final-sample - Math 233 Calculus 3 Fall 2015 Sample Final(1...

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Math 233 Calculus 3 Fall 2015 Sample Final (1) Find the line of intersection between the planes z = 2 x - 4 y +2 and x - y - 2 z = 4. (2) Sketch the level sets of the function f ( x, y, z ) = x 2 - y 2 - z 2 , and calculate the gradient vector at the point (2 , 1 , 1). Use this to find the tangent plane to x 2 = y 2 + z 2 + 2. (3) You are driving anticlockwise around a circular roundabout of radius 10m, at 10m / s. When your car is facing due north, you throw a tennis ball from the car due east at 20m/s, at an angle of π/ 3 from horizontal. Where does the tennis ball land? (4) Let f ( x, y ) = x 2 + 2 y 2 - 2 y + 4. (a) Find the critical points of f in the region x 2 + y 2 < 4, and use the scond derivative test to classify them. (b) Use Lagrange multipliers to find the extreme points on the boundary x 2 + y 2 = 4. (c) Use your answers above to find the extreme values of f on x 2 + y 2 9. (5) Change the order of integration to evaluate Z 8 0 Z 2 3 y cos( x 4 ) dx dy . (6) Write down triple integrals over the following regions. (a) The region inside the sphere x 2 + y 2 + x 2
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