Math 307: Multivariable Calculus
SAMPLE FINAL EXAM
[Note: For the actual open-book Exam as with this one, it wil l be assumed
that you will have your book with you together with a functioning calculator.
Please note that no calculators or books other than your own will be provided
for the test. The questions below are intended to give some idea about the
type and number of questions on the Final Exam. To be fully prepared,
additional practice with the examples and exercises in the book is necessary.]
Instructions. Write clean and for full credit, provide details and computa-
tions, as well as correct notation and diagram labels.
1. Find all the second partial derivatives of the function
f(x, y, z)=3zcos(x2−y).
2. Determine the equation of the tangent plane to the surface xy+yz+zx =3
at the point (1,2,1/3).Simplify your answer as much as possible.
3.Ifz= cos(xy)−xcos ywhere x=u−v2+ 3 and y=u2+v−1,use the
chain rule to compute the partial derivatives ∂z/∂u and ∂z/∂v.
4. Use the method of Lagrange multipliers to compute the maximum and
minimum values of the function f(x, y, z)=4x−2y−2z+ 3 sub ject to the
constraint equation x2+y2+z2=6.
5. Calculate the following iterated integral by properly reversing the order
of integration:
Z1
0Z1
y2
ycos(x2)dxdy
6. (a) Compute the following iterated integral by first converting it to polar
coordinates (ais an unspecified positive constant):
I=Za
0Z√a2
−y2
−√a2
−y2
2
p4+x2+y2dxdy.
(b) Determine a positive value of aso that I=πa.
7. A lamina occupies the region Dbounded by the parabola y=1−x2
and the x-axis. If the mass density ρ(x, y) of the lamina at each point is
1
