Problem Set 5—Vector-Valued Functions

Purpose

This problem set reinforces your understanding of vector-valued functions. It particular, I expect this problem set to develop your understanding of (1) limits and continuity of vector-valued functions, (2) integrals of vector-valued functions, (3) projectile motion, and (4) arc length and vector-valued functions.

Background

This problem set is based on material in sections 13.1 through 13.3 of our textbook. We discussed the basics of vector-valued functions in class on February 10 and February 11. We will discuss integration and arc length on February 17 and 18, respectively.

Activity

Solve each of the following problems:

Problem 1

Let f(t) = ⟨ (t²-1)/(t+1),  sin²t, t²/ln(t²+1) ⟩ Identify any real values of t at which f is not continuous, and give f’s limit as t approaches those values.

Problem 2

Exercise 2 in section 13.2 of our textbook (evaluate the integral from 1 to 2 of ⟨ 6-6t, 3√t, 4/(t²) ⟩).

Problem 3

Exercise 39c in section 13.2 of our textbook (show that dot and cross products involving constant vectors can be “factored” out of definite integrals of vector-valued functions; see textbook for details).

Problem 4

Exercise 28 in section 13.2 of our text (give a mathematical foundation for an experimental result about flying marbles; see the book for details).

Problem 5

Exercise 2 in section 13.3 of our textbook (find the unit tangent vector to r(t) = ⟨ 6sin(2t), 6cos(2t), 5t ⟩; also find the length of r between t = 0 and t = π).

Problem 6

Exercise 10 in section 13.3 of our textbook (find the point on the curve r(t) = ⟨ 12sint, -12cost, 5t ⟩ that is a distance of 13π units along the curve from (0,-12,0) in the direction of decreasing arc length).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.