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Unformatted text preview: MATH 215 – Winter 2005 FINAL EXAM – Solutions Problem 1. (5+5=10 points) In this problem f ( x, y ) = y 2 x x 2 + 2 xy and P is the point P = (2 , 1). (a) In what direction is the rate of change of f greatest at P ? Express your answer in terms of a unit vector. Solution. The gradient of f at the point P is the vector h 1 , 8 i . Its direction is the direction of greatest increase, so the direction expressed as a unit vector is 1 h 1 , 8 i h 1 , 8 i = 1 √ 65 h 1 , 8 i . (b) Suppose ~ r ( t ) = h x ( t ) , y ( t ) i is a parametric curve such that ~ r (0) = h 2 , 1 i , and d dt ~ r (0) = h 3 , 5 i . Find the value of d dt f ( x ( t ) , y ( t ))  t =0 . Solution. By the chain rule, d dt f ( x ( t ) , y ( t )) = dx dt ∂f ∂x + dy dt ∂f ∂y = d~ r dt · ∇ f. At t = 0 this is the dot product h 3 , 5 i · h 1 , 8 i = 37 Problem 2. (8+7=15 points) Suppose that, in an experiment, the temperature of a sample (in degrees Celcius) is given by the function T ( x, y, z ) = 2 y 2 + ze x + 16, where x , y and z are variables one can control in the experiment. (a) Using the linear approximation of the function T at the point ( x , y , z ) = (0 , 1 , 2), find an approximate value of T (0 . 2 , . 9 , 2 . 3). Note that T (0 , 1 , 2) = 20 ◦ , Solution. One computes that, at the point (0 , 1 , 2), the partial derivatives of T are: T x = 2, T y = 4, T z = 1. Therefore, the linearization at that point is L ( x, y, z ) = 2( x 0) + 4( y 1) + 1( z 2) + 20 . Finally, one computes L (0 . 2 , . 9 , 2 . 3) = 19 . 5. (b) Suppose one wants to change ( x , y , z ) a little and yet maintain the temperature at 20 ◦ . Using the linear approximation, find an equation between Δ x , Δ y and Δ z so that T (Δ x, 1 + Δ y, 2 + Δ z ) ∼ = 20 . Solution. By what we just said, L (Δ x, 1 + Δ y, 2 + Δ z ) = 2Δ x + 4Δ y + Δ z + 20 . Therefore, the equation is simply 2Δ x + 4Δ y + Δ z = 0. 1 Problem 3. (4 × 5 = 20 points) For each item, circle the correct answer or indicate if the statement is true or false. Assume that the functions, fields and curves below are smooth. Think carefully before you answer – no partial credit on this one, take your time! (a) Let C be an arc from (0 , 0) to (2 , 1). According to the fundamental theorem for line integrals, R C ( y 1) dx + ( x + 2 y ) dy is equal to: (1) 2 (2) 1 (3) It depends on what C is....
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 Winter '08
 Fish
 Derivative, Rate Of Change, Correct Answer, Manifold, Stokes' theorem

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