MATH 21300: Sample Final Exam C

Question 1

Let $P_1$ be the plane $3x - y + z = 5$ and $P_2$ the plane $x + 2y = -9$.

a) Find the angle between the planes $P_1$ and $P_2$. You may use an inverse trigonometric function in your answer.

b) Find parametric equations for the line in which $P_1$ and $P_2$ intersect.

Question 2

Find the following limit or show that it does not exist: $$\lim_{{(x,y) \to (0,0)}} \frac{xy - x}{x^2 + y^2}$$

Question 3

Let $f(x, y) = \frac{2x}{y+1}$.

a) Find the linearization $L(x, y)$ of the function $f$ at the point $P_0(1, 1)$.

b) Find an upper bound for the magnitude of the error $E$ in the approximation $f(x, y) \approx L(x, y)$ over the rectangle $R: 0.9 \leq x \leq 1.1, 0.8 \leq y \leq 1.2$. You may use decimals and fractions in your answer.

Question 4

Let $f(x, y) = x^2 + y^2 - xy - x$.

a) Find and classify all critical points of $f$.

b) Find the absolute minimum value of the function $f$ on the triangular region bounded by the lines $x = 0$, $y = 1$, and $x - 2y = 0$.

Question 5

a) Change the following Cartesian integral into an equivalent polar integral: $$\int_{0}^{2} \int_{-\sqrt{4 - y^2}}^{\sqrt{4 - y^2}} e^{x^2 + y^2} \, dx \, dy$$

b) Evaluate the polar integral from part a).

Question 6

A solid $E$ is bounded below by the surface $z = y^2$, above by the plane $z = 1$, and on the sides by the planes $x = 0$ and $x = 1$. Find the moment of inertia with respect to the $z$-axis if the mass density $\rho(x, y, z) = 1$ for all $(x, y, z)$ in $E$.

Question 7

Find the circulation of the field $\mathbf{F}(x, y) = 3y \mathbf{i} + 2x \mathbf{j}$ along the ellipse $C$ given by $$\mathbf{r}(t) = \cos(t) \mathbf{i} + 3 \sin(t) \mathbf{j}, \quad 0 \leq t \leq 2\pi.$$

Question 8

Let $E$ be the solid region bounded below by the plane $z = 0$, above by the sphere $x^2 + y^2 + z^2 = 9$, and on the sides by the cylinder $x^2 + y^2 = 4$.

a) Set up the integral that gives the volume of $E$ using cylindrical coordinates and the order of integration $dz \, dr \, d\theta$. Do not evaluate the integral.

b) Set up the integral that gives the volume of $E$ using cylindrical coordinates and the order of integration $dr \, dz \, d\theta$. Do not evaluate the integral.

Question 9

Determine whether the following vector field is conservative in the plane. If it is conservative, find a potential function for $\mathbf{F}$. $$\mathbf{F}(x, y) = \frac{2xy}{1+x^2} \mathbf{i} + \ln(1+x^2) \mathbf{j}.$$

Question 10

Evaluate the line integral $$\oint_C e^{\displaystyle x^2} \, dx + (x^3 - \ln(\sin y + 3)) \, dy,$$ where $C$ is the counterclockwise oriented circle $x^2 + y^2 = 4$. Use whatever method you prefer.