The graph of a differentiable function $f$ is shown above for $- 3 \leq x \leq 3$. The graph of $f$ has horizontal tangent lines at $x = - 1 , x = 1$, and $x = 2$. The areas of regions $A , B , C$, and $D$ are 5, 4, 5, and 3, respectively. Let $g$ be the antiderivative of $f$ such that $g ( 3 ) = 7$.
(a) Find all values of $x$ on the open interval $- 3 < x < 3$ for which the function $g$ has a relative maximum. Justify your answer.
(b) On what open intervals contained in $- 3 < x < 3$ is the graph of $g$ concave up? Give a reason for your answer.
(c) Find the value of $\lim _ { x \rightarrow 0 } \frac { g ( x ) + 1 } { 2 x }$, or state that it does not exist. Show the work that leads to your answer.
(d) Let $h$ be the function defined by $h ( x ) = 3 f ( 2 x + 1 ) + 4$. Find the value of $\int _ { - 2 } ^ { 1 } h ( x ) d x$.

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $[ 0,4 ]$. The areas of the regions bounded by the graph of $f ^ { \prime }$ and the $x$-axis on the intervals $[ 0,1 ] , [ 1,2 ] , [ 2,3 ]$, and $[ 3,4 ]$ are $2, 6, 10$, and $14$, respectively. The graph of $f ^ { \prime }$ has horizontal tangents at $x = 0.6 , x = 1.6$, $x = 2.5$, and $x = 3.5$. It is known that $f ( 2 ) = 5$.
(a) On what open intervals contained in $( 0,4 )$ is the graph of $f$ both decreasing and concave down? Give a reason for your answer.
(b) Find the absolute minimum value of $f$ on the interval $[ 0,4 ]$. Justify your answer.
(c) Evaluate $\int _ { 0 } ^ { 4 } f ( x ) f ^ { \prime } ( x ) \, d x$.
(d) The function $g$ is defined by $g ( x ) = x ^ { 3 } f ( x )$. Find $g ^ { \prime } ( 2 )$. Show the work that leads to your answer.

The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = e ^ { \sin x } - \cos x - 1$ for $0 < x < 9$. On what intervals is $f$ decreasing?
(A) $0 < x < 0.633$ and $4.115 < x < 6.916$
(B) $0 < x < 1.947$ and $5.744 < x < 8.230$
(C) $0.633 < x < 4.115$ and $6.916 < x < 9$
(D) $1.947 < x < 5.744$ and $8.230 < x < 9$

$x$	- 1	0	2	4	5
$f ^ { \prime } ( x )$	11	9	8	5	2

Let $f$ be a twice-differentiable function. Values of $f ^ { \prime }$, the derivative of $f$, at selected values of $x$ are given in the table above. Which of the following statements must be true?
(A) $f$ is increasing for $- 1 \leq x \leq 5$.
(B) The graph of $f$ is concave down for $- 1 < x < 5$.
(C) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime } ( c ) = - \frac { 3 } { 2 }$.
(D) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime \prime } ( c ) = - \frac { 3 } { 2 }$.

Let $f$ be the function with derivative defined by $f ^ { \prime } ( x ) = 2 + ( 2 x - 8 ) \sin ( x + 3 )$.
How many points of inflection does the graph of $f$ have on the interval $0 < x < 9$?
(A) One
(B) Two
(C) Three
(D) Four

19. A polynomial $p ( x )$ has a relative maximum at $( - 2,4 )$, a relative minimum at $( 1,1 )$, a relative maximum at $( 5,7 )$ and no other critical points. How many zeros does $p ( x )$ have?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five

45. Suppose $g ^ { \prime } ( x ) < 0$ for all $x \geq 0$ and $F ( x ) = \int _ { 0 } ^ { x } t g ^ { \prime } ( t ) d t$ for all $x \geq 0$. Which of the following statements is FALSE?
(A) $F$ takes on negative values.
(B) $\quad F$ is continuous for all $x > 0$.
(C) $F ( x ) = x g ( x ) - \int _ { 0 } ^ { x } g ( t ) d t$
(D) $\quad F ^ { \prime } ( x )$ exists for all $x > 0$.
(E) $F$ is an increasing function.

