Math 149 01 02 Dr P D McCray Alternative QUIZ 2 Name Key Due 9 Febuary 2009 Page 1 of 2 EXACT answers only NO numerical approximations 1 Suppose that x and y are differentiable functions of t and that 1 C x D sin xy 2 for all t a 25 points Find dy by implicit differentiation dx d d 1 C x D sin xy 2 dx dx Stewart 3 6 12 1 D cos xy 2 x 2yy 0 C y 2 1 1 D 2xy cos xy 2 y 0 C y 2 cos xy 2 1 y 2 cos xy 2 D 2xy cos xy 2 y 0 1 y 2 cos xy 2 y D 2xy cos xy 2 0 b 25 points Find dx dy in terms of x y and dt dt dy dy dx D dt dx dt dy 1 y 2 cos xy 2 dx D dt 2xy cos xy 2 dt Math 149 01 02 Dr P D McCray Alternative QUIZ 2 Name Due 9 Febuary 2009 Key Page 2 of 2 2 50 points If a snowball melts so that its surface area decreases at a rate of 1 cm 2 min find the rate at which the diameter decreases when the diameter is 10 cm Stewart 3 8 12 Given the rate of decrease of the surface area is 1 cm 2 min If we let t be the time in minutes dS D 1 cm2 and S be the surface area in cm2 then we are given that dt a 5 points b 5 points Unknown the rate of decrease of the diameter when the diameter is 10 cm If we let x be the diameter then we want to find dx dt when x D 10 cm c 10 points d 5 points See separate diagram for a picture of the situation at any time t If the radius is r and the diameter is x then r D x 2 and S D 4 r 2 D 4 x 2 2 D x 2 dS dS dx dx D D 2 x dt dx dt dt e 25 points Finish solving the problem by noting that 1 D dx dS D 2 x dt dt When x D 10 H dx 1 D dt 2 x dx 1 D dt 20 so the rate of decrease is 1 20 cm min AMS LATEX
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