Calculus of One and Several Variables and Linear Algebra Practical Track
MA 001

Spring 2014
Ma 1c Practical, Midterm Exam Sketched Solution
Problem 1. (10 points) Does the following limit exist? If it exists, compute
the limit.
x4
lim
(x,y)0 x4 + x2 + y 2
Solution. We have the inequality
So
x4
x4 +x2 +y 2
whenever (x, y)
x4
x2 + y 2
x4
x4 +
Calculus of One and Several Variables and Linear Algebra Practical Track
MA 001

Spring 2014
#8.2.3. S = cfw_(x, y, z) : x2 + y 2 + z 2 = 1, z 0, S = cfw_(x, y, 0) : x2 + y 2 = 1,
F(x, y, z) = (x, y, z). Very Stokes theorem for S, S and F where S is oriented
as a graph. (Note that the book says S = cfw_(x, y) : x2 + y 2 = 1  which is wrong,
sinc
Calculus of One and Several Variables and Linear Algebra Practical Track
MA 001

Spring 2014
HOMEWORK 7
7.6.4
Let F(x, y, z) = 2xi 2yj + z 2 k. Evaluate
F dS,
S
where S is the cylinder x2 + y 2 = 4 with z [0, 1].
Solution
We parametrize our surface as (, z) = (2 cos , 2 sin , z). Then we get,
= (2 sin , 2 cos , 0),
= (0, 0, 1),
z
Using these we c
Calculus of One and Several Variables and Linear Algebra Practical Track
MA 001

Spring 2014
#7.1.24. Compute the path integral of f (x, y) = y 2 over the graph of y = ex , x [0, 1].
Solution.
We interpret this graph as the path c(t) = (t, et ), where t ranges from 0 to 1. Recall
the formula for computing path integrals:
1
f (c(t)
f (x, y) ds =