The figure above shows the graph of the polar curve $r = 2 + 4 \sin \theta$. What is the area of the shaded region?
(A) 2.174
(B) 2.739
(C) 13.660
(D) 37.699

23. Which of the following gives the area of the region enclosed by the loop of the graph of the polar curve $r = 4 \cos ( 3 \theta )$ shown in the figure above?
(A) $16 \int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 3 } } \cos ( 3 \theta ) d \theta$
(B) $8 \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } \cos ( 3 \theta ) d \theta$
(C) $8 \int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 3 } } \cos ^ { 2 } ( 3 \theta ) d \theta$
(D) $16 \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } \cos ^ { 2 } ( 3 \theta ) d \theta$
(E) $\quad 8 \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } \cos ^ { 2 } ( 3 \theta ) d \theta$

1988 AP Calculus BC: Section I

1. If $c$ is the number that satisfies the conclusion of the Mean Value Theorem for $f ( x ) = x ^ { 3 } - 2 x ^ { 2 }$ on the interval $0 \leq x \leq 2$, then $c =$
   (A) 0
   (B) $\frac { 1 } { 2 }$
   (C) 1
   (D) $\frac { 4 } { 3 }$
   (E) 2
2. The base of a solid is the region in the first quadrant enclosed by the parabola $y = 4 x ^ { 2 }$, the line $x = 1$, and the $x$-axis. Each plane section of the solid perpendicular to the $x$-axis is a square. The volume of the solid is
   (A) $\frac { 4 \pi } { 3 }$
   (B) $\frac { 16 \pi } { 5 }$
   (C) $\frac { 4 } { 3 }$
   (D) $\frac { 16 } { 5 }$
   (E) $\frac { 64 } { 5 }$
3. If $f$ is a function such that $f ^ { \prime } ( x )$ exists for all $x$ and $f ( x ) > 0$ for all $x$, which of the following is NOT necessarily true?
   (A) $\quad \int _ { - 1 } ^ { 1 } f ( x ) d x > 0$
   (B) $\quad \int _ { - 1 } ^ { 1 } 2 f ( x ) d x = 2 \int _ { - 1 } ^ { 1 } f ( x ) d x$
   (C) $\quad \int _ { - 1 } ^ { 1 } f ( x ) d x = 2 \int _ { 0 } ^ { 1 } f ( x ) d x$
   (D) $\quad \int _ { - 1 } ^ { 1 } f ( x ) d x = - \int _ { 1 } ^ { - 1 } f ( x ) d x$
   (E) $\quad \int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x + \int _ { 0 } ^ { 1 } f ( x ) d x$
4. If the graph of $y = x ^ { 3 } + a x ^ { 2 } + b x - 4$ has a point of inflection at $( 1 , - 6 )$, what is the value of $b$ ?
   (A) - 3
   (B) 0
   (C) 1
   (D) 3
   (E) It cannot be determined from the information given.

1988 AP Calculus BC: Section I

1. $\frac { d } { d x } \ln \left| \cos \left( \frac { \pi } { x } \right) \right|$ is
   (A) $\frac { - \pi } { x ^ { 2 } \cos \left( \frac { \pi } { x } \right) }$
   (B) $- \tan \left( \frac { \pi } { x } \right)$
   (C) $\frac { 1 } { \cos \left( \frac { \pi } { x } \right) }$
   (D) $\frac { \pi } { x } \tan \left( \frac { \pi } { x } \right)$
   (E) $\frac { \pi } { x ^ { 2 } } \tan \left( \frac { \pi } { x } \right)$
2. The region $R$ in the first quadrant is enclosed by the lines $x = 0$ and $y = 5$ and the graph of $y = x ^ { 2 } + 1$. The volume of the solid generated when $R$ is revolved about the $y$-axis is
   (A) $6 \pi$
   (B) $8 \pi$
   (C) $\frac { 34 \pi } { 3 }$
   (D) $16 \pi$
   (E) $\frac { 544 \pi } { 15 }$
3. $\sum _ { i = n } ^ { \infty } \left( \frac { 1 } { 3 } \right) ^ { i } =$
   (A) $\frac { 3 } { 2 } - \left( \frac { 1 } { 3 } \right) ^ { n }$
   (B) $\frac { 3 } { 2 } \left[ 1 - \left( \frac { 1 } { 3 } \right) ^ { n } \right]$
   (C) $\frac { 3 } { 2 } \left( \frac { 1 } { 3 } \right) ^ { n }$
   (D) $\frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { n }$
   (E) $\frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$
4. $\int _ { 0 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x =$
   (A) $\frac { 8 } { 3 }$
   (B) $\frac { 16 } { 3 }$
   (C) $\pi$
   (D) $2 \pi$
   (E) $4 \pi$
5. The general solution of the differential equation $y ^ { \prime } = y + x ^ { 2 }$ is $y =$
   (A) $C e ^ { x }$
   (B) $C e ^ { x } + x ^ { 2 }$
   (C) $- x ^ { 2 } - 2 x - 2 + C$
   (D) $e ^ { x } - x ^ { 2 } - 2 x - 2 + C$
   (E) $\quad C e ^ { x } - x ^ { 2 } - 2 x - 2$

