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.

Imagine you have a function $z = x^2 + y^2$. How would you begin to plot this function in a 3-dimensional space?

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.

Given a function with two-dimensional input $(x, y)$ and a vector output, determine the vector at a specific input point such as $(1, 2)$ using the functions $(y^3 - 9y)$ for the x-component and $(x^3 - 9x)$ for the y-component.