Calculus 3

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.

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.

Using the equation $z = 1 - y^2$, sketch the graph and determine its characteristics.

Find the equation of a quadric surface using the general form $Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0$.

Find the traces on the xy, xz, and yz planes for the quadric surface given by the function $2x^2 + 9y^2 + 18z^2 = 18$.

Divide the equation by 144 to identify the quadric surface for the given equation, which results in an ellipsoid.

Complete the square to transform the equation into the standard form of an elliptic paraboloid.

Graph the cone using the equation $\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ with the given axes.

Graph the ellipsoid using the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ with the given axes.

Graph the elliptic paraboloid using the equation $\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$.

Graph the hyperboloid of one sheet using the equation $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$.

Graph the hyperboloid of two sheets and explain why it does not pass through the origin, using its equation with two negative terms.

Graph the hyperbolic paraboloid using the equation $\displaystyle \frac{z}{c} = \frac{x^2}{a^2} - \frac{y^2}{b^2}$.

Solve problems involving different variations of axes given a three-dimensional graph and assess the suitability of the axis variation for the right-hand rule.

Find a parameterization for the cone using cylindrical coordinates, specifically with parameters r and theta, where the height of the cone is $z=r$, and the cone is bounded by $z \leq 3$.