Calculus 3

Find the extrema of the function $f(x, y, z) = xyz$ subject to the constraint $x^2 + y^2 + z^2 = 3$.

Suppose you make TV sets at two different factories, Factory A and Factory B. You produce x TVs at Factory A, y TVs at Factory B, and the cost given by the production of x TVs and y TVs is the function $6x^2 + 12y^2$. You need to produce exactly 90 TV sets a month. Determine how many TVs should be produced at each factory to minimize the cost.

Find the surface area of the part of the surface $z = x^2 + 2y$ that lies above the triangular region with vertices (0, 0), (1, 0), and (1, 1).

Find the surface area of the portion $z = x^2 + y^2$ below the plane $z = 9$.

Find the surface area of the curve $z = \sqrt{4 - x^2}$ above $R$ which is the rectangle where $x$ is between $0$ and $1$ and $y$ is between $0$ and $4$.

Find the surface area of the part of the surface $Z = x^2 + 2y$ that lies above the triangular region with vertices $(0, 0)$, $(1, 0)$, and $(1, 1)$.

Find the surface area of the portion of $Z = x^2 + y^2$ that is below the plane $Z = 9$.

Find the surface area of $Z = \sqrt{4 - x^2}$ which is above the region $R$, a rectangle defined by $x \in [0, 1]$ and $y \in [0, 4]$.

Find the surface area of the part of the function $z = xy$ that lies inside the circle $x^2 + y^2 = 1$ using double integrals.

Compute the surface area of a surface given its parametric description. Use the formula: $\ iint_{\text{Region}} \| \mathbf{r}_u \times \mathbf{r}_v \| \, du \, dv$ where $\mathbf{r}_u$ and $\mathbf{r}_v$ are the partial derivatives of the position vector with respect to the parameters $u$ and $v$.

Compute the flux of a vector field across the surface $z = 1 - x^2 - y^2,$ with $z \geq 0,$ where the vector field is $\mathbf{f}(x, y, z) = (x, y, z).$ The surface is oriented with outward normals from the perspective of starting at the origin and moving out to the surface.

Find the surface area of the portion of the plane described by $z = -x$ that is inside the cylinder $x^2 + y^2 = 4$.

Find the parametric representation of the surface which is the part of the sphere $x^2 + y^2 + z^2 = 4$ that lies above the cone with equation $z = \sqrt{x^2 + y^2}$.

Parametrize the saddle surface using the equation: $z = x \cdot y$. Use parameters $u$ and $v$.

Parametrize the upper hemisphere defined by the equation: $z = \sqrt{4 - x^2 - y^2}$, and identify the domain for $u$ and $v$.

Parametrize the circular cylinder $x^2 + z^2 = 1$, with $u$ ranging from 0 to $2\pi$ and $v$ being any real number.

Parametrize the paraboloid defined by the equation $z = x^2 + y^2$ over the closed unit disk in the xy-plane.

By hand, draw the vector field $\mathbf{F}(x, y) = y\hat{i} - x\hat{j}$ and plot various points to visualize the pattern.

For a vector field given by components $f(x, y) = (y, x)$, sketch the vector field.

Given a gravitational field, represented as $F = -\frac{G m_1 m_2}{r^2} \hat{r}$, where $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, and $r$ is the distance, derive the expression for the gravitational field as a vector field.