SOME PROBLEMS COPIED FROM HANDOUTS - NOT SURE OF THEIR SOURCE
Online Problems/Completion Packet
In single variable calculus, you learned the formula for the length of the curve
defined by for . In this problem, you will show that our new definition for arclength
coincides with this single variable formula.
The curve , for , can be parametrized as for . Prove that the the length of is .
Consider the path
- (a)
- Sketch this path.
- (b)
- Is this path ? Explain.
- (c)
- Find the length of the path.
Consider the path for .
- (a)
- Sketch this path.
- (b)
- Compute .
- (c)
- Explain why we consider this path to have finite length, even though the parameter is unbounded.
Consider the curve shown below.
- (a)
- Is the curvature of the curve greater at or at ? Explain.
- (b)
- Estimate the curvature at and at by sketching the osculating circles at those points.
Each of the graphs below has two curves, and . In each case one of the curves is a
graph of the curvature of the other. Identify which is the original curve and which is
the graph of the curvature. Explain your answers
Find an equation of a parabola that has curvature 4 at the origin. Use the definition
of curvature given in class. Make sure to justify your answer with pictures,
calculations, and other explanations. Hint: it will help to simplify your calculations
at each point. For example, and may appear messy, but can be greatly
simplified.
Written Problems
- (a)
- Parametrize the graph of . At what point does the curve have maximum curvature? What happens to the curvature as ?
- (b)
- Parametrize the graph of . Without calculating , answer these questions: At what point does the curve have maximum curvature? What happens to the curvature as ? (Hint: Use the geometric relationship between the functions in parts a and b.)
Professional Problem
In this problem you will prove that the value of the integral used to compute
arclength is independent of parametrization. First prove the following version of the
chain rule, which will be needed later. Your proof will be evaluated as part of your
solution to the professional problem.
Next, prove the following theorem.
Let and be two smooth and simple (that is, not self-intersecting) parametrizations
of a curve in with the same starting and ending points, so and . Then
Hints:
- Define a new function, .
- Write out the arclength integral in terms of , rewrite as , and use the above proposition and -substitution to transform the left side into the right side.
- exists, so the definition for makes sense. (You’ll need to justify this statement.)
- , and . (You’ll need to justify this statement.)
- is strictly increasing. (You’ll need to justify this statement.)
- . (You’ll need to justify this statement.)