Problem Set 8 — Applications of Partial Derivatives

Purpose

This exercise gives you practice with various applications of partial derivatives, including their use in directional derivatives and gradients, and finding extreme values of multivariable functions. It thus contributes to the following learning outcomes for this course:

• Outcome 3.1. Analyze vector functions to find their derivatives and tangent lines
• Outcome 5. Compute derivatives of functions of 2 and 3 variables
• Outcome 6.1. Apply derivative concepts to find tangent lines to level curves
• Outcome 6.2. Apply derivative concepts to solve optimization problems.

Background

This exercise is based on material in sections 3.6 and 3.7 in our textbook. We covered those sections in classes between October 18 and October 25.

One question asks you to compare tangents to level curves found via gradients to the same tangents found as tangents to parametric or vector-valued curves. Relevant material on vector-valued functions is in section 2.2 of our textbook, covered in class on September 23.

Activity

Solve each of the following problems.

Problem 1

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 2

(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 \(120^\circ\) 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 3

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

Problem 4

(Exercise 10 in the section 13.7E of our book.)

Find the critical points of \(f(x,y) = -x^3 + 4xy - 2y^2 + 1\) and use the second derivative test to classify them as minima, maxima, or saddle points.

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.