t), T = (- sin 2t, cos 2t); grad f = (y,x), 2 = -2 sin2 2t + 2cos2 2t, % = is; zero slope because f = 1on this path 492-1=2(x-4)+3(y-5); f = l+2(x-4)+3(y-5) 51 grad f .T=O;T Section 13.5 The Chain Rule (page 503) 1f, = cfx = c COS(X + ey) 3 f, = 7fx = 7ex+7' 5 3g2*& ax dt + 32% 2 7 Moves left at speed 2 9 2 = 1 (wave moves at speed 1) 11 sf (x + iy) = ftt(x + iy), -@-f(x+iy) = i2ftt(x+iy) so fix + = 0; (x + ~y)~ = (x2 - #) + i(2xy) 13%=2~(1)+2~(2t)=2t+4t~ 15$=y$+x$=-1 17*=ld.+1*=1 x+ydt s+ydt 19 V = STr2h dV 27rrh dr 7rr2 dh = 3GT dt=-- 3 dt+~dt 90 90a+90a 903+90 = Ji m~h; dD 21 % = dz(60) + d7(45) 7T = d- 60 (60) + J- (45) cl 74 mph 23 $=UI%+U~%+U~% 25 g=lwithxandyfixed; %=6 27 ft = fxt + f,W ftt = fxtt + fx + 2fytt + 2f, = (fxxt + fYX(2t))t + + 2(fx,t + f,,(2t))t + 2f, 29
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