Return to Other Sample Exams.
Math 223 Name
Fall 1997
Test 3

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.
(20 points) Let f(x,y) = x2 + xy.
(a)
Find all local minima, local maxima, and saddle points of f(x,y).
(b)
Find the maximum and minimum values, if they exist, of f(x,y) on xy + x = 1.

2.
(10 points) Let z = f(x,y) be given by the table below.

x \( \setminus \) y 2 2.5 3 3.5 4
2 5.41 5.58 5.73 5.87 6
2.5 7.66 7.83 7.98 8.12 8.25
3 10.41 10.58 10.73 10.87 11
3.5 13.66 13.83 13.98 14.12 14.25
4 17.41 17.58 17.73 17.87 18

Give reasonable upper and a lower bounds for \( \displaystyle \int_{R} f \mbox{ } dA \) where R is the region given by \( 3 \leq x \leq 4 \) and \( 2.5 \leq y \leq 3.5 \).

3.
(15 points) Use a triple integral to find the volume of the region above the xy-plane
(i.e. \( z \geq 0 \) ) bounded by x = 0, x = 3, and z = 4 - y2.

4.
(15 points) Evaluate \( \displaystyle \int_{0}^{1} \int_{y}^{1} \sqrt{(x^{2}+3)} \mbox{ } dx dy \).

5.
(10 points) Consider the integral \( \displaystyle \int_{W} x \mbox{ } dV \) where W is the region above the xy-plane between the spheres \( \displaystyle x^{2} + y^{2} + z^{2} = 1 \) and \( \displaystyle x^{2} + y^{2} + z^{2} = 4 \). Write this integral as a triple integral in spherical coordinates, but do not evaluate it.

6.
(20 points)

=3ingraph12.ps

(a)
Consider the curve above. Determine which of the choices below is a parameterization of the curve.
i.
\( x = \sin{t} \), \( y = \cos{t} \), \( \frac{\pi}{2} \leq t \leq \frac{3 \pi}{2} \).
ii.
\( x = \cos{t} \), \( y = \sin{t} \), \( \frac{\pi}{2} \leq t \leq \frac{3 \pi}{2} \).
iii.
\( x = \cos{t} \), \( y = \sin{t} \), \( \frac{- \pi}{2} \leq t \leq \frac{\pi}{2} \).
iv.
\( x = \sin{t} \), \( y = \cos{t} \), \( \frac{- \pi}{2} \leq t \leq \frac{\pi}{2} \).

(b)
Suppose the parameterization you chose above describes the motion of a particle. Find the velocity of the particle at time t = 0.

7.
(10 points) Describe in words the object parameterized by the equations \( x = 2z \cos{\theta} \), \( y = 2z \sin{\theta} \), z = z, \( 0 \leq z \leq 7 \), \( 0 \leq \theta \leq 2 \pi \).

Vector Calculus
8/21/1998