Homework 3
Math 352, Fall 2014
Due Date: Friday, September 19
1. Use the formula
~ (t)
T~ 0 (t) = s0 (t) κg (t) U
to compute the curvature κg (t) of the tractrix ~x(t) = (t − tanh t, sech t) for t > 0.
2. (a) Use equation 1.12 in the textbook to compute the curvature of the ellipse
x 2
a
+
y 2
b
= 1
at each of its four vertices.
(b) Find the equation of the osculating circle for this ellipse at the point (a, 0).
~
3. (a) Find the center C(h)
of the circle that intersects the catenary y = cosh x at the
points (−h, cosh h), (0, 1), and (h, cosh h).
~
(b) Use L’Hôpital’s rule to compute lim C(h).
h→0
(c) Based on your answer to part (b), what is the curvature of the catenary y = cosh x
at the point (0, 1)? Explain.
4. Let ~x(s) be a unit-speed curve of class C 2 (where s ≥ 0), and suppose that:
• ~x(0) = (1, 0),
• ~x 0 (0) = (1, 0), and
• The curvature of ~x(s) is given by the formula κg (s) = (2s)−1/2 .
(a) Find a formula for θ(s).
(b) Find a formula for ~x 0 (s).
(c) Find a formula for ~x(s).
(d) Find a parametrization for this curve that does not involve any square roots.