Calculus 3

Find the gradient of the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ and evaluate it at the point $(-3, 2)$.

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.

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.

Find the linear approximation to this multivariable function at the point $(2, 3)$ using the tangent plane, and then use the linear approximation to estimate the value of the function at $(2.1, 2.99)$.

Find the equation of the tangent plane to the graph of the function f(x, y) = 2 - x^2 - y^2 at the point $\left(\frac{1}{2}, \frac{1}{2}\right)$.

Find the linearization $L(x, y, z)$ of a function of three variables at the point (2, 1, 0).

Linearize the multivariable function $f(x,y) = 1 + x \ln(xy - 5)$ at the point (2, 3).

Given the function $f(x, y) = 1 + x \cdot \ln(xy - 5)$, find the linearization of the function at the point $(2, 3)$.

Estimate the temperature of a pizza at the point $(2.9, 4.2)$ where the temperature function is given by $f(x, y) = 5x^2 \sqrt{x^2 + y^2} - 100$ using the tangent plane at a nearby point.

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.

Given a contour plot with yellow representing higher values and blue representing lower values, visualize what the surface would look like in 3-dimensional space.

Given that $f(x, y) = 10 - 3x^2 - 2y^2 + 8y + 12x$, identify any critical points, saddle points, and any local extrema.

Find and classify the critical points of $f(x,y) = 2x^4 + 2y^4 - 8xy + 12$.

Given the function $f(x, y)$ on the rectangle D, find the absolute extreme values.

A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.)

Find the maximum and minimum values of the function $f(x, y) = 2x^2 + y^2 - y$ given the constraint $x^2 + y^2 \leq 1$.

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.