# Problem 1 (15 pts)

Does the following series converge or diverge? Give reasons for your answer.
(a) (7 pts) $\sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}$

(b) (8 pts) $\sum_{n=1}^{\infty} \frac{n!}{n^n}$ Hint: You may want to use the fact: $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e \approx 2.718$

# Problem 2 (10 pts)

Find the radius and the interval of convergence of the series $\sum_{n=0}^{\infty} \frac{(-3)^nx^n}{\sqrt{n + 1}}$

# Problem 3 (10 pts)

Consider the function $f(x) = \sqrt{1 + x}$.
(a) (5 pts) Approximate $f(x)$ by a Taylor polynomial of degree $1$ centered at $0$. Compute the value of $\sqrt{2}$ using that approximation.

(b) (5 pts) How accurate is this approximation when $0 \le x \le 1$? You can express the approximation error either by fraction or by a three decimal point number. Then, confirm that the approximate value of $\sqrt{2}$ computed in Part (a) is within this error. Note that the precise value of $\sqrt{2}$ up to $4$ digits is $1.414$.

# Problem 4 (10 pts)

Let $P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)$.
(a) (5 pts) Find the area of the triangle $PQR$.
Hint: Length of the cross product of two vectors is equal to the area of a parallelogram formed by those two vectors.

(b) (5 pts) Find the distance from $P$ to the line $QR$.

# Problem 5 (10 pts)

Find the limit if it exists, or show that the limit does not exist.
(a) (5 pts) $\lim_{(x, y) \to (0, 0)} \frac{x^y}{x^4 + y^2}$ Hint: Consider $(x, y) \to (0, 0)$ along the line $y = x$ and along the parabola $y = x^2$.

(b) (5 pts) $\lim_{(x, y) \to (0, 0)} \frac{x^3 + y^3}{x^2 + y^2}$ Hint: Consider the limit in the polar coordinates $(r, \theta)$.

# Problem 6 (10 pts)

Verify that the function of two variables $u(x, t) = e^{-\alpha^2 k^2 t} \sin kx$ is a solution of the heat conduction equation: $u_t = \alpha^2 u_{xx}$

# Problem 7 (10 pts)

Assuming that the equation $xe^y + \sin(xy) + y - \ln 2 = 0$ defines $y$ as a differentiable function of $x$, use the Implicit Differentiation Theorem to find the value of $\frac{dy}{dx}$ at the point $(x, y) = (0, \ln 2)$.

# Problem 8 (15 pts)

Find the direction in which $f(x, y) = \sin x + e^{xy}$.
(a) (5 pts) Increases most rapidly at the point $(0, 1)$.

(b) (5 pts) Decreases most rapidly at the point $(0, 1)$.

(c) (5 pts) Does not change (i.e., is flat) at the point $(0, 1)$.

# Problem 9 (15 pts)

Consider the sphere with radius $r > 0$ in 3D, $x^2 + y^2 + z^2 = r^2$.
(a) (7 pts) Find the tangent plane at the point $\left( \frac{r}{\sqrt{3}}, \frac{r}{\sqrt{3}}, \frac{r}{\sqrt{3}} \right)$ on this sphere.

(b) (8 pts) Show that every normal line to this sphere passes through the center of the sphere, i.e., the origin.
Hint: Pick any point $(a, b, c)$ on this sphere, and consider the normal line at that point.

# Problem 10 (15 pts)

Let $f(x, y) = 2x^2 + y^2$.
(a) (8 pts) Find the linearization at the point $(1, 1)$. Then use it to approximate $f(1.1, 0.9)$. Compare the approximate value with the true value.

(b) (7 pts) Approximate $f(2, 2)$ using the same linearization. Compare the approximate value with the true value. At which point is the linear approximation better, $(1.1, 0.9)$ or $(2, 2)$?

# Problem 11 (15 pts)

Consider the following function over the closed domain $D = \{ (x, y) | -\pi \le x \le \pi, -\pi \le y \le \pi \}$: $f(x, y) = x \cos y$ (a) (7 pts) Find the local maxima, local minima, saddle points of $f$ if any.

Use Lagrange multipliers to find the maximum and minimum values of the function $f(x, y) = e^{xy} \; \text{subject to} \; x^2 + y^2 = 1$ Note that we only consider the real values for $x$ and $y$, not the complex values.