Math 0200 Section 02
March 18, 2012
Review question for Midterm II
1. Consider a thin rectangular plate R in the xy -plane,
0 x a,
0 y b,
with constant density (x, y ) = 1.
(a) Compute the mass m(a, b) of the plate in terms a and b.
(b) Compute the polar
Math 0200 Section 02
April 25, 2012
Dierent notations for line integrals
Let C be a curve in 2D with parametrization r = r(t) = x(t), y (t) , a t b. For a scalar
function f (x, y ), we dene
b
b
f (r(t) |r (t)| dt =
f (x, y ) ds =
C
f (x(t), y (t)
x (t)2 +
Math 200 Section 02
November 5, 2012
Double integration over polar regions with r < 0.
We often use negative values of r when plotting polar curves; this makes formulas simpler, but can
be confusing when it comes to computing double integrals. The best ad
Practice Final Exam. M0200. Spring 2011
Every problem is worth 20 points; some problems are divided into parts; in this case each
part is assigned certain number of points.
Question 1.
2
2
1.a. [10] Find the points on the ellipse x2 + y2 = 1 where the nor
Problem 1
Given the following position vector r(t), determine the velocity v(t), the acceleration a(t),
and the speed v (t):
r(t) = 4 cos t, 4 sin t, 3t .
Solution 1
The velocity and acceleration vectors are given by the rst and second time derivatives of
Name:
Math 0200 - Quiz 3
Problem 1
Let V be the volume of a cylinder, such that V = r2 h. Suppose that the radius r and
height h are changing with respect to time t, but the volume is not. In particular,
V
=0
t
Using the multivariable chain rule, determin
Math 0200 - Quiz 4
Problem 1
Consider a rectangle inscribed inside the unit circle. Using Lagrange multipliers, nd
the maximum area of the rectangle. (Hint: assume the unit circle and the rectangle are
centered about the origin.)
(a) What is the function
Math 0200 - Quiz 5
Problem 1
Consider the following integral.
ln 2
ln 2
2
ex dxdy
I=
0
y
Sketch the region of integration.
Solution 1
The limits of integration tell us the following.
yx
0y
ln 2
ln 2
Any point our region must fall to the right of the curve
Midterm I Solutions
Math 0200 Section 2
February 15, 2012
1. (10 points) Let a = 1, 1, 0 and b = 0, 1, 1 .
(a) (3 points) Find the angle between a and b.
We know that
a b = |a|b| cos .
Solving for cos , we get
cos =
1
ab
10+11+00
=
=
2 + 1 2 12 + 1 2
|a|b
Midterm II Solutions
Math 0200 Section 2
March 21, 2012
1. (15 points) Let f (x, y ) = xy .
(a) (3 points) Find the critical points of f .
To nd critical points we set f (x, y ) = 0:
0 = fx = y,
0 = fy = x.
The only solution to these equations is x = y =
Math 0200 Section 02
March 19, 2012
Review question for Midterm II
1. Consider a thin rectangular plate R in the xy -plane,
0 x a,
0 y b,
with constant density (x, y ) = 1.
(a) Compute the mass m(a, b) of the plate in terms a and b.
Since the plate has co
Math 0200 Section 02
November 2, 2012
Completely optional non-review extra fun differentiation problems
1. Consider the function f ( x, y) = 1 y2 + cos x.
2
(a) Find the critical points of f .
(b) Use the second derivative test to classify the critical po