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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?
a. |
b. |
c. |
d. z = x2 + 2xy + y2 |
- (b)
- If , which of the following are possible
values for the directional derivative of f in the direction making an
angle with the x-axis at the point (2,1)?
a. I only |
b. II only |
c. II and III |
d. I and III |
- (c)
- Suppose at some point a function f has . The rate of maximum decrease of f at this point is in the
direction of the vector:
a. |
b. |
c. |
d. along a vector perpendicular to |
- (d)
- At the point (-1,2) on the graph of z = f(x,y) we know that , fxx = -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 .
- (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 . - 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) = 2xey. 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.
Vector Calculus
8/20/1998