1988 AP Calculus BC: Section I

1. The length of the curve $y = x ^ { 3 }$ from $x = 0$ to $x = 2$ is given by
   (A) $\int _ { 0 } ^ { 2 } \sqrt { 1 + x ^ { 6 } } d x$
   (B) $\int _ { 0 } ^ { 2 } \sqrt { 1 + 3 x ^ { 2 } } d x$
   (C) $\pi \int _ { 0 } ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$
   (D) $2 \pi \int _ { 0 } ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$
   (E) $\int _ { 0 } ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$
2. A curve in the plane is defined parametrically by the equations $x = t ^ { 3 } + t$ and $y = t ^ { 4 } + 2 t ^ { 2 }$. An equation of the line tangent to the curve at $t = 1$ is
   (A) $y = 2 x$
   (B) $y = 8 x$
   (C) $y = 2 x - 1$
   (D) $y = 4 x - 5$
   (E) $y = 8 x + 13$
3. If $k$ is a positive integer, then $\lim _ { x \rightarrow + \infty } \frac { x ^ { k } } { e ^ { x } }$ is
   (A) 0
   (B) 1
   (C) $e$
   (D) $k !$
   (E) nonexistent
4. Let $R$ be the region between the graphs of $y = 1$ and $y = \sin x$ from $x = 0$ to $x = \frac { \pi } { 2 }$. The volume of the solid obtained by revolving $R$ about the $x$-axis is given by
   (A) $2 \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin x d x$
   (B) $2 \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$
   (C) $\pi \int _ { 0 } ^ { \frac { \pi } { 2 } } ( 1 - \sin x ) ^ { 2 } d x$
   (D) $\pi \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } x d x$
   (E) $\quad \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } \left( 1 - \sin ^ { 2 } x \right) d x$
5. A person 2 meters tall walks directly away from a streetlight that is 8 meters above the ground. If the person is walking at a constant rate and the person's shadow is lengthening at the rate of $\frac { 4 } { 9 }$ meter per second, at what rate, in meters per second, is the person walking?
   (A) $\frac { 4 } { 27 }$
   (B) $\frac { 4 } { 9 }$
   (C) $\frac { 3 } { 4 }$
   (D) $\frac { 4 } { 3 }$
   (E) $\frac { 16 } { 9 }$
6. What are all values of $x$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { x ^ { n } } { n }$ converges?
   (A) $- 1 \leq x \leq 1$
   (B) $- 1 < x \leq 1$
   (C) $- 1 \leq x < 1$
   (D) $- 1 < x < 1$
   (E) All real $x$
7. If $\frac { d y } { d x } = y \sec ^ { 2 } x$ and $y = 5$ when $x = 0$, then $y =$
   (A) $e ^ { \tan x } + 4$
   (B) $e ^ { \tan x } + 5$
   (C) $5 e ^ { \tan x }$
   (D) $\quad \tan x + 5$
   (E) $\quad \tan x + 5 e ^ { x }$
8. Let $f$ and $g$ be functions that are differentiable everywhere. If $g$ is the inverse function of $f$ and if $g ( - 2 ) = 5$ and $f ^ { \prime } ( 5 ) = - \frac { 1 } { 2 }$, then $g ^ { \prime } ( - 2 ) =$
   (A) 2
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 1 } { 5 }$
   (D) $- \frac { 1 } { 5 }$
   (E) - 2
9. $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \left[ \sqrt { \frac { 1 } { n } } + \sqrt { \frac { 2 } { n } } + \ldots + \sqrt { \frac { n } { n } } \right] =$
   (A) $\frac { 1 } { 2 } \int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } d x$
   (B) $\int _ { 0 } ^ { 1 } \sqrt { x } d x$
   (C) $\int _ { 0 } ^ { 1 } x d x$
   (D) $\int _ { 1 } ^ { 2 } x d x$
   (E) $\quad 2 \int _ { 1 } ^ { 2 } x \sqrt { x } d x$
10. If $\int _ { 1 } ^ { 4 } f ( x ) d x = 6$, what is the value of $\int _ { 1 } ^ { 4 } f ( 5 - x ) d x$ ?
    (A) 6
    (B) 3
    (C) 0
    (D) - 1
    (E) - 6

1988 AP Calculus BC: Section I

1. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple?
   (A) $\frac { 3 \ln 3 } { \ln 2 }$
   (B) $\frac { 2 \ln 3 } { \ln 2 }$
   (C) $\frac { \ln 3 } { \ln 2 }$
   (D) $\quad \ln \left( \frac { 27 } { 2 } \right)$
   (E) $\quad \ln \left( \frac { 9 } { 2 } \right)$
2. Which of the following series converge? I. $\quad \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { 2 n + 1 }$ II. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n } \left( \frac { 3 } { 2 } \right) ^ { n }$ III. $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$
   (A) I only
   (B) II only
   (C) III only
   (D) I and III only
   (E) I, II, and III
3. What is the area of the largest rectangle that can be inscribed in the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ ?
   (A) $6 \sqrt { 2 }$
   (B) 12
   (C) 24
   (D) $24 \sqrt { 2 }$
   (E) 36

