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 \(y\):
| \(x=1\) | \(x=2\) | \(x=3\) | \(x=4\) | |
|---|---|---|---|---|
| \(y=1\) | 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 \(w = xy^2\), 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
\[PV = kT\]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.