SUNY Geneseo Department of Mathematics

Problem Set 5 — Vector-Valued Functions

Math 223
Spring 2023
Prof. Doug Baldwin

Complete by Friday, March 3
Grade by Friday, March 10

Purpose

This exercise reinforces your understanding of vector-valued functions and their calculus. It therefore contributes to the following learning outcomes for this course:

Outcome 3.1. Analyze vector functions to find their derivatives and tangent lines
Outcome 3.2. Analyze vector functions to find their integrals.
Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is mainly based on material in sections 2.1. and 2.2 of our textbook. We covered that material in classes between February 20 and 24, with an introduction to differentiation, integration, and other vector operations in Mathematica on February 27.

Activity

Solve each of the following problems.

Problem 1

Let

\[\vec{r}(t) = \left\langle t^2 - 1, \frac{2t}{t+1}, \sqrt{t} + 2 \right\rangle\]

Part A

Calculate \(\vec{r}(1)\).

Part B

Does \(\vec{r}(t)\) ever equal the zero vector, \(\langle 0, 0, 0 \rangle\)? If so, give the value(s) of \(t\) at which it does so; if not, show why no value of \(t\) can make \(\vec{r}(t) = \langle 0, 0, 0 \rangle\).

Part C

Use Mathematica to plot \(\vec{r}(t)\) over the interval \(0 \le t \le 4\).

Problem 2

Find the derivatives of the following vector-valued functions. Also confirm your answers by using Mathematica to find each derivative.

\(\vec{f}(t) = \left\langle t^2 + 3t, \frac{t+1}{t}, 2 - t \right\rangle\)
\(\vec{s}(u) = \left\langle \tan(3u), \sec(3u), \sqrt{3u} \right\rangle\)

Problem 3

Evaluate the following integrals by hand, and then confirm your answers by evaluating them with Mathematica.

\(\int \left\langle t^2 + 3t, e^t \cos(e^t), \sqrt{t+1} \right\rangle\,dt\)
\(\int_1^2 \left\langle 1, t, t^2, t^3 \right\rangle\,dt\)

Problem 4

Our textbook says that the derivative of a sum of vector-valued functions is the sum of the derivatives, i.e., that

\[\frac{d}{dt}\left(\vec{r}(t) + \vec{u}(t)\right) = \vec{r}^\prime(t) + \vec{u}^\prime(t)\]

Prove this, using Theorem 2.2.1 (informally, that the derivative of a vector-valued function is the vector of derivatives of the component functions) and what you already know about derivatives of scalar-valued functions.

Problem 5

An ant is crawling along a coil of wire in such a manner that \(t\) seconds after the ant starts crawling it is at position \(\langle 2t, \sin (\pi t), \cos (\pi t) \rangle\), in some coordinate system in which distance is measured in inches.

Part A

Use Mathematica to plot the ant’s path for the first 10 seconds of its journey.

Part B

Find a function for the ant’s velocity as a function of time. (Hint: remember that velocity is the derivative of position.)

Part C

Find a function for the ant’s acceleration as a function of time. (Hint: remember that acceleration is the derivative of velocity.)

Part D

Show that the ant’s acceleration is always perpendicular to its velocity.

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above, and should ordinarily last half an hour. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.