Calculus 3

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.

Sketch the graph for the equation $x \cdot y = 1$ and describe its properties.

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.

Using the equation of a sphere, $x^2 + y^2 + z^2 = 1$, graph the sphere with the given axes.

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 $z \leq 3$.

Parametrize the plane given by the equation $2x - 3y + z = 6$ using two parameters $u$ and $v$.

Parametrize the cylindrical surface given by the equation $x^2 + (y - 2)^2 = 4$.

Parametrize the sphere given by the equation $x^2 + y^2 + z^2 = 9$ using spherical coordinates.

Given the parametrization x = u, y = v, and $z=\sqrt{u^2 + v^2}$, determine the rectangular equation.