105-Mid2Prac1 - ( 1 2 , 1 4 ,f ( 1 2 , 1 4 )) on the hill....

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Math 105 Practice Midterm 1 for Midterm 2 This practice midterm may be harder and/or longer than the real midterm. Not all question will be worth the same number of points. 1. Evaluate Z 0 e - x x dx , or show that it doesn’t exist. 2. Solve the initial value problem y 0 = 1 xy , y (1) = 4. 3. Find an equation for the plane that is parallel to x - 2 y + 6 z = 1 and contains the point (4 , 0 , 2). 4. Sketch the level curves of z = y 2 - 1 4 x 2 at the heights z = - 1 , 0 , 1. 5. Evaluate the limit lim ( x,y ) (0 , 0) 5 x - 2 y 2 x + 2 y 2 , or show that it doesn’t exist. 6. Consider the hill given by the function z = f ( x,y ) = p 1 - x 2 - 4 y 2 . (a) Compute f x and f y . (b) Find the unit vector that gives the direction of steepest ascent at the point
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Unformatted text preview: ( 1 2 , 1 4 ,f ( 1 2 , 1 4 )) on the hill. Also nd a unit vector that gives the direction of no change at that point. (c) Suppose youre walking over the hill along the path that is right above the path ( x ( t ) ,y ( t )) = ( t,t 2 ) in the xy-plane. As you pass the point ( 1 2 , 1 4 ,f ( 1 2 , 1 4 )) , at what rate is your height changing? 7. Find the critical points of f ( x,y ) = 1 2 x 2 + 4 xy + y 3 + 8 y 2 + 3 x + 2, and classify each one as a maximum, minimum or saddle point....
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This note was uploaded on 01/14/2012 for the course MATH 105 taught by Professor Malabikapramanik during the Fall '10 term at The University of British Columbia.

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