Problem Set 7 — Multivariable Functions

Purpose

This exercise develops your understanding of multivariable functions (and has one question to reinforce your understanding of arc length of vector-valued functions). 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 arc length and curvature
• Outcome 4. Compute limits of functions of 2 and 3 variables
• Outcome 5. Compute derivatives of functions of 2 and 3 variables
• 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 3.1 through 3.3 of our textbook. We covered that material in classes between March 6 and 10. The exercise also asks you to plot multivariable functions and find their derivatives with Mathematica. We talked about plotting in class on March 6, and derivatives on March 10.

Arc length is covered in section 2.3 of the textbook, and we talked about it in class on March 1.

Activity

Solve each of the following problems.

Problem 1

If \(\vec{r}(t)\) is a straight line, i.e., it’s a function of the form \(\vec{r}(t) = \langle x_0 + at, y_0 + bt, z_0 + ct \rangle\), you now have two ways to compute the length of the segment between two points, \(\vec{r}(n)\) and \(\vec{r}(m)\): you could calculate it by using the distance formula, since the “curve” is a straight line, or you could use the arc length formula. Show that both approaches give the same answer. Show this in general, i.e., show that it’s true for all values of \(x_0, y_0, z_0, a, b\), and \(c\), not just for specific example(s).

Problem 2

(Exercise 18 in section 13.2E of our textbook.)

Determine whether

exists, and if so what it is.

Problem 3

Find all the (first) partial derivatives of each of the following functions. After finding the derivatives by hand, use Mathematica to check your results. :

Problem 4

(Inspired by exercise 4 in section 13.1E of our textbook.)

An oxygen tank is formed from a right circular cylinder of radius \(r\) and height \(h\), with hemispheres of radius \(r\) at the ends of the cylinder.

Part A

Express the tank’s volume as a function of \(r\) and \(h\). Remember that the volume of a cylinder of radius \(r\) and height \(h\) is \(V= \pi r^2 h\); the volume of a hemisphere of radius \(r\) is \(V = \frac{2}{3} \pi r^3\). Then use your function to find the volume of a tank with radius 3 and height 5.

Part B

Use Mathematica to plot your volume function over the region \(0 \le r \le 6, 0 \le h \le 6\).

Part C

The Geneseo Oxygen Tank company makes a standard tank of radius 3 inches and height 5 inches. If they want to change these dimensions slightly in order to hold more oxygen in their standard tank, will they get more “bang for the buck” by increasing the radius or by increasing the height? In other words, does volume change faster with changes in radius or with changes in height when \(r = 3\) and \(h = 5\)?

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.