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 (
(Hint: While most of this problem set deals with gradients and directional derivatives, this question doesn’t.)
Problem 2
Suppose
Part A
Find the gradient of
Part B
Using the gradient you found in Part A, find a vector tangent to the level curve for
Part C
Find a parametric equation (i.e., a vector-valued function) for the level curve of
Problem 3
(Inspired by exercise 48 in the 13.6E of our textbook.)
The temperature at point
Part A
If the temperature at point
Part B
How fast is the temperature changing as one moves from point
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
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.