1993 AP Calculus AB: Section I

90 Minutes-Scientific Calculator

Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

1. If $f ( x ) = x ^ { \frac { 3 } { 2 } }$, then $f ^ { \prime } ( 4 ) =$
   (A) - 6
   (B) - 3
   (C) 3
   (D) 6
   (E) 8 [Figure]
2. Which of the following represents the area of the shaded region in the figure above?
   (A) $\quad \int _ { c } ^ { d } f ( y ) d y$
   (B) $\quad \int _ { a } ^ { b } ( d - f ( x ) ) d x$
   (C) $f ^ { \prime } ( b ) - f ^ { \prime } ( a )$
   (D) $( b - a ) [ f ( b ) - f ( a ) ]$
   (E) $( d - c ) [ f ( b ) - f ( a ) ]$
3. $\lim _ { n \rightarrow \infty } \frac { 3 n ^ { 3 } - 5 n } { n ^ { 3 } - 2 n ^ { 2 } + 1 }$ is
   (A) - 5
   (B) - 2
   (C) 1
   (D) 3
   (E) nonexistent
4. If $x ^ { 3 } + 3 x y + 2 y ^ { 3 } = 17$, then in terms of $x$ and $y , \frac { d y } { d x } =$
   (A) $- \frac { x ^ { 2 } + y } { x + 2 y ^ { 2 } }$
   (B) $- \frac { x ^ { 2 } + y } { x + y ^ { 2 } }$
   (C) $- \frac { x ^ { 2 } + y } { x + 2 y }$
   (D) $- \frac { x ^ { 2 } + y } { 2 y ^ { 2 } }$
   (E) $\frac { - x ^ { 2 } } { 1 + 2 y ^ { 2 } }$
5. If the function $f$ is continuous for all real numbers and if $f ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }$ when $x \neq - 2$, then $f ( - 2 ) =$
   (A) - 4
   (B) - 2
   (C) - 1
   (D) 0
   (E) 2
6. The area of the region enclosed by the curve $y = \frac { 1 } { x - 1 }$, the $x$-axis, and the lines $x = 3$ and $x = 4$ is
   (A) $\frac { 5 } { 36 }$
   (B) $\ln \frac { 2 } { 3 }$
   (C) $\ln \frac { 4 } { 3 }$
   (D) $\quad \ln \frac { 3 } { 2 }$
   (E) $\quad \ln 6$
7. An equation of the line tangent to the graph of $y = \frac { 2 x + 3 } { 3 x - 2 }$ at the point $( 1,5 )$ is
   (A) $13 x - y = 8$
   (B) $13 x + y = 18$
   (C) $x - 13 y = 64$
   (D) $x + 13 y = 66$
   (E) $- 2 x + 3 y = 13$

1993 AP Calculus AB: Section I

1. If $y = \tan x - \cot x$, then $\frac { d y } { d x } =$
   (A) $\sec x \csc x$
   (B) $\sec x - \csc x$
   (C) $\sec x + \csc x$
   (D) $\sec ^ { 2 } x - \csc ^ { 2 } x$
   (E) $\sec ^ { 2 } x + \csc ^ { 2 } x$
2. If $h$ is the function given by $h ( x ) = f ( g ( x ) )$, where $f ( x ) = 3 x ^ { 2 } - 1$ and $g ( x ) = | x |$, then $h ( x ) =$
   (A) $\quad 3 x ^ { 3 } - | x |$
   (B) $\left| 3 x ^ { 2 } - 1 \right|$
   (C) $3 x ^ { 2 } | x | - 1$
   (D) $3 | x | - 1$
   (E) $\quad 3 x ^ { 2 } - 1$
3. If $f ( x ) = ( x - 1 ) ^ { 2 } \sin x$, then $f ^ { \prime } ( 0 ) =$
   (A) - 2
   (B) - 1
   (C) 0
   (D) 1
   (E) 2
4. The acceleration of a particle moving along the $x$-axis at time $t$ is given by $a ( t ) = 6 t - 2$. If the velocity is 25 when $t = 3$ and the position is 10 when $t = 1$, then the position $x ( t ) =$
   (A) $\quad 9 t ^ { 2 } + 1$
   (B) $3 t ^ { 2 } - 2 t + 4$
   (C) $t ^ { 3 } - t ^ { 2 } + 4 t + 6$
   (D) $t ^ { 3 } - t ^ { 2 } + 9 t - 20$
   (E) $36 t ^ { 3 } - 4 t ^ { 2 } - 77 t + 55$
5. If $f$ and $g$ are continuous functions, and if $f ( x ) \geq 0$ for all real numbers $x$, which of the following must be true? I. $\quad \int _ { a } ^ { b } f ( x ) g ( x ) d x = \left( \int _ { a } ^ { b } f ( x ) d x \right) \left( \int _ { a } ^ { b } g ( x ) d x \right)$ II. $\quad \int _ { a } ^ { b } ( f ( x ) + g ( x ) ) d x = \int _ { a } ^ { b } f ( x ) d x + \int _ { a } ^ { b } g ( x ) d x$ III. $\quad \int _ { a } ^ { b } \sqrt { f ( x ) } d x = \sqrt { \int _ { a } ^ { b } f ( x ) d x }$
   (A) I only
   (B) II only
   (C) III only
   (D) II and III only
   (E) I, II, and III

