Math 20C Final Exam 2006
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Math 20C.
Final Exam
2006
1. (10 points) Find the plane through the point P0 = (2, 1, -1) which is perpendicular to the planes 2x + y - z = 3 and x + 2y + z = 2.
2. (8 points) Decide whether the lim exists. Give reasons your answer.
3. (8 points) Does the function f (x, y , z) = e3x+4y cos(5z) satisfy the Laplace equation fxx + fyy + fzz = 0? Give reasons your answer.
4. (10 points) Find the linear approximation L(x, y) of the function f (x, y) = 尸6 - x2 - y2 at the point (1, 1). Use this approximation to estimate the value of f (0.8, 1.1).
5. (10 points) Find the local maxima, local minima and saddle points of the function f (x, y) = x3 + y3 + 3x2 - 3y2 - 8.
6. (10 points) Use Lagrange multipliers to find the maximum and minimum values of the function f (x, y) = -1
x + 1
y subject to the constraint 1
x2 + 1
y2 = 1.
|
7. Consider the integral |
f (x, y) dA = D |
3
0 |
2(1_ a f (x, y) dy dx.
_2 √ 1_ |
(a) (8 points) Sketch the region of integration.
(b) (8 points) Switch the order of integration in the above integral.
(c) (8 points) Compute the integral f (x, y) dA for the case f (x, y) = xy .
D
8. (10 points) Transform to polar coordinates and then evaluate the integral I 
╱x2 + y2、3/2 dx dy .
9. (10 points) Find the volume of a parallelepiped whose base is a rectangle in the z = 0 plane given by 0 ≤ y ≤ 1 and 0 ≤ x ≤ 2, while the top side lies in the plane x+y+z = 3.
z
3
3 y
2022-03-14