The graphs of the circles $x^2 + y^2 = 2$ and $(x-1)^2 + y^2 = 1$ intersect at the points $(1,1)$ and $(1,-1)$. Let $R$ be the shaded region in the first quadrant bounded by the two circles and the $x$-axis.
(a) Set up an expression involving one or more integrals with respect to $x$ that represents the area of $R$.
(b) Set up an expression involving one or more integrals with respect to $y$ that represents the area of $R$.
(c) The polar equations of the circles are $r = \sqrt{2}$ and $r = 2\cos\theta$, respectively. Set up an expression involving one or more integrals with respect to the polar angle $\theta$ that represents the area of $R$.

The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r = \theta + \sin ( 2 \theta )$ for $0 \leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by $\frac { d r } { d \theta } = 1 + 2 \cos ( 2 \theta )$.
(a) Find the area bounded by the curve and the $x$-axis.
(b) Find the angle $\theta$ that corresponds to the point on the curve with $x$-coordinate $-2$.
(c) For $\frac { \pi } { 3 } < \theta < \frac { 2 \pi } { 3 } , \frac { d r } { d \theta }$ is negative. What does this fact say about $r$ ? What does this fact say about the curve?
(d) Find the value of $\theta$ in the interval $0 \leq \theta \leq \frac { \pi } { 2 }$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.

The graphs of the polar curves $r = 2$ and $r = 3 + 2\cos\theta$ are shown in the figure above. The curves intersect when $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$.
(a) Let $R$ be the region that is inside the graph of $r = 2$ and also inside the graph of $r = 3 + 2\cos\theta$, as shaded in the figure above. Find the area of $R$.
(b) A particle moving with nonzero velocity along the polar curve given by $r = 3 + 2\cos\theta$ has position $(x(t), y(t))$ at time $t$, with $\theta = 0$ when $t = 0$. This particle moves along the curve so that $\frac{dr}{dt} = \frac{dr}{d\theta}$. Find the value of $\frac{dr}{dt}$ at $\theta = \frac{\pi}{3}$ and interpret your answer in terms of the motion of the particle.
(c) For the particle described in part (b), $\frac{dy}{dt} = \frac{dy}{d\theta}$. Find the value of $\frac{dy}{dt}$ at $\theta = \frac{\pi}{3}$ and interpret your answer in terms of the motion of the particle.

What is the slope of the line tangent to the polar curve $r = 1 + 2 \sin \theta$ at $\theta = 0$ ?
(A) 2
(B) $\frac { 1 } { 2 }$
(C) 0
(D) $- \frac { 1 } { 2 }$
(E) - 2

The figure above shows the graphs of the polar curves $r = 2 \cos ( 3 \theta )$ and $r = 2$. What is the sum of the areas of the shaded regions?
(A) 0.858
(B) 3.142
(C) 8.566
(D) 9.425
(E) 15.708

The graphs of the polar curves $r = 3$ and $r = 4 - 2 \sin \theta$ are shown in the figure. The curves intersect when $\theta = \frac { \pi } { 6 }$ and $\theta = \frac { 5 \pi } { 6 }$.
(a) Let $S$ be the shaded region that is inside the graph of $r = 3$ and also inside the graph of $r = 4 - 2 \sin \theta$. Find the area of $S$.
(b) A particle moves along the polar curve $r = 4 - 2 \sin \theta$ so that at time $t$ seconds, $\theta = t ^ { 2 }$. Find the time $t$ in the interval $1 \leq t \leq 2$ for which the $x$-coordinate of the particle's position is $-1$.
(c) For the particle described in part (b), find the position vector in terms of $t$. Find the velocity vector at time $t = 1.5$.

The graphs of the polar curves $r = 3$ and $r = 3 - 2 \sin ( 2 \theta )$ are shown in the figure above for $0 \leq \theta \leq \pi$.
(a) Let $R$ be the shaded region that is inside the graph of $r = 3$ and inside the graph of $r = 3 - 2 \sin ( 2 \theta )$. Find the area of $R$.
(b) For the curve $r = 3 - 2 \sin ( 2 \theta )$, find the value of $\frac { d x } { d \theta }$ at $\theta = \frac { \pi } { 6 }$.
(c) The distance between the two curves changes for $0 < \theta < \frac { \pi } { 2 }$. Find the rate at which the distance between the two curves is changing with respect to $\theta$ when $\theta = \frac { \pi } { 3 }$.
(d) A particle is moving along the curve $r = 3 - 2 \sin ( 2 \theta )$ so that $\frac { d \theta } { d t } = 3$ for all times $t \geq 0$. Find the value of $\frac { d r } { d t }$ at $\theta = \frac { \pi } { 6 }$.

