Prelim 1 Math 1920 September 27, 2012
7 questions; 100 points. For full credit, you must show your work and justify your answers! You
may use anything that has been covered in class or in the book, as long as you show clearly what
you are using. No calculators or notes allowed.
1. (15 points) The velocity of a certain particle at time tis given by
v(t)=t, π cos(πt),1.
At time 0, the particle is at the point (0,1,−1).
(a) Find the position r(t) of the particle at time t.
(b) Find the speed of the particle at time 3.
(c) Set up an integral for the distance traveled by the particle between the points (0,1,−1)
and (8,1,3). (Please don’t attempt to evaluate the integral.)
2. (15 points) Let L1be the line through the point (0,1,1) and parallel to the vector 1,−1,1.
Let L2be the line through the points (5,−3,6) and (−1,1,0).
(a) Parametrize the line L1, and also parametrize the line L2.
(b) Is there a plane containing the lines L1and L2? If so, find an equation for this plane,
and explain how you know it contains both lines. Otherwise, explain why no such plane
exists.
3. (15 points)
(a) Find two vectors vand wsuch that v×w=i+k.
(b) Find the area of the triangle with vertices (1,1,1),(−1,3,−2),and (1,0,3).
4. (10 points) Find a parametrization for the curve which is the intersection of the cylinder
y2+z2= 4 and the surface x=y2z. Give bounds on your parameter between which the
curve is traced exactly once.
5. (15 points) Determine the limit as (x, y)→(0,0) of x2y2
x4+y4or show that it does not exist.
6. (15 points) Consider the function f(x, y)=sin x
y.
(a) Plot the level curves z=cof this function for c=0,c=1, and c=−1.
(b) Does the limit
lim
(x,y)→(π,0)
sin x
y
exist or not? In either case, explain your reasoning. If it does exist, what is its value?
7. (15 points) Let Lbe the line segment in the xy plane joining the points (0,−√3) and (1,0).
(a) Give an equation in polar coordinates r=f(θ)forL. What are the bounds on θ?
(b) Find the polar coordinates of the point of Lthat is closest to the origin.
