Mathematics 241 -- Multivariable Calculus
Review Sheet for Exam 1
September 21, 2007
General Information
The first exam for the course will be given Friday, September 28,
as announced in the course syllabus. It will cover the material
we have discussed in class (including the labs)
through and including the material on partial derivatives from
from class on Friday, September 21.
The topics are:
- Coordinates in R3, subsets
defined by coordinate equations or inequalities.
- Vectors, the dot product, lengths and angles
- Cross products, applications to volume.
- Equations of lines and planes
- Parametric curves and motion (know how to parametrize
the line through a given point with a given direction vector, the
circle in a plane with a given center and radius, and an ellipse
with given center, major and minor axes),
- Polar, cylindrical, spherical coordinates.
- Level curves, contour curves, and the graph z = f(x,y);
slices in other coordinate planes, graphs of more general equations
(see question III in the sample exam questions below).
- Limits for f : Rn -> R.
- Partial derivatives for f : Rn -> R.
There will be 4 or 5 problems, each with several parts. Some may
ask for a graph or the result of a calculation; others may ask for
a description or explanation of some phenomenon (similar to some
questions from labs).
I will happy to schedule an evening or afternoon review session
before the exam. Wednesday evening would be good.
Suggested Review Problems
For practice, it would be a very good idea to
try a few of the odd-numbered problems from all the sections
of the text that we have covered. (Look for ones similar to questions from
the problem sets so far.)
Sample Exam Questions
Note: the actual exam will not be this long.
I.
- A) Find the equation of the plane in R3 containing
the point P = (1,1,1) and the line through
Q = (0,1,-1) with direction vector v = (-4,1,2).
- B) Find the parametric equations of the line in R3
passing through the point (1,2,3) and perpendicular to the plane
5 x + 2 y - 7 z = -12.
- C) Two lines with direction vectors v = (0,1,2) and
w = (4,3,2) meet at (0,0,0). Find the acute angle between
the two lines.
- D) Show that if u = (u1,u2,u3),
v = (v1,v2,v3), and
w = (w1,w2,w3)
are general vectors in space, then the following cross product formula holds:
u x (v + w) = u x v + u x w.
- E) Find the volume of the parallelpiped with one corner at the origin
and sides parallel to the vectors (1,1,1),(0,2,3),(2,0,1).
II. In this problem, α refers to the parametric curve
α(t) = ((1 + cos(t))cos(t), (1 + cos(t))sin(t))
(called a cardioid)
- A) Show that for all t, the distance from the origin
to the point α(t) is given by 1 + cos(t).
- B) Explain why this curve could be described by the polar equation
r = 1 + cos(θ).
- C) At how many different times t in [0,2 π] is
the y-coordinate of α(t) equal to zero?
Find them.
- D) Give a parametrization for any one circle whose interior
completely contains the cardioid. Explain how
you determined your center and radius.
- E) What would you see if you plotted r = 1 + cos(θ)
cylindrical in cylindrical coordinates
III.
- A) What are the spherical coordinates of the point with rectangular
coordinates (4,2,0)?
- B) Describe the slices of the spherical graph
ρ = θ2 in planes θ = const.
What does the whole surface look like?
- C) Describe the slices of the spherical graph
ρ = φ2 in planes z = const.
What does the whole surface look like?
IV.
- A) Consider the set
Q = {(x,y,z) : x2 - y2/4 + z2 = 1}.
Identify the slices of Q in planes parallel to each of the three
coordinate planes, and use that information to generate a rough
sketch of Q.
- B) Same question for
R = {(x,y,z) : x2 - y2/4 - z2 = 1}.
V. Consider the function f : R2 -> R
defined by f(x,y) = (x4 - y4)/(x2 - y2)
and f(0,0) = 3.
- A) Does lim(x,y)->(0,0)f(x,y) exist? Why or why not?
- B) Is f(x,y) continuous at (0,0)? Why or why not?
- C) Is f(x,y) continuous at (1,1)? Why or why not?
- D) Sketch the level curves of f(x,y) for c = 0,1,4.
- E) Find ∂ f/∂ x and ∂ f/∂ y.
Are your formulas valid if (x,y) = (0,0)?