The figure shows the polar curves $r = f(\theta) = 1 + \sin\theta\cos(2\theta)$ and $r = g(\theta) = 2\cos\theta$ for $0 \leq \theta \leq \frac{\pi}{2}$. Let $R$ be the region in the first quadrant bounded by the curve $r = f(\theta)$ and the $x$-axis. Let $S$ be the region in the first quadrant bounded by the curve $r = f(\theta)$, the curve $r = g(\theta)$, and the $x$-axis.
(a) Find the area of $R$.
(b) The ray $\theta = k$, where $0 < k < \frac{\pi}{2}$, divides $S$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.
(c) For each $\theta$, $0 \leq \theta \leq \frac{\pi}{2}$, let $w(\theta)$ be the distance between the points with polar coordinates $(f(\theta), \theta)$ and $(g(\theta), \theta)$. Write an expression for $w(\theta)$. Find $w_A$, the average value of $w(\theta)$ over the interval $0 \leq \theta \leq \frac{\pi}{2}$.
(d) Using the information from part (c), find the value of $\theta$ for which $w(\theta) = w_A$. Is the function $w(\theta)$ increasing or decreasing at that value of $\theta$? Give a reason for your answer.

The graphs of the polar curves $r = 4$ and $r = 3 + 2 \cos \theta$ are shown in the figure above. The curves intersect at $\theta = \frac { \pi } { 3 }$ and $\theta = \frac { 5 \pi } { 3 }$.
(a) Let $R$ be the shaded region that is inside the graph of $r = 4$ and also outside the graph of $r = 3 + 2 \cos \theta$, as shown in the figure above. Write an expression involving an integral for the area of $R$.
(b) Find the slope of the line tangent to the graph of $r = 3 + 2 \cos \theta$ at $\theta = \frac { \pi } { 2 }$.
(c) A particle moves along the portion of the curve $r = 3 + 2 \cos \theta$ for $0 < \theta < \frac { \pi } { 2 }$. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle $\theta$ changes with respect to time at the instant when the position of the particle corresponds to $\theta = \frac { \pi } { 3 }$. Indicate units of measure.

Let $S$ be the region bounded by the graph of the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$, as shown in the figure above.
(a) Find the area of $S$.
(b) What is the average distance from the origin to a point on the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$?
(c) There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.
(d) For $k > 0$, let $A ( k )$ be the area of the portion of region $S$ that is also inside the circle $r = k \cos \theta$. Find $\lim _ { k \rightarrow \infty } A ( k )$.

Curve $C$ is defined by the polar equation $r ( \theta ) = 2 \sin ^ { 2 } \theta$ for $0 \leq \theta \leq \pi$. Curve $C$ and the semicircle $r = \frac { 1 } { 2 }$ for $0 \leq \theta \leq \pi$ are shown in the $x y$-plane.
(Note: Your calculator should be in radian mode.)
A. Find the rate of change of $r$ with respect to $\theta$ at the point on curve $C$ where $\theta = 1.3$. Show the setup for your calculations.
B. Find the area of the region that lies inside curve $C$ but outside the graph of the polar equation $r = \frac { 1 } { 2 }$. Show the setup for your calculations.
C. It can be shown that $\frac { d x } { d \theta } = 4 \sin \theta \cos ^ { 2 } \theta - 2 \sin ^ { 3 } \theta$ for curve $C$. For $0 \leq \theta \leq \frac { \pi } { 2 }$, find the value of $\theta$ that corresponds to the point on curve $C$ that is farthest from the $y$-axis. Justify your answer.
D. A particle travels along curve $C$ so that $\frac { d \theta } { d t } = 15$ for all times $t$. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where $\theta = 1.3$. Show the setup for your calculations.

