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
MA0200 LECTURE NOTES - WEEK 12
FRANCESCO DI PLINIO
These notes summarize the topics treated during Week 12.
Topics.
12.1 The theorem of Stokes.
12.2 The divergence theorem
12.3 Incompressible vector fields
12.1 The theorem of Stokes. The theorem of Stokes
MA0200 HOMEWORK 8, DUE FRI 03/25/16
8.1. Find the volume enclosed between the surfaces
z = x 2 + y 2,
z = 2x + 2 y + 2
using triple integrals (hint: polar coordinates centered at (1, 1).
8.2. The moment of inertia of a solid E R3 having density = (x 1 , x
MA0200 TA SESSION WEEK 4 - FEBRUARY 17, 2016
TOPICS
Linear approximation
Tangent plane
Directional Derivatives
Chain rule
4.1. Three numbers between zero and one are rounded to the nearest tenth (first decimal
digit) and then their squares are multiplied
MA0200 HOMEWORK 6, DUE FRI 03/11/16
6.1. Find the second order Taylor polynomial for the following functions at the indicated
points.
1
1
a. f (x, y) = x 2 y 3 at (x 0 , y0 ) = (1, 1)
b. f (x 1 , x 2 , x 3 ) = x 1 x 2 e x 3 at (1, 1, 1).
6.2. Find the cri
MA0200 LECTURE NOTES - WEEK 9
FRANCESCO DI PLINIO
These notes summarize the topics treated during Week 9.
Topics.
9.1
9.2
9.3
9.4
Line integrals of scalar functions.
Vector fields and work integrals
Conservative vector fields. The curl.
The Gauss-Green fo
MA0200 TA SESSION WEEK 3 - FEBRUARY 10, 2016
TOPICS
Motion in space
Domain, range and level curves
Limits in Rn
Partial Derivatives
2.1 Projectile motion. A basketball player shoots a free throw letting go of the ball at
height h above the floor level. Th
MA0200 TA SESSION WEEK 2 - FEBRUARY 4, 2016
TOPICS
Cross product and determinants
Linear transformations
Equations of lines and planes
Curves in space and tangent, normal, binormal vectors.
2.1. Find the volume of the prism with sides given by the vectors
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MA0200 HOMEWORK 4, DUE WED 02/24/16
4.1. Two numbers x, y with 0 x, y 1 are rounded to the nearest tenth, and then x 3 y 3
is computed. Give an estimate of the error introduced by the rounding by using linear
approximation at the point (x, y).
4.2. Find t
MA0200 HOMEWORK 2, DUE FRI 02/12/16
2.1. Let P = (0, 1, 2). Find
(a) the plane 1 passing through P and perpendicular
to the line x = y = z 2;
p
(b) the distance between P and the plane 2 : 6x 3 y + z = 0.
2.2. Consider the lines of parametric equations
t
MA0200 HOMEWORK 1, DUE FRI 02/05/16
~ as in
~ and b
1.1. A boat sitting at point A is being pulled along a river with two ropes a
~ find k.
Figure 1. Knowing that the boat is moving horizontally, and |~
a| = k|b|,
FIGURE 1
~
a
6
A
4
b
1.2. Find the angle
MA0200 TA SESSION WEEK 11 - APRIL 14, 2016
TOPICS
Flux and divergence
Parametrization of surfaces
Surface integral, mass and barycenters
Surface integrals of vector fields
~ : R2 R2 ,
11.1. Recall that for a 2-dimensional vector field F
~ = x F1 + x F2
di
MA0200 TA SESSION WEEK 7 - MARCH 10, 2016
TOPICS
Riemann sums
Double integrals and iterated integrals
Change of variable
Polar coordinates
6.1. Find a Riemann sum approximation of the following integrals using a n m grid. Use
the midpoint rule.
a. f (x, y
MA0200 HOMEWORK 11, DUE MON 04/25/16
11.1. Let C : x 2 + y 2 = 1 be the unit circle with outward unit normal. Compute the flux
Z
~ Nds
~
F
C
4~
~(x, y) = x y~i + x j both directly and using the divergence theorem.
of the vector field F
3
11.2. Let a be an
MA0200 LECTURE NOTES - WEEK 11
FRANCESCO DI PLINIO
These notes summarize the topics treated during Week 11.
Topics.
11.1 Divergence of a vector field. The divergence theorem in the plane.
11.2 Parametrization and surface integrals
11.3 Orientation and sur
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
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 =
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
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
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
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
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
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
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
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 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
MAOZOO 8132016 MlDTERM l . 02/25/16
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INSTRUCTIONS. There are 5 problems plus one extra credit problem. There are 80
points at disposal and the exam is graded on a scale of 75. Answer each problem in the
co
MA0200 HOMEWORK 7, DUE FRI 03/18/16
7.1. Find a Riemann sum approximation, using the midpoint rule with n = 2, m = 2, of
Z
x 2 2 y dxd y,
D = cfw_0 x 1, 0 y 1
D
and compare it to the actual value of the integral. What do you notice?
7.2. Find the volume o