Problem Set 4—Vector-Valued Functions

Purpose

This problem set reinforces your understanding of vector-valued functions and their derivatives and integrals.

Background

This problem set is based on material in sections 13.1 through 13.3 of our textbook. We discussed, or will discuss, this material in class between September 23 and September 30.

Activity

Solve each of the following problems:

Problem 1

Where does the line r(t) = ⟨ t, 2t, -t ⟩ intersect the sphere x² + y² + z² = 1?

Problem 2

Exercise 20 in section 13.1 of our textbook (find an equation for the line tangent to ⟨ t², 2t-1, t³ ⟩ at t = 2).

Problem 3

Exercise 12 in section 13.2 of our textbook (solve the initial value problem dr/dt = ⟨ 180t, 180t-16t² ⟩ and r(0) = ⟨0,100⟩).

Problem 4

Exercise 6 in section 13.3 of our textbook (find the unit tangent vector to the curve r(t) = ⟨ 6t³, -2t³, -3t³ ⟩; also find the length of the curve from t = 1 to t = 2).

Problem 5

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—or even negative—arc length).

Problem 6

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

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.