On the coordinate plane, for a natural number $a > 1$, consider the two curves $y = 4 ^ { x } , y = a ^ { - x + 4 }$ and the line $y = 1$. Find the number of values of $a$ such that the number of points with integer coordinates inside or on the boundary of the region enclosed by these curves and line is between 20 and 40 (inclusive). [4 points]

12. In the rectangular coordinate system xOyz, with the coordinate origin as the pole and the positive x-axis as the polar axis, if the polar equation of curve C is $\rho = 3 \sin \theta$, then the rectangular coordinate equation of curve C is $\_\_\_\_$

15. The parametric equation of line $l$ is $\left\{ \begin{array} { c } x = - 1 + t \\ y = 1 + t \end{array} \right.$ (where $t$ is the parameter). With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho ^ { 2 } \cos 2 \theta = 4 \left( \rho > 0 , \frac { 3 \pi } { 4 } < \theta < \frac { 5 \pi } { 4 } \right)$. The polar coordinates of the intersection point of line $l$ and curve $C$ are $\_\_\_\_$ .

16. (Elective 4-4: Coordinate Systems and Parametric Equations)
[Figure]
Figure for Question 15
In the rectangular coordinate system $xOy$, establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of line $l$ is $\rho(\sin\theta - 3\cos\theta) = 0$. The parametric equation of curve $C$ is $\begin{cases} x = t - \frac{1}{t}, \\ y = t + \frac{1}{t} \end{cases}$ (where $t$ is the parameter). If $l$ and $C$ intersect at points $A$ and $B$, then $|AB| = $ $\_\_\_\_$ .
III. Solution Questions: This section has 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps.

16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B } , \overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$, then the eccentricity of $C$ is $\_\_\_\_$.
III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17-21 are required for all students. Questions 22 and 23 are optional; students should choose one to answer. If more than one is answered, only the first one will be graded.
(I) Required Questions: Total 60 points.

16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B }$ and $\overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$ , then the eccentricity of $C$ is $\_\_\_\_$ .
Section III: Solution Questions: Total 70 points. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22 and 23 are optional; choose one to answer.

(I) Required Questions: Total 60 points

22. Solution: (1) Since $-1 < \frac{1-t^2}{1+t^2} \leq 1$ and $x^2 + \left(\frac{y}{2}\right)^2 = \left(\frac{1-t^2}{1+t^2}\right)^2 + \frac{4t^2}{(1+t^2)^2} = 1$, the rectangular coordinate equation of $C$ is $x^2 + \frac{y^2}{4} = 1$ $(x \neq -1)$. The rectangular coordinate equation of $l$ is $2x + \sqrt{3}y + 11 = 0$.
(2) From (1) we can set the parametric equation of $C$ as $\left\{\begin{array}{l} x = \cos\alpha, \\ y = 2\sin\alpha \end{array}\right.$ ($\alpha$ is the parameter, $-\pi < \alpha < \pi$). The distance from a point on $C$ to $l$ is $\frac{|2\cos\alpha + 2\sqrt{3}\sin\alpha + 11|}{\sqrt{7}} = \frac{4\cos\left(\alpha - \frac{\pi}{3}\right) + 11}{\sqrt{7}}$.
When $\alpha = -\frac{2\pi}{3}$, $4\cos\left(\alpha - \frac{\pi}{3}\right) + 11$ attains its minimum value of 7, therefore the minimum distance from a point on $C$ to $l$ is $\sqrt{7}$.

[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is $$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$
(1) When $k = 1$ , what type of curve is $C _ { 1 }$?
(2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .

[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C$ is $\left\{ \begin{array} { l } x = 2 - t - t ^ { 2 } , \\ y = 2 - 3 t + t ^ { 2 } \end{array} ( t \right.$ is a parameter and $t \neq 1 )$. $C$ intersects the coordinate axes at points $A , B$.
(1) Find $| A B |$;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system and find the polar equation of line $A B$ .

Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. The curve $C_a$ in polar coordinates $(\rho, \theta)$ in the frame $\mathcal{R}'$ satisfies $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$ Simplify this equation when $a = 1$. Study and sketch the shape of the curve $C_1$.

Let $$S = \left\{ \left( \theta \sin \frac { \pi \theta } { 1 + \theta } , \frac { 1 } { \theta } \cos \frac { \pi \theta } { 1 + \theta } \right) : \theta \in \mathbb { R } , \theta > 0 \right\}$$ and $$T = \left\{ ( x , y ) : x \in \mathbb { R } , y \in \mathbb { R } , x y = \frac { 1 } { 2 } \right\}$$ How many elements does $S \cap T$ have?
(A) 0
(B) 1
(C) 2
(D) 3