WebWork Exam Review Solutions (pdf)

The problem is something like the following: The maximum value of $f(x,y)$ subject to the constraint $g(x,y)=250$ is 6000. The method of Lagrange multipliers gives $\lambda=35$. Find an approximate value for the maximum of $f(x,y)$ subject to the constraint $g(x,y)=252$.

Hint: The main idea here is that the functions satisfy the relation $\nabla f=35\nabla g$ at the maximum point where $g(x,y)=250$ and, very importantly, this implies that the gradients are parallel. Hence, if one increases $g$ by 2, one will increase $f$ by some very easy to calculate value. How fast is $f$ increasing compared to $g$??

Lots of people are asking about this problem. It states something along the lines of: If $z=\sin(x^2+y^2)$, $x=v\cos(u)$, and $y=v\sin(u)$, find $\partial z/\partial u$ and $\partial z/\partial v$. The variables are restricted to domains on which the functions are defined.

There is an easy way and a hard way to do this. Most of you are doing it the hard way, and your answer ends up being different than what WebWork is expecting.

First, notice that

\[z=\sin((v\cos(u))^2+(v\sin(u))^2)\]

and that this will simplify to a function of a single variable. Now, after doing the simplification, take your derivatives. If you try to take the derivatives too early, you get something that is technically correct, but far too complicated to be applicably correct.

There’s a ridiculously easy way to solve this problem that has nothing to do with dot products. The hint is to use unit vectors in the same direction as the vectors you are given. What happens if you add those unit vectors together? (This approach doesn’t work unless you have unit vectors. Why?)

Please log in to WebWork and change your password if you have not done so already. If you have not done this by tonight, I will e-mail you. If you don’t do this by Wednesday, I will suspend your account.

There are several bugs in WebWork that have been fixed:

Review 12.1 12.2: Problem 12 WebWork was not accepting the correct answer for certain variations of this problem. That has been fixed, but in the process you may have seen your specific problem change a bit.

Sections 12.5 12.6: Problem 12 and Sections 12.5 12.6: Problem 13 had a formatting error, making it challenging to read the problems. That has been fixed.