Calculus 3: Multivariable functions

Calculate the gradient vector for a given function $f(x, y)$ and describe its significance in the context of a 3D graph.

What direction should you travel to keep your height constant (i.e. travel on a contour aka a level curve)?

For the function $f(x, y) = x^2y - xy^2$ at the point $(1, -1)$, find the direction and rate of greatest increase, greatest decrease, and a direction of no change.

Explain and visualize different types of multivariable functions.

The limit as X and Y approaches 5 and 5 of $\displaystyle \frac{x^2 - y^2}{x - y}$

The limit as X and Y approaches the origin of $\frac{x^2 + y^2}{x+y}$

The limit of multivariable function given by the expression with three variables x, y, and z, using parametric curves for variables substitution.

Using the Extreme Value Theorem, find the global maximum and minimum values of a multivariable function $f(x, y)$ on a domain that is closed and bounded, either in the interior or along the boundary.

Imagine one input is constant (e.g., $S = \pi$) and another input (T) varies. Determine the resulting shape in the output space.

Consider the matrix A, which is $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. Find the eigenvalues and corresponding eigenvectors.