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FIRST HOUR EXAM, MATH 223, Section 3

Instructor: Kris Green, September 12, 1996

1.

Ture or False (2 points each):

(a)

The component of $\vec{v} = 2\hat{i} - 3\hat{j} + 2\hat{k}$ in the direction of $\vec{w} = \hat{i} + \hat{j} + \hat{k}$ is $\vec{w}$.

(b)

Two contours of z = f(x,y) corresponding to different values of z can never cross.

(c)

The two planes 3z - 2x + y = 5 and -6z + 4x - 2y = 0 are parallel.

(d)

If f(x,y) is a linear function then f(3,4) - f(3,5) = f(4,4) - f(4,5).

2.

The following are worth 6 points each.

(a)

Given the vectors in figure 1, draw the vector $2\vec{u} - \vec{v}$.

(b)

Given that $\vec{w} = 3\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{s} = -2\hat{i} + 3\hat{j} +2\hat{k}$ find:

i.

$\vec{w} \cdot \vec{s}$

ii.

$\vec{w} \times \vec{s}$

iii.

The angle between $\vec{w}$ and $\vec{s}$.

3.

(20 points) Consider the contour diagram of the linear function shown in figure 2.

(a)

Find the equation of the plane z = f(x,y) corresponding to this diagram.

(b)

Find a unit normal vector to this plane.

4.

(20 points) Consider the function z = f(x,y) = e^xy.

(a)

Draw a contour diagram and label the contours for z = 0.5, 1, 1.5.

(b)

What are the domain and range of f.

(c)

Is f(x,y) increasing or decreasing as a function of y at the point (1,0)?

5.

(20 points) Fox television uses a linear function to decide what to do with its programs. A show's score is a function of both its Neilson Rating, R, and its production cost per episode, C. If a show score is 1000 or above, then it is placed into a prime time slot. If its score is below 500, then it is cancelled and sold to UPN. Some examples are: (1) a show with R = 80 and $C < \$1$ million receives a prime time slot, (2) for any value of R, a show is kept on the air if its cost is below $50,000, and (3) a show with a Neilson rating of 80 that costs more than $2 million per episode will be cancelled. Note that the lowest Neilson rating is R = 0.

(a)

Use the three cases above (1)-(3) to help determine the linear function that Fox uses to determine the score of a show.

(b)

If the show Married with Pets costs $1.5 million per epsiode and receives a Neilson rating of 60, what will happen to it?

6.

(20 points) If a burner on a stove is turned off at time t=0, then the function describing the temperature at a point (x,y) on the burner at time t is z = e^-(x2+y2+t) where the point (0,0) corresponds to the center of the burner. Draw a contours of constant z (in x and y) at time t=0 and describe (in words) how the contour diagram wll change as t increases.