Calculus 3: Cylinders and quadric surfaces

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 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.

Sketch the quadric surface for the equation $x^2 + z^2 = 1$.

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 $\displaystyle \frac{z}{c} = \frac{x^2}{a^2} - \frac{y^2}{b^2}$.