1. Briefly compare and contrast the definition of "vector-valued function" and the definition of an ordinary function y = f (x).
2. How do you determine whether a curve C determined by r(t) is "smooth"? Give an example of a function r(t) that determines a non-smooth curve C, and explain why it fails to be smooth. Your example should be different from the examples we discussed in class.
3. Give the domain of r(t) = ln(t + 2) i + sin-1(t/5) j + (1/t-1)k. Justify your answers.
4. If r'(t) = (tcos2t, e-3t, √5t+4) and r(0) = (2,4,6) , find r(t).
5. Let r(t) = (sin 2t,3cos2t,et/4), with - ∞
(a) Explain why it is important to consider the domain of r(t) when sketching the graph of C, the space curve determined by r(t).
(b) Is C smooth? Explain how you know.
(c) Sketch the space curve C determined by 4(t), and show r(π), v(π), and a(π) on a BIG graph of C. You'll need lots of space for this graph - make sure your scale is appropriate. Be sure your graph of C displays enough of the important characteristics of the curve to be able to infer the shape of the entire graph of C for all values oft in the domain of r. If you wish, you may include a sentence or two to further describe C in addition to your graph.
6. Consider the vector-valued function r(t) = at + b sinωt j + b cosωt k, where a, b, and co are positive constants, that describes the position (in units of length from the origin) of a particle t units of time after the object starts moving.
(a) Describe the graph of C, the space curve determined by r(t).
(b) How does changing the value of a (while leaving b and ω fixed) change the graph of C?
(c) How dots changing the value of b (while leaving a and ω fixed) change the graph of C?
(d) How does changing the value of ω (while leaving a and b fixed) change the graph of C?
(e) Find the speed of the particle moving along Cat any time t.
(f) Find the arclength of C for one coil. (Note that your answer will involve a, b, and ω.)
(g) Based on your answer to part (e), is the particle speeding up or slowing down as t increases? Explain.