Problem Set 9 — Gradients and Directional Derivatives

Purpose

This exercise mainly develops your understanding of directional derivatives and gradients. It therefore contributes to the following learning outcomes for this course:

• Outcome 5. Compute derivatives of functions of 2 and 3 variables
• Outcome 6.1. Apply derivative concepts to find tangent lines to level curves.

Background

This exercise is mainly based on material in sections 3.3 through 3.6 of our textbook. We covered that material in classes between March 10 and 29.

Activity

Solve each of the following problems.

Problem 1

(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.

(Hint: While most of this problem set deals with gradients and directional derivatives, this question doesn’t.)

Problem 2

Suppose \(g(x,y) = x^2 + y^2\).

Part A

Find the gradient of \(g\).

Part B

Using the gradient you found in Part A, find a vector tangent to the level curve for \(g\) that passes through point \((\sqrt{2}, -\sqrt{2}, 4)\).

Part C

Find a parametric equation (i.e., a vector-valued function) for the level curve of \(g\) through point \((\sqrt{2}, -\sqrt{2}, 4)\). Find the tangent vector to that function at that point, and verify that it’s parallel to the vector you found in Part B.

Problem 3

(Inspired by exercise 48 in the 13.6E of our textbook.)

The temperature at point \((x,y,z)\) in a metal sphere is inversely proportional to distance from the origin.

Part A

If the temperature at point \((1,2,2)\) is \(100^\circ\) C, find the exact formula for temperature as a function of \(x\), \(y\), and \(z\).

Part B

How fast is the temperature changing as one moves from point \((1,2,2)\) in the direction towards point \((2,1,3)\)? Assume distances are in centimeters if you want to attach units to your answer.

Problem 4

In many situations when one wants a single thing that acts as “the” derivative of a multivariable function, that single thing is the gradient. As such, you would expect gradients to behave in ways similar to derivatives. As an example of gradients behaving like derivatives, show that they obey a sum law analogous to the one for derivatives. More specifically, show that if \(f(x,y)\) and \(g(x,y)\) are differentiable multivariable functions, then

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.