Problem Set 7—Partial Derivatives

Purpose

This problem set helps you understand partial derivatives. In particular, this problem set should develop your ability to (1) calculate first- and second-order partial derivatives, (2) use the chain rule to calculate partial derivatives, and (3) calculate directional derivatives.

Background

This problem set is based on material in sections 14.3 through 14.5 of our textbook. We discussed partial derivatives (section 14.3) in class on March 8 and the chain rule (section 14.4) on March 9. We will discuss directional derivatives (section 14.5) on March 10.

Activity

Solve each of the following problems:

Problem 1

Exercise 6 in section 14.3 of our textbook (find the first-order partial derivatives of f(x,y) = (2x - 3y)³).

Problem 2

Exercise 46 in section 14.3 of our textbook (find all the second-order partial derivatives of s(x,y) = tan^-1(y/x)).

Problem 3

Exercise 82 in section 14.3 of our textbook (show that w = cos(2x+2ct) is a solution to the wave equation; see text for details of the wave equation).

Problem 4

Exercise 2 in section 14.4 of our textbook (find dw/dt in terms of t given that w = x² + y², x = cost + sint, and y = cost - sint; see the textbook for additional requirements).

Problem 5

Exercise 40 in section 14.4 of our textbook (find ∂w/∂t and ∂w/∂s given that w = f(x,y) = f(ts²,s/t) and ∂f/∂x = xy and ∂f/∂y = x²/2).

Problem 6

Exercise 12 in section 14.5 of our textbook (find the derivative of 2x² + y² in the direction of ⟨3,-4⟩ at point (-1,1)).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.