1993 AP Calculus AB: Section I

1. The fundamental period of $2 \cos ( 3 x )$ is
   (A) $\frac { 2 \pi } { 3 }$
   (B) $2 \pi$
   (C) $6 \pi$
   (D) 2
   (E) 3
2. $\int \frac { 3 x ^ { 2 } } { \sqrt { x ^ { 3 } + 1 } } d x =$
   (A) $2 \sqrt { x ^ { 3 } + 1 } + C$
   (B) $\frac { 3 } { 2 } \sqrt { x ^ { 3 } + 1 } + C$
   (C) $\sqrt { x ^ { 3 } + 1 } + C$
   (D) $\quad \ln \sqrt { x ^ { 3 } + 1 } + C$
   (E) $\quad \ln \left( x ^ { 3 } + 1 \right) + C$
3. For what value of $x$ does the function $f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$ have a relative maximum?
   (A) - 3
   (B) $- \frac { 7 } { 3 }$
   (C) $- \frac { 5 } { 2 }$
   (D) $\frac { 7 } { 3 }$
   (E) $\frac { 5 } { 2 }$
4. The slope of the line normal to the graph of $y = 2 \ln ( \sec x )$ at $x = \frac { \pi } { 4 }$ is
   (A) - 2
   (B) $- \frac { 1 } { 2 }$
   (C) $\frac { 1 } { 2 }$
   (D) 2
   (E) nonexistent
5. $\int \left( x ^ { 2 } + 1 \right) ^ { 2 } d x =$
   (A) $\frac { \left( x ^ { 2 } + 1 \right) ^ { 3 } } { 3 } + C$
   (B) $\frac { \left( x ^ { 2 } + 1 \right) ^ { 3 } } { 6 x } + C$
   (C) $\left( \frac { x ^ { 3 } } { 3 } + x \right) ^ { 2 } + C$
   (D) $\frac { 2 x \left( x ^ { 2 } + 1 \right) ^ { 3 } } { 3 } + C$
   (E) $\frac { x ^ { 5 } } { 5 } + \frac { 2 x ^ { 3 } } { 3 } + x + C$
6. If $f ( x ) = \sin \left( \frac { x } { 2 } \right)$, then there exists a number $c$ in the interval $\frac { \pi } { 2 } < x < \frac { 3 \pi } { 2 }$ that satisfies the conclusion of the Mean Value Theorem. Which of the following could be $c$ ?
   (A) $\frac { 2 \pi } { 3 }$
   (B) $\frac { 3 \pi } { 4 }$
   (C) $\frac { 5 \pi } { 6 }$
   (D) $\pi$
   (E) $\frac { 3 \pi } { 2 }$
7. Let $f$ be the function defined by $f ( x ) = \left\{ \begin{array} { l l } x ^ { 3 } & \text { for } x \leq 0 , \\ x & \text { for } x > 0 . \end{array} \right.$ Which of the following statements about $f$ is true?
   (A) $\quad f$ is an odd function.
   (B) $f$ is discontinuous at $x = 0$.
   (C) $f$ has a relative maximum.
   (D) $\quad f ^ { \prime } ( 0 ) = 0$
   (E) $\quad f ^ { \prime } ( x ) > 0$ for $x \neq 0$

1993 AP Calculus AB: Section I

1. Let $R$ be the region in the first quadrant enclosed by the graph of $y = ( x + 1 ) ^ { \frac { 1 } { 3 } }$, the line $x = 7$, the $x$-axis, and the $y$-axis. The volume of the solid generated when $R$ is revolved about the $y$-axis is given by
   (A) $\pi \int _ { 0 } ^ { 7 } ( x + 1 ) ^ { \frac { 2 } { 3 } } d x$
   (B) $2 \pi \int _ { 0 } ^ { 7 } x ( x + 1 ) ^ { \frac { 1 } { 3 } } d x$
   (C) $\pi \int _ { 0 } ^ { 2 } ( x + 1 ) ^ { \frac { 2 } { 3 } } d x$
   (D) $2 \pi \int _ { 0 } ^ { 2 } x ( x + 1 ) ^ { \frac { 1 } { 3 } } d x$
   (E) $\pi \int _ { 0 } ^ { 7 } \left( y ^ { 3 } - 1 \right) ^ { 2 } d y$
2. At what value of $x$ does the graph of $y = \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 3 } }$ have a point of inflection?
   (A) 0
   (B) 1
   (C) 2
   (D) 3
   (E) At no value of $x$
3. An antiderivative for $\frac { 1 } { x ^ { 2 } - 2 x + 2 }$ is
   (A) $\quad - \left( x ^ { 2 } - 2 x + 2 \right) ^ { - 2 }$
   (B) $\quad \ln \left( x ^ { 2 } - 2 x + 2 \right)$
   (C) $\quad \ln \left| \frac { x - 2 } { x + 1 } \right|$
   (D) $\quad \operatorname { arcsec } ( x - 1 )$
   (E) $\quad \arctan ( x - 1 )$
4. How many critical points does the function $f ( x ) = ( x + 2 ) ^ { 5 } ( x - 3 ) ^ { 4 }$ have?
   (A) One
   (B) Two
   (C) Three
   (D) Five
   (E) Nine
5. If $f ( x ) = \left( x ^ { 2 } - 2 x - 1 \right) ^ { \frac { 2 } { 3 } }$, then $f ^ { \prime } ( 0 )$ is
   (A) $\frac { 4 } { 3 }$
   (B) 0
   (C) $- \frac { 2 } { 3 }$
   (D) $- \frac { 4 } { 3 }$
   (E) - 2

