Problem Set 7—Directional Derivatives and Tangents

Purpose

This problem set develops your understanding of gradients, directional derivatives, and tangents to surfaces. It also reinforces your understanding of partial derivatives and the chain rule for them.

Background

Directional derivatives and gradients are covered in section 14.5 of our text, and will be discussed in class on October 21. Tangents to multi-variable functions are discussed in section 14.6, and will be covered in class on October 23 or 26.

Activity

Solve each of the following problems:

Problem 1

Exercise 26 in section 14.4 of our textbook (use the text’s Theorem 8 to find dy/dx given that y is defined by the equation xy + y² - 3x - 3 = 0).

Problem 2

Exercise 2 in section 14.5 of our textbook (find the gradient of ln( x² + y² ) and sketch with the level curve through (1,1)).

Problem 3

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)).

Problem 4

Exercise 20 in section 14.5 of our textbook (find the directions in which the function f(x,y) = x²y + e^xy siny changes most rapidly at point (1,0); find the derivatives of f in these directions).

Problem 5

Exercise 2 in section 14.6 of our textbook (find the tangent plane and normal line at (1,1,1) to the surface x² + y² - z² = 18)

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.