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

1.

Multiple Choice (5 points each):

(a)

Whish of the following functions could not have a single global maximum?

$$z = \ln (xy) f(x,y) = \cos (x^2 + y^2)$$

$$g(x,y) = \exp (-x^2 - y^2)$$ d. z = x² + 2xy + y²

(b)

If $\vert\vert\nabla f(2,1) \vert\vert = 3$, which of the following are possible values for the directional derivative of f in the direction making an angle $\theta = \pi/4$ with the x-axis at the point (2,1)?

I. 0 II. 4 III. -2

a. I only b. II only c. II and III d. I and III

(c)

Suppose at some point a function f has $\nabla f = 2\hat{i} - 3\hat{j}$. The rate of maximum decrease of f at this point is in the direction of the vector:

$$3\hat{i}+ 2\hat{j} -2\hat{i}+3\hat{j}$$

$$4\hat{i} - 6\hat{j} 2\hat{i} - 3\hat{j}$$

(d)

At the point (-1,2) on the graph of z = f(x,y) we know that $\nabla f = 3\hat{i} + \hat{j}$, f_xx = -2 and the discrimminant, D = 1. Thus the point (-1,2) is

a. a local maximum b. a local minimum

c. a saddle point d. not a critical point

2.

Consider the function $f(x,y) = \ln (2x + 3y)$.

(a)

Find the first partial derivatives of f.

(b)

Find all the second partial derivatives of f.

(c)

Find the normal vector to the surface z = f(x,y) at the point (-1,1,0).

3.

Find all the global maxima and minima of the function

g(x,y) = x² - 8x + 2y² + 19

on the disk $x^2 + y^2 \le 25$.

4.

We want to design a rectangular milk carton box of width w, height h and length l that will hold 512 cc of milk. (cc = cubic centimeter)

(a)

Write an expression for the volume in terms of w, l, h. Solve this expression for h.

(b)

Now suppose that it costs 1 cent/square cm for the material on the sides of the box and 2 cents/square cm for the material forming the tops and bottom. Let C(w,l,h) represent the total cost of the box. Find an expression for C. Drawing a picture may help.

(c)

Substitute the expression for h from part (a) into C. Now set up, but do not solve the algebraic equations to find the dimensions of the box that minimize the total cost of the box.

5.

Let f(x,y) = 2xe^y. Use the chain rule and the graphs provided for x(t) and y(t) to determine whether the sign of df/dt is positive, negative, or zero at t = 1. Write your answer in full sentences with a complete description of the steps you take to solve the problem.