Math203, Demo Exam #2 (Chapters 15-17)
Directions: Answer all of the questions. The problems will have their weights assigned next
to them. Show all work and reasoning for maximum credit. No notes, cellular phones,
pagers, calculators, books or friends. Good luck.
1) Definitions.
a. Describe how you would find all of the potential maxima and minima for f(x,y).
How could you distinguish between a maximum, minimum, or a saddle?
b. Describe the differences between the rectangular, cylindrical, and spherical
coordinate systems. When are times when a particular coordinate system is a better
choice than the other two. Provide an example of each.
c. Describe what a Jacobian is. Then use it to derive the
sin
2
Jacobian in the
spherical system.
d. Define a gradient field. Describe how you would draw one. Describe what it
means. Provide an example.
e. Provide a sample parameterization of motion in 3 dimensional space. Also
describe how you would find the velocity and total distance from this.
2) Maximize Revenue given the following: R(x,y)=x(1000-0.1x2)+y(200-0.02y) subject to
x>0, y>0, and 8x+3y<2400.
3) For each of the figures below, create an integral to describe the region. Choose the
appropriate coordinate system and describe why you chose it.
4) Integrate the integrals that you found in the problem above. Use the tables in the back
of the book if necessary.
5) Map the following surface area onto 2 dimensions: a 4-sided pyramid, but without the
base. Assume that the base is units b wide and b long. And the pyramid is of height h.