1993 AP Calculus AB: Section I

1. $\frac { d } { d x } \left( 2 ^ { x } \right) =$
   (A) $2 ^ { x - 1 }$
   (B) $\left( 2 ^ { x - 1 } \right) x$
   (C) $\left( 2 ^ { x } \right) \ln 2$
   (D) $\left( 2 ^ { x - 1 } \right) \ln 2$
   (E) $\frac { 2 x } { \ln 2 }$
2. A particle moves along a line so that at time $t$, where $0 \leq t \leq \pi$, its position is given by $s ( t ) = - 4 \cos t - \frac { t ^ { 2 } } { 2 } + 10$. What is the velocity of the particle when its acceleration is zero?
   (A) $\quad - 5.19$
   (B) 0.74
   (C) 1.32
   (D) 2.55
   (E) 8.13
3. The function $f$ given by $f ( x ) = x ^ { 3 } + 12 x - 24$ is
   (A) increasing for $x < - 2$, decreasing for $- 2 < x < 2$, increasing for $x > 2$
   (B) decreasing for $x < 0$, increasing for $x > 0$
   (C) increasing for all $x$
   (D) decreasing for all $x$
   (E) decreasing for $x < - 2$, increasing for $- 2 < x < 2$, decreasing for $x > 2$
4. $\int _ { 1 } ^ { 500 } \left( 13 ^ { x } - 11 ^ { x } \right) d x + \int _ { 2 } ^ { 500 } \left( 11 ^ { x } - 13 ^ { x } \right) d x =$
   (A) 0.000
   (B) 14.946
   (C) 34.415
   (D) 46.000
   (E) 136.364
5. $\lim _ { \theta \rightarrow 0 } \frac { 1 - \cos \theta } { 2 \sin ^ { 2 } \theta }$ is
   (A) 0
   (B) $\frac { 1 } { 8 }$
   (C) $\frac { 1 } { 4 }$
   (D) 1
   (E) nonexistent
6. The region enclosed by the $x$-axis, the line $x = 3$, and the curve $y = \sqrt { x }$ is rotated about the $x$-axis. What is the volume of the solid generated?
   (A) $3 \pi$
   (B) $2 \sqrt { 3 } \pi$
   (C) $\frac { 9 } { 2 } \pi$
   (D) $9 \pi$
   (E) $\frac { 36 \sqrt { 3 } } { 5 } \pi$

1993 AP Calculus AB: Section I

1. If $f ( x ) = e ^ { 3 \ln \left( x ^ { 2 } \right) }$, then $f ^ { \prime } ( x ) =$
   (A) $e ^ { 3 \ln \left( x ^ { 2 } \right) }$
   (B) $\frac { 3 } { x ^ { 2 } } e ^ { 3 \ln \left( x ^ { 2 } \right) }$
   (C) $6 ( \ln x ) e ^ { 3 \ln \left( x ^ { 2 } \right) }$
   (D) $5 x ^ { 4 }$
   (E) $6 x ^ { 5 }$
2. $\int _ { 0 } ^ { \sqrt { 3 } } \frac { d x } { \sqrt { 4 - x ^ { 2 } } } =$
   (A) $\frac { \pi } { 3 }$
   (B) $\frac { \pi } { 4 }$
   (C) $\frac { \pi } { 6 }$
   (D) $\frac { 1 } { 2 } \ln 2$
   (E) $- \ln 2$
3. If $\frac { d y } { d x } = 2 y ^ { 2 }$ and if $y = - 1$ when $x = 1$, then when $x = 2 , y =$
   (A) $- \frac { 2 } { 3 }$
   (B) $- \frac { 1 } { 3 }$
   (C) 0
   (D) $\frac { 1 } { 3 }$
   (E) $\frac { 2 } { 3 }$
4. The top of a 25 -foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?
   (A) $- \frac { 7 } { 8 }$ feet per minute
   (B) $- \frac { 7 } { 24 }$ feet per minute
   (C) $\frac { 7 } { 24 }$ feet per minute
   (D) $\frac { 7 } { 8 }$ feet per minute
   (E) $\frac { 21 } { 25 }$ feet per minute
5. If the graph of $y = \frac { a x + b } { x + c }$ has a horizontal asymptote $y = 2$ and a vertical asymptote $x = - 3$, then $a + c =$
   (A) - 5
   (B) - 1
   (C) 0
   (D) 1
   (E) 5

1993 AP Calculus AB: Section I

1. If the definite integral $\int _ { 0 } ^ { 2 } e ^ { x ^ { 2 } } d x$ is first approximated by using two inscribed rectangles of equal width and then approximated by using the trapezoidal rule with $n = 2$, the difference between the two approximations is
   (A) 53.60
   (B) 30.51
   (C) 27.80
   (D) 26.80
   (E) 12.78
2. If $f$ is a differentiable function, then $f ^ { \prime } ( a )$ is given by which of the following? I. $\quad \lim _ { h \rightarrow 0 } \frac { f ( a + h ) - f ( a ) } { h }$ II. $\lim _ { x \rightarrow a } \frac { f ( x ) - f ( a ) } { x - a }$ III. $\quad \lim _ { x \rightarrow a } \frac { f ( x + h ) - f ( x ) } { h }$
   (A) I only
   (B) II only
   (C) I and II only
   (D) I and III only
   (E) I, II, and III
3. If the second derivative of $f$ is given by $f ^ { \prime \prime } ( x ) = 2 x - \cos x$, which of the following could be $f ( x )$ ?
   (A) $\frac { x ^ { 3 } } { 3 } + \cos x - x + 1$
   (B) $\frac { x ^ { 3 } } { 3 } - \cos x - x + 1$
   (C) $x ^ { 3 } + \cos x - x + 1$
   (D) $x ^ { 2 } - \sin x + 1$
   (E) $\quad x ^ { 2 } + \sin x + 1$
4. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is
   (A) $\frac { 1 } { \pi }$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 2 } { \pi }$
   (D) 1
   (E) 2 [Figure]
5. The graph of $y = f ( x )$ is shown in the figure above. Which of the following could be the graph of $y = f ( | x | )$ ?
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]
   (E) [Figure]
6. $\frac { d } { d x } \int _ { 0 } ^ { x } \cos ( 2 \pi u ) d u$ is
   (A) 0
   (B) $\frac { 1 } { 2 \pi } \sin x$
   (C) $\frac { 1 } { 2 \pi } \cos ( 2 \pi x )$
   (D) $\cos ( 2 \pi x )$
   (E) $\quad 2 \pi \cos ( 2 \pi x )$
7. A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 3 months old?
   (A) 4.2 pounds
   (B) 4.6 pounds
   (C) 4.8 pounds
   (D) 5.6 pounds
   (E) 6.5 pounds

