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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, fxx = 2 and grad . 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 is . This answer is incorrect
because:
a. should be a scalar |
b. should be a unit
vector |
c. neither (a) nor (b) is the correct reason |
d. there is not enough
information |
- (c)
- Suppose that f(1,-1) = 5, fx (1,-1) = 2, fy(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
a. grad f(a,b) |
b. |
c. | fx(a,b) + fy(a,b)| |
d. |
- 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 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 in the direction of the vector .
- 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.
Vector Calculus
8/20/1998