Solutions to Problem Set 13

Math 311-01/02

2-15-2018

[Curvature problems]

1. Find the equation of the osculating plane to

A point on the plane is

The velocity vector is

The acceleration vector is

The cross product of and is perpendicular to the plane:

The plane is

2. Find the arc length parametrization of the circle .

The circle may be parametrized by

I need to find the length s of the curve from to .

I have

I'll switch variables in the integral to u, since I want the result to come out in terms of t:

Solve for t in terms of s to get .

Substitute this into the original parametric equations:

3. By considering a graph, determine the point on at which the curvature is largest.

The largest value of the curvature occurs at the vertex .

4. For what values of x does have curvature 0?

The curvature is

Thus, when . I have

The curvature is 0 when . Note that this is an inflection point of the graph.

5. Find the curvature of at .

Hence,

The curvature is

6. Find the curvature of at .

Hence,

The curvature is

7. Find the curvature of at .

I'll compute the pieces that go into the curvature formula first:

The curvature is

8. Find the curvature of at .

I'll compute the pieces that go into the curvature formula first:

The curvature is

Now that I'm here, where am I? - Janis Joplin

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