Mathematics 241, section 1 -- Multivariable Calculus
Review Sheet for Exam 1
September 24, 2010
General Information
The first exam for the course will be given Friday, October 1,
as announced in the course syllabus. It will cover the material
we have discussed in class (including the discussions and lab)
through and including the material on tangent vectors and tangent
vectors to parametric curves.
The topics are:
- Coordinates in R3, subsets
defined by coordinate equations, slices, etc.
- Vectors, the dot product, lengths and angles
- The cross product in R3
- Equations of lines and planes
- Parametric curves and motion (know how to parametrize any segment of
the line through a given point with a given direction vector, any arc of the
circle in the plane with a given center and radius, and any arc of an ellipse
with given center, major and minor axes),
- Tangent vectors and tangent lines to parametric curves
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 discussions and labs).
I will happy to try to schedule a review session
before the exam, but there are some constraints. Wednesday evening is not
good for me because of the Chamber Orchestra concert. Thursday morning
at 8:00am is a possibility(!)
Suggested Review Problems
From the text (in addition to the problems from the problem sets):
- Section 1.1/5,6,11
- Section 1.2/4,5,6,7,9
- Section 1.3/1bde, 7bd, 8,10
- Section 1.4/1,3,6,9,questions like 11 and permutations thereof
- End of Chapter Exercises Chapter 1/15, 16 (we did not discuss
this but you should be able to figure out the idea if you
think about it)
- Section 2.1/1,2,5bd, 7bd, 9, 10
- Section 2.2/1c, 2bd, 3, 8
Sample Exam Questions
Disclaimer: The following questions indicate the range of topics
that may be covered and the ``style'' of the questions I might ask. The
actual exam questions may be posed differently and may combine
the topics we have discussed in different ways. It will also be
a bit shorter.
I. Consider the set
Q = {(x,y,z) in R3 : x2 - y2/4 - z2 = 1}.
- A) Identify the slices of Q in planes parallel to the three
coordinate planes. Take x,y,z = -2,-1,0,1,2 (at least).
- B) Use that information to generate a rough sketch of Q or
a verbal description. (This is
called a hyperboloid of two sheets. Can you see why?)
- C) Show that the image of the parametric curve
α(t) = ((et + e-t)/2, 0,
(et - e-t)/2) for t in R
lies entirely in 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) The points P,Q,Q + v from part A are three corners
of a parallelogram in the plane you found.
Find the fourth corner and the area of that parallelogram.
- 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
u x (v + w) = u x v + u x w.
III.
- A) Sketch the parametric curve α(t) = (4 cos(π t), 3 sin(π t))
for 0 ≤ t ≤ 1. Show the starting and final points and indicate the
direction the curve is traced with an arrow.
- B) Give a parametrization of the circle with center (1,3) and
radius r = 4 so that one full circuit of the circle is traced out clockwise for 0 ≤ t ≤ 4 π.
IV. All parts of this problem refer to the parametric curve
α(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 α(t) is given by 1 + cos(t).
- B) At how many different times t in [0,2 π] is
the y-coordinate of α(t) equal to zero?
Find them.
- C) Find a parametrization of the tangent line to the
cardioid at the point α(π/4).
- D) Give a parametrization for any one circle whose interior
region completely contains the cardioid. Explain how
you determined your center and radius.