3c-spring2011-exam_2_sample

3c-spring2011-exam_2_sample - Math 3C Sample Exam #2 Laney...

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Unformatted text preview: Math 3C Sample Exam #2 Laney College, Spring 2011 Fred Bourgoin 1. Find f x (1 , 2) and f y (1 , 2) if f ( x, y ) = x 3 + 3 x 2 y- 2 y 2 . 2. Find the differential of f ( x, y ) = radicalbig x 2 + y 3 at the point (1 , 2). Then use it to estimate f (1 . 04 , 1 . 98). 3. Compute the gradient of f ( r, h ) = 2 rh + r 2 at the point (2 , 3). 4. Optimize f ( x, y ) = 2 x 2 + y 2 + 2 subject to the constraint x 2 + 4 y 2- 4 = 0. 5. Find a vector normal to the surface x 2- xyz = 3 at the point (- 1 , 1 , 2). 6. Find the critical points of f ( x, y ) = x 4 + 2 y 2- 4 xy , and determine whether each is a saddle point, a local minimum, or a local maximum. 7. Let f ( x, y ) = 3 xy + y 2 . (a) What is the rate of change of f at the point (2 , 3) in the direction vectorv = 3 vector i- vector j ? (b) What is the direction of maximum rate of change of f at (2 , 3)? (c) What is the maximum rate of change? 8. Use the given contour diagram of z = f ( x, y ) to decide the sign (positive, negative, or zero) of each of the following: f x ( P ), f y ( P ), f xx ( P ), f yy ( P ), and f xy ( P ). 1 2 3 4 5 P 9. Find z u and z v if z = xe y , where x = u 2 + v 2 and y = u 2- v 2 . 10. Find the extrema of f ( x, y ) = 2 x 2 +3 y 2- 4 x- 5 subject to the constraint x 2 + y 2 16. 1 Math 3C Sample Exam #2 Solutions Laney College, Fall 2010 Fred Bourgoin 1. Find f x (1 , 2) and f y (1 , 2) if f ( x, y ) = x 3 + 3 x 2 y- 2 y 2 ....
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This note was uploaded on 04/19/2011 for the course MTH 110 taught by Professor Helenius during the Spring '08 term at Grand Valley State University.

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3c-spring2011-exam_2_sample - Math 3C Sample Exam #2 Laney...

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