Math 223 Partial Derivatives

Purpose

This exercise is a chance to practice working with partial derivatives and some of their applications. It thus contributes to the following learning outcomes for this course:

Outcome 4. Compute limits of functions of 2 and 3 variables
Outcome 5. Compute derivatives of functions of 2 and 3 variables.

Background

This exercise is based on material in sections 3.2 through 3.5 in our textbook. We talked about those sections in classes between October 4 and October 17.

Activity

Solve each of the following problems.

Problem 1

Suppose $g (x, y) = \frac{x}{\sqrt{y}}$ .

Part A

Use the limit definition of partial derivatives to calculate $\frac{\partial g}{\partial x}$ and $\frac{\partial g}{\partial y}$ .

Part B

Calculate each of the second derivatives of $g$ . You do not have to use the limit definition.

Problem 2

The following table gives values for function $f (x, y)$ for certain values of $x$ and $y$ :

Values of $f (x, y)$ for certain $x$ and $y$ .
	$x = 1$	$x = 2$	$x = 3$	$x = 4$
$y = 1$	1.5	2	2.5	3
$y = 2$	2.5	3	3.5	4
$y = 3$	3.5	4	4.5	5
$y = 4$	4.5	5	5.5	6

Estimate the value of $f (2.1, 3.05)$ . Be prepared to explain during our meeting what if any connections you made between this problem and things we’ve talked about in class.

Problem 3

(Based on exercise 4 in section 13.5E of our textbook.) Suppose $w = x y^{2}$ , and that $x = 5 \cos (2 t)$ and $y = 5 \sin (2 t)$ . Use the chain rule to find $\frac{d w}{d t}$ . Then substitute the definitions of $x$ and $y$ into the equation for $w$ , to get an equation in terms of $t$ from which you can calculate $\frac{d w}{d t}$ directly. Verify that both ways of calculating the derivatives produce the same answer.

Problem 4

(Exercise 38 in section 13.5E of our textbook.)

The equation

P V = k T

relates the pressure ( $P$ ), volume ( $V$ ) and temperature ( $T$ ) of a gas. Find $\frac{d P}{d t}$ given information about $V$ , $T$ , and their derivatives with respect to time ( $t$ ). See the textbook for the details, except treat temperature as kelvins, not degrees Fahrenheit.

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.