MA330: Assignment 3
Required Reading.
• Read Chapter 4, § 37, 38, 40, & 41
To be turned in March 29th at the start of class.
1. Textbook, page 93, #1
2. Textbook, page 93, #2
3. Textbook, page 94, #5
4. Textbook, page 104, #1
5. Textbook, page 104, #2a
6. Textbook, page 104, #9
7. Given a plane curve r(t) = hx(t), y(t)i, use the methods shown in class to derive the formula
κ=
|ẋÿ − ẍẏ|
(ẋ2 + ẏ 2 )
3/2
.
An object moving near Earth under the influence of gravity obeys r(t) = hx0 + v0x t, y0 + v0y t − gt2 /2i. Compute the
curvature at the instant when the y velocity is 0. Simplify your answer.
8. If a curve is written in the form y = f (x), it can be parameterized as r(t) = hx(t), f (x(t))i. Use the result of
problem 7 to derive the formula
κ=
|f 00 (x)|
3/2
(1 + f 0 (x)2 )
.
When cable is suspended between two fixed endpoints, it’s resting shape under the influence of gravity is a catenary,
y = a cosh(x/a) for x ∈ [−L, L], where a and L are positive. Compute the curvature at the center of the cable.
Simplify your answer.
9. If a curve is written in polar coordinates as r = f (θ) then it can be parameterized r(θ) = hf (θ) cos θ, f (θ) sin θi. Use
the result of problem 7 to derive the formula
κ=
|f 2 + 2(f 0 )2 − f f 00 |
3/2
(f 2 + (f 0 )2 )
.
Use this to compute the curvature of the logarithmic spiral r = aebθ , where a > 0 and b are real constants. Simplify
your answer.
1