Math 241, Spring 2006- EXAM 1. NAME:
Instructions. Closed book, closed notes, no calculators. No credit for
answers without justification- show all work. 50-min test.
1.(List 1:3,5) [10] For the parameterized helix r(t) = (3 cos t, 4t, 3 sin t), t ∈
[0,2π], find: (1) the total length; (ii) parametric equations for the tangent
line at the point (3/√2, π, 3/√2)
2.(List 1:10) [10] A particle moves with constant speed v= 4 on the
ellipse x2+ 2y2= 3 (counterclockwise). When the particle is at the point
(1,1), find: (i) the velocity vector; (ii) the direction of the acceleration vector
(recall directions are given by unit vectors).
3.(List 3:8)[5] Let f(x, y) = x3+y3−6xy. The origin is a critical point.
Let r(t) be a smooth curve satisfying r(0) = (0,0),r0(0) = (1,1). Does the
function of one variable g(t) = f(r(t)) have a local max, local min or an
inflection point when t= 0?
4.(List 3:5)[5] Find a unit normal vector to the ellipsoid x2+2y2+4z2=
7, at the point (1,1,1).
5.(List 2:8)[5] Let f(x, y, z ) = xy2z−3(z6= 0); assume the values of
x, y, z are known up to an error, so that |∆x
x| ≤ 0.005, with the same relative
error estimates for yand z. Estimate the relative error |∆f
f|in the computed
value of f.
6.(List 3:7,14)[10] Let f(x, y) = x4+y4−4xy. (i) Locate a saddle point
of f(if there is one); (ii) Write down the degree 2 Taylor polynomial of f
at (1,1).
7. (Lecture: 2/10)[5] Explain why the function in problem 6 is guaran-
teed to have an absolute (global) minimum, over all (x, y)∈R2.
1