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Math 223 Name
Fall 1998
Test 2

Directions: Please read each question carefully and show all your work in doing each question. No partial credit will be given if you do not show your work. If you have any questions about the test, please ask me. Your work should of course be your own. Good Luck!

1.
(31 points) Consider the surface given by z = f(x,y) = x e(x+y-4).
(a)
Find \( \nabla f \).
(b)
Find the directional derivative of f at the point P = (3,1) in the direction of \( \vec{v} = \vec{i} + 2 \vec{j} \).

(c)
Give a vector in the direction of the most negative directional derivative at the point
P = (3,1).

(d)
Give a vector in the direction of zero directional derivative at the point P = (3,1).

(e)
Consider the points P = (3,1) and Q = (5,-1). Would you expect the contours of f to be closer together at P than they are at Q? Why or why not?

2.
(15 points)

=3ingraph7.ps

(a)
Consider the graph of the surface z = f(x,y) shown above. In the space provided, draw the coordinate axes for the xy-plane and then draw a vector in the direction of the gradient of f(,x,y) at the point P = (0.5,0.5).
(b)
Is the magnitude of the gradient larger at P = (0.5,0.5) or Q = (1,1)? Explain your reasoning.

(c)
Would you expect \( \displaystyle \frac{\partial^{2} z}{\partial y \partial x} \) to be positive or negative at R = (1,0.5)? Explain your reasoning.

3.
(20 points) Compute all second order partial derivatives of the function
z = f(x,y) = e(x<<53>>2-y).

4.
(24 points) Evaluate the following definite integrals.
Hint: The integral may not be easy to evaluate in the form given.

(a)
\( \displaystyle \int_{0}^{1} \int_{x}^{1} e^{(y^{2})} dy dx \).
(b)
\( \displaystyle \int_{0}^{1} \int_{-\sqrt{4-y^{2}}}^{0} dxdy \).

5.
(10 points) Write, but do not evaluate, a triple integral in spherical coordinates that gives the volume between a sphere of radius 1 and a sphere of radius 4.


Vector Calculus
10/13/1998