Calculus III, Final Examination, May 11, 2009
1. Let R be the region inside the triangle with vertices (1, 0), (1, 0), (0, 1).
Write R as a region of type I and of type II.
Solution: Type I: 1 x 1, 0 y 1 |x|.
Type II: 0 y 1, y 1 x 1 y .
2. Evaluate the do
Problem Set 25, Due
(1) Find a one-to-one function T that maps S = cfw_(u, v ) : 0 u 1, 0 v 1
onto the parallelogram with vertices at (0, 0), (1, 4), (2, 5), and (3, 9).
We follow the method outlined in class. The idea is to rst nd parametric equations fo
Problem Set 23, Due
(1) Consider the solid S bounded below by the x-y plane and above by the
paraboloid z = 1 x2 y 2 . Compute
xy 2 z dV.
S
This part of the exercise was solved on the previous assignment.
If its density at the point (x, y, z ) is x2 + y 2
Problem Set 14, Due Sixteenth Class Meeting
(1) Suppose that f : (, ) (, ) is twice dierentiable. Dene
g (t, x) = f (at + x). Show that gtt (t, x) = a2 gxx . This is the case of
travelling waves.
According to the chain rule for functions of one variable a
Problem Set 13, Due Fifteenth Class Meeting
(1) Find the best linear approximation for f (x, y ) = x2 +3xy + y 2 +2x +4y 7 at
the point (2, 7). Now give an equation for the tangent plane to the surface
z = f (x, y ) at the point (2, 7, f (2, 7).
The best
Problem Set 6, Due Seventh Class Meeting
(1) Show that if the double cone x2 + y 2 = z 2 is intersected with the plane
y + z = 1 then we must have y = (1 x2 )/2.
Solving the equation of the plane for z and substituting into the equation
of the cone we hav
Problem Set 5, Due Sixth Class Meeting
(1) Find an equation for the plane passing through the points (1, 3, 4), (2, 7, 5)
and (6, 11, 5). How far is the point (3, 6, 4) from this plane? Find an
equation for the line passing through (3, 6, 4) and that is p
Problem Set 8, Due Ninth Class Meeting
(1) Graph the curve R(t) =< t cos(t), t sin(t), 0 >. Find the length of this curve
for the section when t [0, 2 ].
The graph is a spiral lying in the x-y plane. See the solution webpage
for a plot when t [4, 4 ]. For
Problem Set 9, Due Tenth Class Meeting
In this sequence of exercises we will discuss motion in space with air resistance. We
will regard this as motion in the x-y plane with the positive y axis in the direction
of up.
(1) We rst need to be able to solve a
1
Problem Set 11, Due Twelfth Class Meeting
Recall t the denition of limit says:
We say that f (x, y ) converges to L as (x, y ) approaches (a, b) if for every t > 0
we can nd d > 0 so that 0 < (x a)2 + (y b)2 implies |f (x, y ) L| < t.
1. Use the denitio
Problem Set 9, Due Tenth Class Meeting
(1) Draw the level curves for the function f (x, y ) = x2 + y 2 2x + 4y . Find a
level curve that consists of one point.
Before we try to draw the level curves we will complete the square:
f (x, y )
f (x, y ) + 1 + 4
Problem Set 22, Due
(1) Compute the integral of f (x, y, z ) = xyz 2 over the cube with vertices at the
points (0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1).
This cube is the set C = cfw_(x, y, z ) R3 : 0 x 1, 0 y
Problem Set 21, Due
(1) Locate the centroid of a quarter disk of constant density and radius R. It
may help to choose a simple coordinate system.
This calculation should be done in polar coordinates, and we should
locate the quarter disk with its vertex a
Problem Set 18, Due
2
3
(1) Let f (x, y ) = x + y , and let B = cfw_(x, y ) : 0 x 1, 0 y 2. Sketch
the set B and, by dividing B into 8 squares of equal area and using their
midpoints, estimate
f (x, y ) dA.
B
Now compute this integral exactly using Fubini
Calculus III, Math 233, Homework 1
(1) Consider the triangle with vertices P (1, 2, 3), Q(2, 1, 4), R(0, 1, 1).
Find the angle at each vertex of the triangle.
Solution: Let the angles at P, Q, R be denoted by , , , respectively.
Then
cos =
PQ PR
=
(3)(1)
Calculus III, Math 233
Homework 2
(1) A function of two variables is given by f (x, y ) = 1 + x2 + y 2 . Find the
domain and range, and sketch the level curves f (x, y ) = k for k = 1, 2, 3.
Solution: The domain of f consists of all points (x, y ) in the
Calculus III, Math 233
Homework 3
(1) Evaluate the double integral R y 2 dA, where R is the region bounded
by y = x2 and y = 2 x.
Solution: The equation x2 = 2 x has solutions x = 2, 1. The limits for
the region are 2 x 1, x2 y 2 x.
2x
1
y 2 dA =
y 2 dy d
Calculus III
Review for First Test on Chapter 10
Thursday, February 26
1. The three vectors a = x i j + k, b = j + 2k, c = 2i j + k form the
edges of a parallelepiped (box), where x is unknown. For which value
of x does the volume of the parallelepiped eq
Calculus III, First Test
1. Show that the vectors a = 1, 3, 1 and b = 1, 1, 2 are orthogonal.
Solution:
a b = 1 1 + 3 (1) + 1 2 = 1 + 3 2 = 0.
2. Find two unit vectors orthogonal to the vectors a = 1, 1, 3 and
b = 2, 3, 1 .
Solution:
ijk
a b = 1 1 3 = 8,
Review for Second Test on Chapter 11, April 7
1. Find and sketch the domain of the functions
f (x, y ) =
ln(1 + x) + ln(1 x)
x2 + y 2 1
2. Find the domain of the function f (x, y ) = 1 x2 y y and the
level curve f (x, y ) = 0. Sketch the domain and the le
Calculus III, Second Test
1. Find the domain and range of the function f (x, y ) = 1 y x, and
the level curve f (x, y ) = 1. Sketch the domain and the level curve.
Solution: The domain is D = cfw_(x, y ) : y 1 x and the range is R =
[0, ). The level curve
Review for Final Exam
Monday, May 11, 12:302:30, 2009
1. Find the double integral R (x + y ) dA over the region R bounded by
the parabolas y = x2 + 1 and y = x2 x + 2.
2. Find the centroid of the region bounded by the curves y = 4x x2
and y = 2x.
3. Evalu
Problem Set 1, Due Second Class Meeting
(1) Find the midpoint and length of the line segment joining the points (1, 2, 3)
and (7, 12, 13). Now give an equation for the sphere for which this segment
is a diameter.
Let P = (1, 2, 3) and Q = (7, 12, 13). The
Problem Set 20, Due
(1) Let M > 0, and consider D(M ) = cfw_(x, y ) : x2 + y 2 M 2 . Sketch D(2).
What set do you get if you let M ?
D(2) is disk of radius 2 centered at the origin. As M the set D(M )
expands to be the whole plane.
(2) Continuation: Now,
Problem Set 4, Due Fifth Class Meeting
(1) Find the volume of the parallelopiped generated by the vectors
< 1, 2, 3 >, < 5, 9, 11 > and < 2, 3, 5 >. We know from class that the
volume is given by
| < 1, 2, 3 > < 5, 9, 11 > < 2, 3, 5 > | =
| < 5, 4, 1 > <