Calculus 3

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?

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.

Given a multivariable function $f(x,y) = x^2 + y^2$, find where the partial derivatives are equal to zero to identify candidates for maximums or minimums.

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.

Find the maximum and minimum of the function f(x, y) = xy + 1 subject to the constraint $x^2 + y^2 = 1$ using Lagrange multipliers.

Find the local extrema of the function $f(x, y) = x^3 - 12xy + 8y^3$ using the second derivative test.

Given an ellipsoid represented by the equation $\frac{x^2}{1} + \frac{y^2}{4} + \frac{z^2}{9} = 1$, determine the lengths of the axes in the coordinate planes.

For the cone represented by the equation $\displaystyle \frac{x^2}{4} + \frac{y^2}{9} = \frac{z^2}{16}$, determine the intersection traces with the coordinate planes.

For a circular paraboloid given by $z = x^2 + y^2$, determine its axis of symmetry and describe the shape of its traces in the coordinate planes.

Analyze the hyperbolic paraboloid represented by the equation $z = x^2 - y^2$, and determine the shape of its traces in the coordinate planes.

For a hyperboloid of one sheet given by $x^2 + y^2 - z^2 = 1$, identify the axis of symmetry and describe the coordinate plane traces.

For a hyperboloid of two sheets represented by $z^2 - x^2 - y^2 = 1$, analyze its traces in the coordinate planes and describe its shape.