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Unformatted text preview: Math 3C — Exam #1 Sample Laney College, Spring 2011 Fred Bourgoin 1. (14 points) Consider the function f ( x,y ) = x 2 + 3 y 2 + 1. (a) Sketch a contour diagram for f with at least 3 level curves. (b) Sketch the graph of f . (c) Find a function g ( x,y,z ) such that the graph of f is a level surface of g . (Don’t forget to indicate which level surface of g it is.) 2. (12 points) Consider the points P = (3 , 2 , − 5) and Q = ( − 1 , 3 , − 2) in R 3 . (a) Write the vector −−→ PQ in terms of vector i , vector j , and vector k . (b) Find the magnitude of −−→ QP . (c) What is the distance between P and Q ? 3. (10 points) Suppose P and Q are as in the preceding problem, and let R = (2 , 6 , 4). (a) What is the area of the triangle △ PQR ? (b) Find an equation for the plane through P , Q , and R . 4. (8 points) Find the angle between the planes z = 2 x − y + 3 and x − 3 y + 2 z = 5. 5. (8 points) Sketch the domain of f ( x,y ) = radicalbig 1 + x − y 2 . 6. (10 points) Shortly after takeoff, a plane is climbing northwest through still air at an airspeed of 200 km/hr, and rising at a rate of 300 m/min. Resolve its velocity vector into components. The xaxis points east, the yaxis points north, and the zaxis points up. 7. (10 points) The following table gives values of a function h ( x,y ) at 20 points. y − 3 − 2 − 1 1 − 3 − 1 1 3 3 − 4 − 2 2 x 5 − 5 − 3 − 1 1 7 − 6 − 4 − 2 9 − 7 − 5 − 3 − 1 (a) Could h be a linear function? Why, or why not? (b) Find a possible expression for h . 1 8. (6 points) Describe the level surfaces of f ( x,y,z ) = 1 radicalbig x 2 + 2 y 2 + z 2 . 9. (16 points) Let vectoru = 2 vector i + vector j − 3 vector k , vectorv = vector i − vector j + 2 vector k , and vectorw = vector i − 5 vector j − vector k . (a) Compute 2 vectoru − 3 vectorv . (b) Are two of the vectors perpendicular? Which ones? Justify. (c) Find vectors vectorw 1 and vectorw 2 such that vectorw = vectorw 1 + vectorw 2 , vectorw 1 is parallel to vectorv , and vectorw 2 is perpendicular to vectorv . 10. (6 points) Find the volume of the parallelepiped defined by the vectors vectoru = 2 vector i + vector j − 3 vector k , vectorv = vector i − vector j + 2 vector k , and vectorw = vector i − 5 vector j − vector k of the preceding problem....
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 Spring '08
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 Math, Vector Motors, Laney College, Fred Bourgoin

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