MATH 21300: Sample Final Exam A

Question 1

Show that the following limit does not exist: $$\lim_{{(x, y) \to (0, 0)}} \frac{x^2\cos(x+y)}{x^2+y^2}$$

Question 2

Find $\alpha$ and $\beta$ to make the following statement as precise as possible: If at some point $(x_0, y_0)$, $\frac{\partial f}{\partial x}(x_0, y_0) = 3$ and $\frac{\partial f}{\partial y}(x_0, y_0) = -4$, then $$\text{for any unit vector } \mathbf{u}, \quad \alpha \leq D_{\mathbf{u}} f(x_0, y_0) \leq \beta$$ where $D_{\mathbf{u}} f$ is the directional derivative of $f$ in the direction of $\mathbf{u}$.

Question 3

The cylindrical solid enclosed by $x^2 + y^2 = 4$, $z = 0$ and $z = 5$ has mass density given by $\rho(x, y, z) = 8 - \arctan(z^2)$. Set up integrals to find the mass and $z$-coordinate of the center of mass. You do not need to actually do any of the integrals.

Question 4

Find and classify the critical points of the function $f(x, y) = x^3 - 12x + 3y^2 - 6y$.

Question 5

Evaluate $\iint_T (2x + 6y) \, dA$, where $T$ is the triangular region in the $xy$-plane with corners at the origin, $(0, 4)$ and $(2, 0)$.

Question 6

Find the volume of the solid between the $xy$-plane and the surface $z = 12x^2y + 4$ enclosed by the planes $x = -2$, $x = 1$, $y = 2$ and $y = 4$.

Question 7

Let $\mathbf{F} = (2ye^{2xy} + 2) \mathbf{i} + (2xe^{2xy} + 4y) \mathbf{j}$. Find the work done along the path $\mathbf{r}(t) = (t^2 + t^3) \mathbf{i} + (t^2 + t^7) \mathbf{j}$, with $t \in [0, 1]$. It may help to notice that $\mathbf{F}$ is conservative.

Question 8

Let $S$ be the sphere of radius 5 centered at the point $(11, 9, 7)$. Think of the sphere as a globe with the north pole at the point with the largest $z$-value. Let $C$ be the curve which is the equator of $S$, travelled clockwise when viewed from above. 1. Parameterize the curve $C$. 2. For $\mathbf{F} = \cos(y^2) \mathbf{i} + \ln(x) \mathbf{j} + (z - 3) \mathbf{k}$, set up the following integral: $$\int_C \mathbf{F} \cdot \mathbf{T} \, ds$$ Leave your integral for part (b) in terms of your parameter from part (a). Do not evaluate your integral.

Question 9

Sketch the region of integration, change the order of integration, and evaluate: $$\int_0^9 \int_{x/3}^{\sqrt{x}} 6x \, dy \, dx$$

Question 10

1. Evaluate $\int_C y \, dx$ directly as a line integral, where $C$ is an ellipse parameterized by $x = 3 \cos(t)$, $y = 2 \sin(t)$, with $0 \leq t \leq 2\pi$.
2. Apply Green's theorem to find the area enclosed by the curve $C$ of part (a).