1993 AP Calculus AB: Section I

1. $\int x f ( x ) d x =$
   (A) $x f ( x ) - \int x f ^ { \prime } ( x ) d x$
   (B) $\frac { x ^ { 2 } } { 2 } f ( x ) - \int \frac { x ^ { 2 } } { 2 } f ^ { \prime } ( x ) d x$
   (C) $x f ( x ) - \frac { x ^ { 2 } } { 2 } f ( x ) + C$
   (D) $x f ( x ) - \int f ^ { \prime } ( x ) d x$
   (E) $\frac { x ^ { 2 } } { 2 } \int f ( x ) d x$
2. What is the minimum value of $f ( x ) = x \ln x$ ?
   (A) $- e$
   (B) - 1
   (C) $- \frac { 1 } { e }$
   (D) 0
   (E) $f ( x )$ has no minimum value.
3. If Newton's method is used to approximate the real root of $x ^ { 3 } + x - 1 = 0$, then a first approximation $x _ { 1 } = 1$ would lead to a third approximation of $x _ { 3 } =$
   (A) 0.682
   (B) 0.686
   (C) 0.694
   (D) 0.750
   (E) 1.637

1993 AP Calculus BC: Section I

90 Minutes-Scientific Calculator

Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

1. The area of the region enclosed by the graphs of $y = x ^ { 2 }$ and $y = x$ is
   (A) $\frac { 1 } { 6 }$
   (B) $\frac { 1 } { 3 }$
   (C) $\frac { 1 } { 2 }$
   (D) $\frac { 5 } { 6 }$
   (E) 1
2. If $f ( x ) = 2 x ^ { 2 } + 1$, then $\lim _ { x \rightarrow 0 } \frac { f ( x ) - f ( 0 ) } { x ^ { 2 } }$ is
   (A) 0
   (B) 1
   (C) 2
   (D) 4
   (E) nonexistent
3. If $p$ is a polynomial of degree $n , n > 0$, what is the degree of the polynomial $Q ( x ) = \int _ { 0 } ^ { x } p ( t ) d t$ ?
   (A) 0
   (B) 1
   (C) $\quad n - 1$
   (D) $n$
   (E) $n + 1$
4. A particle moves along the curve $x y = 10$. If $x = 2$ and $\frac { d y } { d t } = 3$, what is the value of $\frac { d x } { d t }$ ?
   (A) $- \frac { 5 } { 2 }$
   (B) $- \frac { 6 } { 5 }$
   (C) 0
   (D) $\frac { 4 } { 5 }$
   (E) $\frac { 6 } { 5 }$
5. Which of the following represents the graph of the polar curve $r = 2 \sec \theta$ ?
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]
   (E) [Figure]
6. If $x = t ^ { 2 } + 1$ and $y = t ^ { 3 }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } =$
   (A) $\frac { 3 } { 4 t }$
   (B) $\frac { 3 } { 2 t }$
   (C) $3 t$
   (D) $6 t$
   (E) $\frac { 3 } { 2 }$
7. $\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x ^ { 4 } } d x =$
   (A) $\frac { 1 } { 4 } ( e - 1 )$
   (B) $\frac { 1 } { 4 } e$
   (C) $e - 1$
   (D) $e$
   (E) $4 ( e - 1 )$
8. If $f ( x ) = \ln \left( e ^ { 2 x } \right)$, then $f ^ { \prime } ( x ) =$
   (A) 1
   (B) 2
   (C) $2 x$
   (D) $e ^ { - 2 x }$
   (E) $2 e ^ { - 2 x }$
9. If $f ( x ) = 1 + x ^ { \frac { 2 } { 3 } }$, which of the following is NOT true?
   (A) $\quad f$ is continuous for all real numbers.
   (B) $f$ has a minimum at $x = 0$.
   (C) $f$ is increasing for $x > 0$.
   (D) $f ^ { \prime } ( x )$ exists for all $x$.
   (E) $\quad f ^ { \prime \prime } ( x )$ is negative for $x > 0$.
10. Which of the following functions are continuous at $x = 1$ ? I. $\ln x$ II. $e ^ { x }$ III. $\ln \left( e ^ { x } - 1 \right)$
    (A) I only
    (B) II only
    (C) I and II only
    (D) II and III only
    (E) I, II, and III
11. $\int _ { 4 } ^ { \infty } \frac { - 2 x } { \sqrt [ 3 ] { 9 - x ^ { 2 } } } d x$ is
    (A) $7 ^ { \frac { 2 } { 3 } }$
    (B) $\frac { 3 } { 2 } \left( 7 ^ { \frac { 2 } { 3 } } \right)$
    (C) $9 ^ { \frac { 2 } { 3 } } + 7 ^ { \frac { 2 } { 3 } }$
    (D) $\frac { 3 } { 2 } \left( 9 ^ { \frac { 2 } { 3 } } + 7 ^ { \frac { 2 } { 3 } } \right)$
    (E) nonexistent
12. The position of a particle moving along the $x$-axis is $x ( t ) = \sin ( 2 t ) - \cos ( 3 t )$ for time $t \geq 0$. When $t = \pi$, the acceleration of the particle is
    (A) 9
    (B) $\frac { 1 } { 9 }$
    (C) 0
    (D) $- \frac { 1 } { 9 }$
    (E) - 9
13. If $\frac { d y } { d x } = x ^ { 2 } y$, then $y$ could be
    (A) $3 \ln \left( \frac { x } { 3 } \right)$
    (B) $e ^ { \frac { x ^ { 3 } } { 3 } } + 7$
    (C) $2 e ^ { \frac { x ^ { 3 } } { 3 } }$
    (D) $3 e ^ { 2 x }$
    (E) $\frac { x ^ { 3 } } { 3 } + 1$

