MATH 260
FINAL EXAM
1.
(8pts) Consider the planes given by the equations 
and 
. Find the equation of the line of intersection of these two
planes.
2.
(18pts) Consider the points 
and 
a)
Find the vectors a=
and b=
.
b)
Compute a
b and a
b.
c)
What is the angle between a
and b?
d) Compute the vector projection of a on b.
e) Find the all lines determined by these three points.
f) Do these three point determine a plane?
3.
(15pts) Consider the function of two variables 
.
a) Find all critical points and classify them as local max, local min, or saddle points.
b) Find the absolute maximum and minimum of this function on the triangle with vertices (-4,4), (4,4) and (-4,-4).
4.
(10pts) Find the volume inside both the sphere 
and the cylinder 
where 
. Check your answer.
5.
(9pts) Consider the following matrix:
. Find all eigenvalues of the matrix and for one of the
eigenvalues, find a corresponding eigenvector.
6. (10pts) Consider the following system of linear equations:
Write down all solutions to this system.
7.
(10pts) You wish to make a cylinder that will hold 2
gallons of liquid. Find the radius and the height
of the cylinder that minimizes the amount of material to be used.
8.
(6pts) Determine if the vectors 
, and 
are linearly
independent in R
. Do they form a basis for R
?
9.
(9pts) Consider the function 
:
a) Find the direction of greatest increase at the point (1,-1,-1).
b) What is the directional derivative in this direction (i.e. what is the greatest increase)?
c) Find the tangent plane to this surface at this point?
10.
(5pts) Let 
be a function of
three variables such that 
and 
. Find 
.
11.
(Extra credit 10pts). Show that if three vectors in R
do not all lie in the same plane, then they form a basis for R
. Let A be an 
matrix with entries in R.
If 
is an eigenvalue for A, show that all eigenvectors
corresponding to 
form a subspace of R
(this is called the eigenspace
for 
.