Pellissippi State Community College



Things one should know from chapter 10:

1. Computations: Given vector-valued functions r(t) and s(t) one should be able to:

a. compute the derivative of r(t).

b. compute integrals (definite and indefinite) of r(t).

c. find r(t) x s(t), r(t) ( s(t), Dt (r(t) x s(t)), Dt(r(t) ( s(t)).

d. recover r(t) from r'(t) when given initial conditions on r(t) (such as r(0)).

e. trace the path given by r(t) when that path lies along well recognized graphs such as circles, ellipses, parabolas, straight lines, etc. The emphasis here is on the direction of movement along the path and what portion of the graph is actually traced out by r(t). If one places r'(t) with its tail on r(t), in what direction will it point? Given the normal and tangential components of acceleration, in what direction will a(t) point?

f. compute T, N, and B. For a specific path, if one changes the direction of motion along that path, will either T or N change directions ? How, if at all, will this change the direction of B ? If one changes the direction of motion along the path how, if at all, will the direction of r' change? What is T(N, B (N, and T(B ? What is the osculating plane and the normal plane at a point along the path of r?

g. find the velocity and acceleration vector-valued functions from a given vector-valued position function r(t) or, conversely, recover r(t) from a given velocity or acceleration vector-valued function. Also, there are the typical word problem applications that one does with these functions. (We worked out one in class involving whether a baseball would clear the outfield fence.)

h. compute the curvature k of a curve. You should be able to compute with all of the different forms of k. (I will provide the formulas but not specify which to use.)

i. compute the tangential and normal components of acceleration. I will provide the formulas but you should be able 'interprete' them - how do sudden changes in speed affect the tangential component of acceleration, what are the physical factors which will increase or decrease the normal component of acceleration, etc.

2. Given the trace of a path which has both an indication of the direction of movement along with a specific position vector r(t), you should be able to sketch in r'(t), N(t), and T(t). Given the tangential and normal components of the acceleration vector, you should be able to sketch in a(t). You should also be able to sketch in the osculating circle at that point. What is the radius of the osculating circle?

3. Under what circumstances will the curvature k of a curve always be zero? Under what circumstances will the curvature k of a curve be constant? What is the curvature of a circle of radius r?

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