Problem Set 4 — Vector-Valued Functions

Purpose

This exercise develops your understanding of vector-valued functions and their limits and derivatives. It contributes to the following learning outcomes for this course:

• Outcome 1. Represent vectors analytically and geometrically, and perform vector calculations
• Outcome 3.1 Analyze vector functions to find their derivatives and tangent lines
• 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 based on material in sections 2.1 and the beginning of 2.2 in our textbook. We talked about those sections in classes between September 20 and 23. We talked about plotting vector-valued functions with Mathematica on September 20, and talked about using it to find derivatives on September 23.

Activity

Solve each of the following problems.

Problem 1

Let

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

Suppose

Part A

Use the limit definition of the derivative of a vector-valued function to find \(\vec{f}^\prime(t)\).

Part B

Use 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 to find \(\vec{f}^\prime(t)\). Hopefully this step will confirm the derivative you found in Part A.

Part C

Further confirm your answers to Parts A and B by calculating \(\vec{f}^\prime(t)\) with Mathematica.

Problem 3

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

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 4

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. Do the necessary calculations by hand, but then check them with Mathematica. (Hint: remember that velocity is the derivative of position.)

Part C

Find a function for the ant’s acceleration as a function of time. Do the necessary calculations by hand, but then check them with Mathematica. (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. 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.