Mathematics 41, section 1 -- Analysis 3
Review Sheet for Exam
February 9, 1998
General Information
The first exam for the course will be given Friday, February 14,
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 critical points of
vector fields from class Monday, February 9. The topics are
- 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
- Vector fields, flow lines, and critical points
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 1.1/2,3,4,9,14 (14 is an Extra Credit-type
problem!)
- Section 1.2/3,5,6,7,10
- Section 1.3/1,4,8
- Section 1.4/2,3,5,7,9
- Section 2.2/1,2 (the tangent line is the line through the
point with the direction vector v = alpha'(t0)),8
- Section 2.4/1,2,3,4,12
Sample Exam
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) 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.
II. All parts of this problem refer to the parametric curve
alpha(t) = ((1+ cos(t))sin(t),-(1 + cos(t))cos(t))
(a cardioid, in honor of Valentine's Day).
- 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.
III.
- A) Show that the parametric curves
alpha(t) = (et, et)
and
beta(t) = (e-t,-e-t)
are both flow lines of the vector field F(x,y) = (y,x).
- B) Show that for any scalars c,d in R, the
curve given by
gamma(t) = c alpha(t) + d beta(t)
is also a flow line of F.
- C) What do parts A and B say about the type of the
critical point of F(x,y) = (y,x) at the origin
(0,0)? Explain.