No Title

Math 223 Name
Fall 1998
Test 1

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.

(5 points) Consider the function z = f(x,y) given by the table below. Determine whether or not the function is linear and explain your reasoning.

x $\setminus$ y	-2	0	2
-2	4	2	1
0	3	1	0
2	2	0	-1

2.

(10 points) Consider the surface given by z = f(x,y) = 2x - y². Make a contour plot showing contours for z = -1,0,1,2.

3.

(20 points) Consider the plane given by z = 1 + x - 2y and the point Q = (1,2,3). In this problem you will compute the shortest distance from the point Q to the plane.

(a): Find any point, P, on the plane and write the displacement vector $\vec{PQ}$ in components.
(b): Find a unit normal vector to the plane.
(c): Find the component of $\vec{PQ}$ parallel to the unit normal vector you found above, i.e. find $\vec{(PQ)}_{parallel}$ .
(d): Find the length of $\vec{(PQ)}_{parallel}$ .

4.

(20 points) You are given vectors $\vec{v} = \vec{i} + \vec{j} + 3 \vec{k}$ and $\vec{w} = 2 \vec{i} - \vec{j} + 2 \vec{k}$ . Let $\theta$ be the smallest angle between $\vec{v}$ and $\vec{w}$ .

(a): Use the definitions of the dot product to find $\theta$ .
(b): Use the definitions of the cross product to find $\theta$ .
Note: You should have discovered in part (a) that $0 \leq \theta \leq \frac{\pi}{2}$ .

5.

(20 points) Find an equation of the plane containing the points
$(1,2,1) \mbox{, } (2,4,2)$ and (-1,-2,5).

6.

(15 points)

=3.5ingraph10.ps

Consider the surface z = f(x,y) above.

(a): Sketch a cross-section of the surface with x = 1.5.
(b): Is f_x(1,1) positive or negative? Explain your reasoning.
(c): Is f_x(1,1) greater than or less than f_y(1.5,1.5)? Explain your reasoning.

7.

(10 points) Consider $f(x,y) = x^{2} - \cos{(2y)} + e^{xy}$ .

(a): Find $\frac{\partial f}{\partial x}$ .
(b): Find $\frac{\partial f}{\partial y}\vert _{(1,2)}$ .

Vector Calculus
10/13/1998