Mathematics 41, section 1 -- Multivariable Calculus
Review Sheet for Exam
September 20, 1999
General Information
The first exam for the course will be given Friday, September 24,
as announced in the course syllabus. It will cover the material
we have discussed in class (including the discussions and labs)
through and including the material on tangent vectors and
tangent lines to parametric curves from class on Friday, September 17.
The topics are:
- Coordinates in R3, subsets
defined by coordinate equations, slices, etc.
- Vectors, the dot product, lengths and angles
- The cross product, 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),
- Tangent vectors and lines to parametric curves
There will be 3 or 4 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 discussions and labs).
I will happy to schedule an evening or afternoon review session
before the exam. Wednesday evening would be good.
Suggested Review Problems
From the text:
- Section 1.1/4,5,6,11,13
- Section 1.2/3,5,6,7,10
- Section 1.3/1,3,4,5,7,10
- Section 1.4/5,6,7,8,13,18,24
- Section 2.1/1,3,4 (Typo Alert: the parametrization
should be alpha(t) = (a cos(t), b sin(t)).) Number
17 would make a good Extra Credit problem!
- Section 2.2/1,2 (the tangent line is the line through the
point with the direction vector v = alpha'(t0)),8
Sample Exam
I. Consider the set
Q = {(x,y,z) : x2 - y2/4 + z2 = 1}.
Identify the slices of Q in planes parallel to the three
coordinate planes, and use that information to generate a rough
sketch of Q.
II.
- 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) 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.
- C) Show that if u = (u1,u2,u3),
v = (v1,v2,v3), and
w = (w1,w2,w3)
are general vectors in space, then
u x (v + w) = u x v + u x w.
III. All parts of this problem refer to the parametric curve
alpha(t) = ((1+ cos(t))sin(t),-(1 + cos(t))cos(t))
(called a cardioid)
- A) Show that for all t, the distance from the origin
to the point alpha(t) is given by 1 + cos(t).
- B) At how many different times t in [0,2 pi] is
the y-coordinate of alpha(t) equal to zero?
Find them.
- C) Find a parametrization of the tangent line to the
cardioid at the point alpha(pi/4).
- D) Give a parametrization for any one circle whose interior
region completely contains the cardioid. Explain how
you determined your center and radius.