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Math 2XX3 - Midterm Test

Please submit scans / photographs of your full solutions (if handwritten) or pdfs (if typed) to the Crowdmark dropbox by the above date. You are welcome to discuss these problems with your classmates, but your submission must be your own work and written in your own words.  Also please include an acknowledgement naming any classmate who helped you and citing any outside sources you consulted. You are not required to use exactly the same notations that are used in the lectures and course notes, but if you wish to modify or introduce your own notation you must first clearly define it.

1. Assume that C(t) is a nondegenerate curve.

(a)  Suppose that C(t) lies on the surface of the sphere centered at the origin with radius R > 0 for all t _ e. Show that the derivative (velocity vector) CR (t) is always orthogonal to C(t) (position vector).

(b)  Conversely, show that if the position and velocity vectors are always orthogonal then

the path C(t) always remains on a sphere.

2.  Consider the two smooth surfaces given by the graphs of the following functions:

f(x, y) = x2 + y2

g(x, y) =a1 - x2 - y2 .

Can any part of the surface given by f be isometrically deformed into any open part of the surface given by g? Why or why not?

3. Under what conditions does equality hold for the following relations?

(a) In the Cauchy-Schwarz inequality.

(b) In the triangle inequality.

Justify your answers.

4. In an automobile race along a straight road, car A passed car B twice.  Prove that at some time t during the race their accelerations were equal (the acceleration of a path C(t) is CRR (t)). Justify your answer, and explicitly mention what assumptions you are making about how C(t) models the physical event.