1993 AP Calculus BC: Section I

1. The derivative of $f$ is $x ^ { 4 } ( x - 2 ) ( x + 3 )$. At how many points will the graph of $f$ have a relative maximum?
   (A) None
   (B) One
   (C) Two
   (D) Three
   (E) Four
2. If $f ( x ) = e ^ { \tan ^ { 2 } x }$, then $f ^ { \prime } ( x ) =$
   (A) $e ^ { \tan ^ { 2 } x }$
   (B) $\sec ^ { 2 } x e ^ { \tan ^ { 2 } x }$
   (C) $\tan ^ { 2 } x e ^ { \tan ^ { 2 } x - 1 }$
   (D) $2 \tan x \sec ^ { 2 } x e ^ { \tan ^ { 2 } x }$
   (E) $\quad 2 \tan x e ^ { \tan ^ { 2 } x }$
3. Which of the following series diverge? I. $\quad \sum _ { k = 3 } ^ { \infty } \frac { 2 } { k ^ { 2 } + 1 }$ II. $\sum _ { k = 1 } ^ { \infty } \left( \frac { 6 } { 7 } \right) ^ { k }$ III. $\sum _ { k = 2 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k }$
   (A) None
   (B) II only
   (C) III only
   (D) I and III
   (E) II and III
4. The slope of the line tangent to the graph of $\ln ( x y ) = x$ at the point where $x = 1$ is
   (A) 0
   (B) 1
   (C) $e$
   (D) $e ^ { 2 }$
   (E) $1 - e$
5. If $e ^ { f ( x ) } = 1 + x ^ { 2 }$, then $f ^ { \prime } ( x ) =$
   (A) $\frac { 1 } { 1 + x ^ { 2 } }$
   (B) $\frac { 2 x } { 1 + x ^ { 2 } }$
   (C) $2 x \left( 1 + x ^ { 2 } \right)$
   (D) $\quad 2 x \left( e ^ { 1 + x ^ { 2 } } \right)$
   (E) $\quad 2 x \ln \left( 1 + x ^ { 2 } \right)$ [Figure]
6. The shaded region $R$, shown in the figure above, is rotated about the $y$-axis to form a solid whose volume is 10 cubic units. Of the following, which best approximates $k$ ?
   (A) 1.51
   (B) 2.09
   (C) 2.49
   (D) 4.18
   (E) 4.77
7. A particle moves along the $x$-axis so that at any time $t \geq 0$ the acceleration of the particle is $a ( t ) = e ^ { - 2 t }$. If at $t = 0$ the velocity of the particle is $\frac { 5 } { 2 }$ and its position is $\frac { 17 } { 4 }$, then its position at any time $t > 0$ is $x ( t ) =$
   (A) $- \frac { e ^ { - 2 t } } { 2 } + 3$
   (B) $\frac { e ^ { - 2 t } } { 4 } + 4$
   (C) $\quad 4 e ^ { - 2 t } + \frac { 9 } { 2 } t + \frac { 1 } { 4 }$
   (D) $\frac { e ^ { - 2 t } } { 2 } + 3 t + \frac { 15 } { 4 }$
   (E) $\frac { e ^ { - 2 t } } { 4 } + 3 t + 4$
8. The value of the derivative of $y = \frac { \sqrt [ 3 ] { x ^ { 2 } + 8 } } { \sqrt [ 4 ] { 2 x + 1 } }$ at $x = 0$ is
   (A) - 1
   (B) $- \frac { 1 } { 2 }$
   (C) 0
   (D) $\frac { 1 } { 2 }$
   (E) 1

1993 AP Calculus BC: Section I

1. If $f ( x ) = x ^ { 2 } e ^ { x }$, then the graph of $f$ is decreasing for all $x$ such that
   (A) $\quad x < - 2$
   (B) $- 2 < x < 0$
   (C) $x > - 2$
   (D) $x < 0$
   (E) $\quad x > 0$
2. The length of the curve determined by the equations $x = t ^ { 2 }$ and $y = t$ from $t = 0$ to $t = 4$ is
   (A) $\quad \int _ { 0 } ^ { 4 } \sqrt { 4 t + 1 } d t$
   (B) $2 \int _ { 0 } ^ { 4 } \sqrt { t ^ { 2 } + 1 } d t$
   (C) $\int _ { 0 } ^ { 4 } \sqrt { 2 t ^ { 2 } + 1 } d t$
   (D) $\int _ { 0 } ^ { 4 } \sqrt { 4 t ^ { 2 } + 1 } d t$
   (E) $2 \pi \int _ { 0 } ^ { 4 } \sqrt { 4 t ^ { 2 } + 1 } d t$
3. Let $f$ and $g$ be functions that are differentiable for all real numbers, with $g ( x ) \neq 0$ for $x \neq 0$.

