calc exam2_review - MATH 2300 Calculus III Fall 2010...

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MATH 2300 - Calculus III Exam 2 Review Fall 2010 NAME: ................................................................... Student ID: ......................................... In order to receive full credit, SHOW ALL YOUR WORK. 1. (6 pts) Indicate whether the following statements are true (T) or false (F). (Circle one) T F (a) integraldisplay 1 0 integraldisplay 1 x f ( x, y ) dydx = integraldisplay 1 x integraldisplay 1 0 f ( x, y ) dxdy . T F (b) The maximum rate of change of z = f ( x, y ) is given by |∇ f | . T F (c) integraldisplay 2 1 integraldisplay 4 3 xydydx = x integraldisplay 2 1 integraldisplay 4 3 ydydx . 2. (12 points) Find the local maxima, minima, and saddle points of z = f ( x, y ) = xy 2 + x 2 2 xy , using the Second Derivatives Test. 1
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3. (10 points) If w = xy + yz + xz , x = s cos t , y = cos( st ) and z = t , use the multivariate chain rule to find ∂w ∂s when s = 1 and t = π . 4. (10 points) Sketch the region of integration and rewrite the double integral with the order of integration reversed: integraldisplay 4 0 integraldisplay x x/ 2 f ( x, y ) dy dx 5. (12 points) A hiker is walking on a mountain path when it begins to rain. If the surface of the mountain is modeled by z = 1 3 x 2 5 y 2 , (where x , y and z are in miles) and the rain begins when the hiker is at the
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