Math 149 – 01 – 02 AlternativeQUIZ 2 Name: KeyDr. P. D. McCray Due 9 Febuary 2009 Page1of2EXACT answers only: NO numerical approximations!1. Suppose that x and y are differentiable functions of t and that 1 C x D sin.xy2/ for all t.(a) (25 points) Finddydxby implicit differentiation. (Stewart, 3.6 # 12)ddx.1 C x/ Dddxsin.xy2/) 1 D Œcos.xy2/.x 2yy0C y2 1/ )1 D 2xy cos.xy2/y0C y2cos.xy2/ ) 1 y2cos.xy2/ D 2xy cos.xy2/y0)y0D1 y2cos.xy2/2xy cos.xy2/(b) (25 points) Finddydtin terms of x, y; anddxdt.dydtDdydxdxdt)dydtD1 y2cos.xy2/2xy cos.xy2/dxdtMath 149 – 01 – 02 AlternativeQUIZ 2 Name: KeyDr. P. D. McCray Due 9 Febuary 2009 Page2of22. (50 points) If a snowball melts so that its surface area decreases at a rate of 1 cm2/min, find the rate at whichthe diameter decreases when the diameter is 10 cm. (Stewart, 3.8 # 12)(a) (5 points) Given: the rate of decrease of the surface area is 1 cm2/min. If we let t be the time (in minutes)and S be the surface area ( in cm2), then we are given thatdSdtD1 cm2.(b) (5 points) Unknown: the rate of decrease of the diameter when the diameter is 10 cm. If we let x be thediameter, then we want to find dx=dt when x D 10 cm.(c) (10 points) See separate diagram for a picture of the situation at any time t.(d) (5 points) If the radius is r and the diameter is x, then r D x=2 andS D 4r2D 4.x=2/2D x2)dSdtDdSdxdxdtD 2xdxdt.(e) (25 points) Finish solving the problem by noting that1 DdSdtD 2xdxdtH)dxdtD12xWhen x D 10dxdtD120;so the rate of decrease is 1 =20
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