Homework 4
Math 352, Fall 2014
Due Date: Friday, September 26
1. In PerpendicularAnimation.gif, the black circle is the unit circle centered at the origin,
and the red point is (3, 0). Find parametric equations for the indicated curve.
Z
2. Evaluate
3x2 + 6xy dx + 3x2 + 2y dy, where C is the curve ~x(t) = (2 sin t, 3 sin 5t)
C
for 0 ≤ t ≤ π/2.
3. For 0 ≤ t ≤ 1, let L(t) be the line segment from the point (0, 1 − t) to the point (t, 0).
(a) Find an equation for the line containing L(t). Your answer should have the form
y = m(t)x + b(t), where m(t) and b(t) are functions of t.
(b) Together, all of the line segments L(t) fill a region in the plane, as shown in
EnvelopeAnimation.gif. Find parametric equations for the top boundary curve.
4. Find a differentiable function f (x) for −π/2 < x < π/2 such that f (0) = f 0 (0) = 0
and the curvature of the graph y = f (x) is κg (x) = cos x.
Hint: Use the formula
κg (x) =
f 00 (x)
1 + f 0 (x)2
3/2
that we derived in class for the curvature of the graph of a function. Feel free to use a
calculator or computer (e.g. Wolfram Alpha) to evaluate any integrals that arise.