Math 2030, Winter 2011 Sample Problems for Test 2

10 downloads 577 Views 68KB Size Report
Math 2030, Winter 2011. Sample Problems for Test 2. R. Bruner. March 1, 2011. 1. Find all first and second partial derivatives of sin(x) cos(y). 2. Find ∂z/∂x and  ...
Math 2030, Winter 2011 Sample Problems for Test 2 R. Bruner March 1, 2011 1. Find all first and second partial derivatives of sin(x) cos(y). 2. Find ∂z/∂x and ∂z/∂y if sin(x) + cos(y) + sin(z) = 1. 3. Find the tangent plane to z = x2 /y at (x, y) = (2, 3). 4. Find the direction of maximum increase of f (x, y, z) = xy + xz + yz at (x, y, z) = − (1, 2, 3). Find a direction → u in which D− → u f = 0 at (1, 2, 3). 5. Find the tangent plane to xy + xz + yz = 11 at (x, y, z) = (1, 2, 3). 6. Suppose that f (x, y) = x2 /y, ∂x/∂s = 3, ∂x/∂t = 1, ∂y/∂s = 2, and ∂y/∂t = 4. Find ∂f /∂s and ∂f /∂t. 7. The height, length and width of a rectangular box have been measured with relative error at most .01. What is the relative error in our knowledge of the volume? (Use the differential to estimate this.) 8. Find all first and second partial derivatives of x2 sin(y). 9. Find ∂z/∂x and ∂z/∂y if xy + yz + z 2 = 3. 10. Find the tangent plane to z = x2 y + xy 3 at (x, y) = (2, 3). 11. Find the tangent plane to x2 y + y 2 z + z 2 x = 23 at (x, y, z) = (1, 2, 3). 12. Find the direction of maximum decrease of f (x, y, z) = x2 y + y 2 z + z 2 x at (x, y, z) = − (1, 2, 3). Find a direction → u in which D− → u f = 0 at (1, 2, 3). 13. Suppose that f (x, y) = x2 y 3 , ∂x/∂s = 3, ∂x/∂t = 1, ∂y/∂s = 2, and ∂y/∂t = 4. Find ∂f /∂s and ∂f /∂t. 14. Find and classify the critical points of x3 − 3xy + 21 y 2 . 15. Find the absolute maximum and minimum values of x + y 2 on the disk x2 + y 2 ≤ 4.

———— The End ———–