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 \(w = xy^2\), 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.