Math 211 Exam 1 September 27, 2006
Read each problem carefully. Please show all your work for each problem! Use only those
methods discussed thus far in class. Always simplify when possible. No calculators!
1. (18 points) For the curve defined by
r(t) = (1 + t2)i+ (−3+2t)j+t3k,
find:
(a) the derivative r0(t) and the unit tangent vector T(t);
(b) an equation of the tangent line through point (2,−1,1).
(c) Is this curve smooth?
2. (16 points) Reduce the equation
x2−2x+y2+ 4z2+ 2y−16z+ 14 = 0
to the standard form. Then classify this surface and sketch it.
3. (16 points) Compute (in (a), aand bare constant vectors):
(a) d
dt [(a+tb)×(b−ta)].
(b) Zπ
0e−ti−√tj+ cos tkdt.
4. (16 points) Given the two planes
2x−z= 3,2x+ 3y+z= 0,
find:
(a) the cosine of the angle between them;
(b) an equation for the line of their intersection.
5. (16 points)
(a) Plot the point Pwhose cylindrical coordinates are (√2, π/4,3) and then find its
rectangular coordinates.
(b) Describe in words the surface whose equation in polar coordinates is θ=2π
3.
6. (18 points)
(a) Find a vector and a parametric equation of the line passing through points
P(1,2,3) and Q(2,1,5).
(b) Find two different planes whose intersection is the line
r(t) = (1 + t)i+ (2 −t)j+(3+2t)k.
Write equations for each plane in the form ax +by +cz =d.