Calculus 3

Given the parametrization $x=3u\sin v$, $y=3u\cos v$, and $z=u^2$, determine the rectangular equation.

Using two parameters, parametrize the surface of a torus (doughnut shape) in three dimensions, considering a circle of radius $a$ at a distance $b$ from the center (z-axis), and varying the parameters to map the surface of the torus.

How do you actually go from an s and a t that goes from $0$ to $2\pi$, in both cases, and turn it into a three-dimensional position vector-valued function that would define this surface?

Evaluate the function f at T = 0 and S = $\pi$ and determine the resulting point in three-dimensional space.

Imagine one input is constant (e.g., $S = \pi$) and another input (T) varies. Determine the resulting shape in the output space.

Determine the resulting shape when both T and S vary freely in the function representing a torus.

Consider the surface given by $x = u \cos(v), y = u \sin(v), z = v$. Set up the integral for the surface integral of a function $f$ over this surface.

For a vector field $F = \langle x^2, xy, z \rangle$ and the surface given by $z = x + y^2$, find the flux of the field through this surface over the range $x$ from 0 to 1 and $y$ from 0 to 1.

Find the equation for the tangent plane to the function $Z = f(x, y)$ at a given point $(a, b)$.

Find the equation for the tangent plane to the implicit function $F(x, y, z) = 0$ at a point $(a, b, c)$.

Using the linear approximation, approximate the value of the function at a given point near $(a, b)$.

Create the equation of a tangent plane to a given surface at a given point.

Make a tangent plane to a surface at a specified point by calculating the partial derivatives with respect to x and y and then substituting into the tangent plane equation, similar to the previously discussed method.

The base radius and height of a right circular cone are 10 cm and 25 cm, with a possible error in measurement of as much as 0.1 cm in each. Use differentials to estimate the maximum error in the calculated volume of the cone.

The dimensions of a rectangular box are 75 cm, 60 cm, and 40 cm, with each measurement correct to within 0.2 cm. Use differentials to estimate the largest possible error when the volume of the box is calculated.

Find the equation of a tangent plane to the circular paraboloid $z = x^2 + y^2$ at the point (-1, -1, 2).

Find the equation of the tangent plane to the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ at the point $(2, 4, 4)$.

Find a plane tangent to the ellipsoid $\frac{x^2}{1} + \frac{y^2}{25} + \frac{z^2}{9} = 1$ at the point $(-3/5, 4, 0)$.

Find the equation of the tangent plane to the surface $z = 4x^2 - y^2 + 2y$ at $(-1, 2, 4)$.

Estimate the value of $z = \ln(x-2y)$ at the point (3.1, 0.9) using a tangent plane approximation.