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SECOND HOUR EXAM, MATH 223
Instructor: Kris Green, March 11, 1997

1.

Multiple Choice (5 points each):

(a)

At the point (-1,2) on the graph of f(x,y) the discriminant, D = 3, f_xx = 2 and grad $f(-1,2) = 5\hat{i} - 2\hat{j}$. The point (-1,2) is

a. a local maximum b. a local minimum

c. a saddle point d. none of these

(b)

A student answers that the directional derivative of f(x,y) in the direction of $\vec{u}$ is $4\hat{i} - \hat{j}$. This answer is incorrect because:

$$f_{\vec{u}} f_{\vec{u}}$$

c. neither (a) nor (b) is the correct reason d. there is not enough information

(c)

Suppose that f(1,-1) = 5, f_x (1,-1) = 2, f_y(1,-1) = -1. The equation of the plane that is tangent to f at (1,-1) is

a. z = -(x-1) + 2(y+1) + 5 b. z = 2(x+1) - (y+1) + 5

c. z = 2(x-1) - (y+1) + 5 d. z = -(x+1) + 2(y-1) + 5

(d)

The maximum rate of change of z = f(x,y) at the point (a,b) is

$$\vert\vert \nabla f(a,b) \vert\vert$$ a. grad f(a,b)

$$\nabla f(a,b) \cdot \vec{u}$$ c. | f_x(a,b) + f_y(a,b)|

2.

Short answer. One or two sentences per answer are sufficient.

(a)

Consider the function z = f(x,y). Construct a new function g(x,y,z) so that the level surface g = 0 corresponds to the surface z = f(x,y).

(b)

Relate the gradient of g to the gradient of f using the function g constructed in the previous part.

(c)

Using local linearity, explain the relationship between grad g and the surface z = f(x,y). Note that this is a geometric version of the previous part.

3.

For the function $f(x,y) = \cos (xy)$ calculate the following:

(a)

Find all first and second partial derivatives of f.

(b)

Find the gradient of f.

(c)

Find the directional derivative of f at the point $(\sqrt{\pi}, \sqrt{\pi}/2)$ in the direction of the vector $\vec{v} = 3\hat{i} - 4\hat{j}$.

4.

Find and classify all th critical points of the function g(x,y) = x³ + y³ +3xy.

5.

Once more, to the detriment of all vector calculus students, Fox television has decided to consult the University of Arizona. This time, they have decided that the rating of a TV show is a function of the number of scences containing violence, V, and the number of sexually explicit scenes, S according to the function:

R(S,V) = 19 S^(1/12) V^(1/2).

Fox would like us to decide the optimum number of each type of scence to include in a show. However, each sexual scene costs $1000 to produce and each violent scene costs $3000. If the total budget of the show is $28,000, answer the following using complete sentences.

(a)

Set up and explain the system of equations needed to solve the constrained optimization problem described above.

(b)

Solve the system in the previous part.

(c)

Find and interpret the Lagrange multiplier for this problem.