Problem Set 7 — 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 $y$:

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 $x=5\cos (2t)$ and $y=5\sin (2t)$. Use the chain rule to find $\frac{dw}{dt}$. 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 $\frac{dw}{dt}$ 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

relates the pressure ($P$), volume ($V$) and temperature ($T$) of a gas. Find $\frac{dP}{dt}$ 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.