If $\lim _ { x \rightarrow 0 } f ( x ) = \lim _ { x \rightarrow 0 } g ( x ) = 0$ and $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) }$ exists, then $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { g ( x ) }$ is
(A) 0
(B) $\frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) }$
(C) $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) }$
(D) $\frac { f ^ { \prime } ( x ) g ( x ) - f ( x ) g ^ { \prime } ( x ) } { ( f ( x ) ) ^ { 2 } }$
(E) nonexistent

40. The area of the region enclosed by the polar curve $r = 1 - \cos \theta$ is
(A) $\frac { 3 } { 4 } \pi$
(B) $\pi$
(C) $\frac { 3 } { 2 } \pi$
(D) $2 \pi$
(E) $3 \pi$

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.

The graph of the polar curve $r = 1 - 2 \cos \theta$ for $0 \leq \theta \leq \pi$ is shown above. Let $S$ be the shaded region in the third quadrant bounded by the curve and the $x$-axis.
(a) Write an integral expression for the area of $S$.
(b) Write expressions for $\frac { d x } { d \theta }$ and $\frac { d y } { d \theta }$ in terms of $\theta$.
(c) Write an equation in terms of $x$ and $y$ for the line tangent to the graph of the polar curve at the point where $\theta = \frac { \pi } { 2 }$. Show the computations that lead to your answer.

The polar curve $r$ is given by $r ( \theta ) = 3 \theta + \sin \theta$, where $0 \leq \theta \leq 2 \pi$.
(a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of $r$.
(b) For $\frac { \pi } { 2 } \leq \theta \leq \pi$, there is one point $P$ on the polar curve $r$ with $x$-coordinate - 3 . Find the angle $\theta$ that corresponds to point $P$. Find the $y$-coordinate of point $P$. Show the work that leads to your answers.
(c) A particle is traveling along the polar curve $r$ so that its position at time $t$ is $( x ( t ) , y ( t ) )$ and such that $\frac { d \theta } { d t } = 2$. Find $\frac { d y } { d t }$ at the instant that $\theta = \frac { 2 \pi } { 3 }$, and interpret the meaning of your answer in the context of the problem.

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.

As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
Under a rotation transformation by angle $\theta$ about the origin, the line $l$ is mapped to a line passing through the focus F of the hyperbola $C$. What is the value of $\sin 2 \theta$? [4 points]
(1) $- \frac { 2 } { 3 }$
(2) $- \frac { 5 } { 9 }$
(3) $- \frac { 4 } { 9 }$
(4) $- \frac { 1 } { 3 }$
(5) $- \frac { 2 } { 9 }$

7. In the polar coordinate system, the distance from point $M \left( 4 , \frac { \pi } { 3 } \right)$ to the line $l : \rho ( 2 \cos \theta + \sin \theta ) = 4$ is $d =$ $\_\_\_\_$.

11. In the polar coordinate system, the distance from the point $\left( 2 , \frac { \pi } { 3 } \right)$ to the line $\rho ( \cos \theta + \sqrt { 3 } \sin \theta ) = 6$ 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.

22. Solution: (1) From the given conditions, the polar coordinate equations of the circles containing arcs $AB$, $BC$, $CD$ are $\rho = 2\cos\theta$, $\rho = 2\sin\theta$, $\rho = -2\cos\theta$ respectively.
Thus the polar coordinate equation of $M_1$ is $\rho = 2\cos\theta \left(0 \leq \theta \leq \frac{\pi}{4}\right)$, the polar coordinate equation of $M_2$ is $\rho = 2\sin\theta \left(\frac{\pi}{4} \leq \theta \leq \frac{3\pi}{4}\right)$,
the polar coordinate equation of $M_3$ is $\rho = -2\cos\theta \left(\frac{3\pi}{4} \leq \theta \leq \pi\right)$.
(2) Let $P(\rho, \theta)$. From the given conditions and (1), we have
If $0 \leq \theta \leq \frac{\pi}{4}$, then $2\cos\theta = \sqrt{3}$, solving gives $\theta = \frac{\pi}{6}$;
If $\frac{\pi}{4} \leq \theta \leq \frac{3\pi}{4}$, then $2\sin\theta = \sqrt{3}$, solving gives $\theta = \frac{\pi}{3}$ or $\theta = \frac{2\pi}{3}$;
If $\frac{3\pi}{4} \leq \theta \leq \pi$, then $-2\cos\theta = \sqrt{3}$, solving gives $\theta = \frac{5\pi}{6}$.
In summary, the polar coordinates of $P$ are $\left(\sqrt{3}, \frac{\pi}{6}\right)$ or $\left(\sqrt{3}, \frac{\pi}{3}\right)$ or $\left(\sqrt{3}, \frac{2\pi}{3}\right)$ or $\left(\sqrt{3}, \frac{5\pi}{6}\right)$.

[Elective 4-4: Coordinate Systems and Parametric Equations]
Given $P(2,1)$ and line $l : \left\{ \begin{array}{l} x = 2 + t\cos\alpha \\ y = 1 + t\sin\alpha \end{array} \right.$ ($t$ is a parameter). Line $l$ intersects the positive $x$-axis and positive $y$-axis at points $A , B$ respectively, with $|PA| \cdot |PB| = 4$ .
(1) Find the value of $\alpha$ ;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, find the polar equation of line $l$ .