Final Exam Solutions, Calculus III, Math 2210-90, Fall 2005, Bob Palais
Show all your work on the exam for full credit. You may use graphing (or regular)
calculators.
1. (10 points) Let V= 2I+ 6J−3K, and W= 2I+ 0J−1K.
a) Find a multiple of W,αW, whose projection on Vis equal to V.
b) Find the distance from Wto the line tV.
c) What is the cosine of the angle between Vand 0I+ 0J+ 1K?
2. (10 points) A particle moves counterclockwise in a circle of radius 1 in the x−
z−plane according to the equation
R(t) = cos 2tI+ sin 2tK
where trepresents time.
a) Find the unit tangent vector of the particle at time t,T(t).
b) Find the unit normal, N(t) and the curvature κ(t) = 1
||X′(t)|| ||T′(t)||.
3. (10 points) Let R(t) = (3t+t3)I+ (−2 + 7t−t2)J+ (1 −5t+t4)K.
a) Find the parametric and symmetric forms of the line tangent to R(t) at t= 0.
b) Find the equation of the plane tangent to R(t) at t= 0. (The plane tangent to a
curve , or ‘osculating plane’ at a point is parallel to its tangent and normal vectors at that
point.)
4. (10 points) Let f(x, y, z) = 1
2(( x
2)2+y2+z2).
a) At the point (1,1,1), in which direction uwith ||u|| = 1 is fincreasing most
rapidly? How fast is fchanging in this direction at that point?
b) Find df
dt along l(t) = (1 + 4t, 1 + t2,1−t) at t= 0. Is ltangent to the surface
f(x, y, z) = 9
8at (1,1,1)?
5. (10 points) Let f(x, y) = x3y−12xy + 2y2.
a) ∇f=
b) What are the critical points of f?
c) What kind of critical points are they?
