MATH 21300: Sample Final Exam B

Question 1

Compute the limit $$\lim_{{(x, y) \to (0, 0)}} \frac{x^3y^2 + 2y^3}{x^3 + y^3}$$ or prove that it does not exist.

Question 2

For parts (a), (b), and (c), let $f(x, y) = x^2 - 3xy$.

a) At the point with $x = 1$ and $y = -1$, compute the unit vector pointing in the direction of greatest increase of the function $f(x, y)$ and compute the rate of increase in that direction.

b) Compute an equation for the plane tangent to the surface given by the equation $z = f(x, y)$ at the point in space with $x = 1$ and $y = -1$.

c) Find the rate at which $f(x, y)$ is changing at $(1, -1)$ in the direction toward the point $(5, 2)$.

Question 3

Let $E$ be the solid bounded by $y = x^2$, $y = x$, $x = z$, and $z = 0$ whose mass density is given by $\rho(x, y, z) = x$. Sketch $E$ and find its mass.

Question 4

Find and classify the absolute extrema of the function $f(x, y) = x^2 - y^2$ over the region $x^2 + y^2 \leq 1$.

Question 5

Compute $$\iiint_H z \, dV,$$ where $H$ is the solid region bounded above by the $xy$-plane and below by the sphere of radius 4 centered at the origin.

Question 6

Let $f(x, y) = e^{3x - y} \cos(x - 1)$. Estimate $f(0.98, 3.01)$ using differentials (linear approximation).

Question 7

Change the following triple integral to cylindrical coordinates and then to spherical coordinates: $$\int_{-3}^{3} \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} z\sqrt{x^2 + y^2 + z^2} \, dz \, dy \, dx.$$ Now use one of the three integrals to compute the common value.

Question 8

Evaluate $$\oint_C \arctan(x) \, dx + (3x - 4 - 5y) \, dy,$$ where $C$ is the circle of radius 4 centered at $(2, 5)$ parameterized counterclockwise.

Question 9

The fluid flow in a region is given by a vector field $\mathbf{F} = (x - 2y) \mathbf{i} + (2x + 6y) \mathbf{j}$. Compute the total outward flux of the fluid passing through a rectangular box, with opposite corners at the origin and at $(4, 2)$. Is there more flow into or out of the box?

Question 10

Consider the following curves: $$A: \begin{cases} x = 3 + \cos(t) \\ y = 3 + \sin(t) \end{cases}, \quad B: \begin{cases} x = 3 \sin(t) \\ y = 3 \cos(t) \end{cases}, \quad C: \begin{cases} x = 12 \cos(t) \\ y = 9 \sin(t) \end{cases}, \quad t \in [0, 2\pi].$$ Suppose we have a vector field $\mathbf{F}$ defined on all of the plane except the points $(3, 3)$ and $(0, 0)$. Also suppose that we know that $\nabla \cdot \mathbf{F} = 0$ everywhere on the plane except those two points. If $$\oint_A \mathbf{F} \cdot \mathbf{n}\, d{s} = 2\pi \quad \text{and} \quad \oint_B \mathbf{F} \cdot \mathbf{n}\, d{s} = -1,$$ then what is $\oint_C \mathbf{F} \cdot \mathbf{n}\, d{s}$?