Math 234 — Study guide 3rd midterm

Topics covered on the third midterm

Finding a function from its first partial derivatives

1. What does Clairaut’s theorem say?
2. How do you find $f(x, y)$ if you are given $f_x$ and $f_y$? How do you know if such a function exists?

Finding critical points from the zero set of a function

1. What is a critical point of $f$?
2. What are sufficient conditions that guarantee that a function $f(x, y)$ will have a maximal and also a minimal value on some region $E$?
3. Given a drawing of the zero set of a function $f$ and the sign of the function off the zeroset, you can predict that certain points are critical points of $f$, and that some regions will contain at least one critical point of $f$. Explain.
4. If a function $f$ attains its maximal value on some region $E$ at a point $P$ in $E$, must $P$ then be a critical point?
5. You should be able to find critical points of a function like those in the homework problems and the old exams posted below.

Taylor expansion and the second derivative test

1. Compute the Taylor expansion of second order of a function at some point $(a,b)$.
2. What is special about the Taylor expansion at $(a,b)$ if $(a,b)$ is a critical point of $f$?
3. Know how to use the Taylor expansion to determine if a critical point of a function is a local maximum, minimum, saddle point, or none of these. In the case of a saddle point, find the two tangent lines to the level set of the function.
4. Explain how and why the second derivative test works.

Optimization with constraints

1. If a function $f(x, y)$ attains it maximal value among all points $(x, y)$ satisfying $g(x, y)=0$ at some point $(a,b)$, then what equations must $(a, b)$ satisfy?
2. Given a drawing of the level set $g(x, y)=C$ and the gradient $\vec\nabla f$of the function $f$ on the level set determine which points are local maxima and which are local minima on the level set. Be able to explain your answer.
3. Be able to solve Lagrange’s equations for $f$ and $g$ as in the homework problems and the old exams below.

Old exams