Homework 2
Math 352, Fall 2011
Instructions: Solve at least two of the following three problems. Your solutions must be
written in LATEX.
Due Date: Monday, September 19
1. Let γ : [0, ∞) → R2 be the unit-speed curve with γ(0) = γ̇(0) = (1, 0) whose signed
curvature is κs (s) = (2s)−1/2 .
(a) Find a formula for γ̇(s). (Hint: Start by finding a turning angle for γ.)
(b) Use your answer to part (a) to find a formula for γ(s).
(c) Make a plot of γ(s) for 0 ≤ s ≤ 4π. Do you recognize this curve?
2. Let f : R2 → R be a smooth function, let C ∈ R, and let γ(t) = γ1 (t), γ2 (t) be a
regular parametrization of the level curve f (x, y) = C.
(a) Use the Chain Rule to prove that fx (γ(t)) γ̇1 (t) + fy (γ(t)) γ̇2 (t) = 0 for all t.
(b) Take the derivative of the equation in part (a) with respect to t.
(c) Let p = γ(0), and suppose that fx (p) = 0 and fy (p) 6= 0. Use parts (a) and (b)
to find a formula for the curvature of γ at p in terms of fy (p) and fxx (p).
(Hint: What is γ̇2 (0)?)
(d) The curve x2 y + y 3 + x2 − 3y 2 = 0 is known as the trisectrix of Maclaurin:
3
2
1
0
-1
-3
-2
-1
0
1
2
3
Use your formula from part (c) to find the curvature of the trisectrix at the
point (0, 3).
3. Find a smooth function f : (−π/2, π/2) → R such f (0) = f 0 (0) = 0 and the curvature
of the graph y = f (x) is κ(x) = cos x.