1985 AP Calculus AB: Section I

90 Minutes-No Calculator

Notes: (1) In this examination, $\ln x$ denotes the natural logarithm of $x$ (that is, logarithm to the base $e$ ).
(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. $\int _ { 1 } ^ { 2 } x ^ { - 3 } d x =$
   (A) $- \frac { 7 } { 8 }$
   (B) $- \frac { 3 } { 4 }$
   (C) $\frac { 15 } { 64 }$
   (D) $\frac { 3 } { 8 }$
   (E) $\frac { 15 } { 16 }$
2. If $f ( x ) = ( 2 x + 1 ) ^ { 4 }$, then the 4th derivative of $f ( x )$ at $x = 0$ is
   (A) 0
   (B) 24
   (C) 48
   (D) 240
   (E) 384
3. If $y = \frac { 3 } { 4 + x ^ { 2 } }$, then $\frac { d y } { d x } =$
   (A) $\frac { - 6 x } { \left( 4 + x ^ { 2 } \right) ^ { 2 } }$
   (B) $\frac { 3 x } { \left( 4 + x ^ { 2 } \right) ^ { 2 } }$
   (C) $\frac { 6 x } { \left( 4 + x ^ { 2 } \right) ^ { 2 } }$
   (D) $\frac { - 3 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } }$
   (E) $\frac { 3 } { 2 x }$
4. If $\frac { d y } { d x } = \cos ( 2 x )$, then $y =$
   (A) $\quad - \frac { 1 } { 2 } \cos ( 2 x ) + C$
   (B) $- \frac { 1 } { 2 } \cos ^ { 2 } ( 2 x ) + C$
   (C) $\frac { 1 } { 2 } \sin ( 2 x ) + C$
   (D) $\frac { 1 } { 2 } \sin ^ { 2 } ( 2 x ) + C$
   (E) $\quad - \frac { 1 } { 2 } \sin ( 2 x ) + C$
5. $\lim _ { n \rightarrow \infty } \frac { 4 n ^ { 2 } } { n ^ { 2 } + 10,000 n }$ is
   (A) 0
   (B) $\frac { 1 } { 2,500 }$
   (C) 1
   (D) 4
   (E) nonexistent

1985 AP Calculus AB: Section I

1. If $f ( x ) = x$, then $f ^ { \prime } ( 5 ) =$
   (A) 0
   (B) $\frac { 1 } { 5 }$
   (C) 1
   (D) 5
   (E) $\frac { 25 } { 2 }$
2. Which of the following is equal to $\ln 4$ ?
   (A) $\quad \ln 3 + \ln 1$
   (B) $\frac { \ln 8 } { \ln 2 }$
   (C) $\quad \int _ { 1 } ^ { 4 } e ^ { t } d t$
   (D) $\quad \int _ { 1 } ^ { 4 } \ln x d x$
   (E) $\quad \int _ { 1 } ^ { 4 } \frac { 1 } { t } d t$
3. The slope of the line tangent to the graph of $y = \ln \left( \frac { x } { 2 } \right)$ at $x = 4$ is
   (A) $\frac { 1 } { 8 }$
   (B) $\frac { 1 } { 4 }$
   (C) $\frac { 1 } { 2 }$
   (D) 1
   (E) 4
4. If $\int _ { - 1 } ^ { 1 } e ^ { - x ^ { 2 } } d x = k$, then $\int _ { - 1 } ^ { 0 } e ^ { - x ^ { 2 } } d x =$
   (A) $- 2 k$
   (B) $- k$
   (C) $- \frac { k } { 2 }$
   (D) $\frac { k } { 2 }$
   (E) $2 k$
5. If $y = 10 ^ { \left( x ^ { 2 } - 1 \right) }$, then $\frac { d y } { d x } =$
   (A) $\quad ( \ln 10 ) 10 ^ { \left( x ^ { 2 } - 1 \right) }$
   (B) $\quad ( 2 x ) 10 ^ { \left( x ^ { 2 } - 1 \right) }$
   (C) $\left( x ^ { 2 } - 1 \right) 10 ^ { \left( x ^ { 2 } - 2 \right) }$
   (D) $\quad 2 x ( \ln 10 ) 10 ^ { \left( x ^ { 2 } - 1 \right) }$
   (E) $\quad x ^ { 2 } ( \ln 10 ) 10 ^ { \left( x ^ { 2 } - 1 \right) }$
6. The position of a particle moving along a straight line at any time $t$ is given by $s ( t ) = t ^ { 2 } + 4 t + 4$. What is the acceleration of the particle when $t = 4$ ?
   (A) 0
   (B) 2
   (C) 4
   (D) 8
   (E) 12
7. If $f ( g ( x ) ) = \ln \left( x ^ { 2 } + 4 \right) , f ( x ) = \ln \left( x ^ { 2 } \right)$, and $g ( x ) > 0$ for all real $x$, then $g ( x ) =$
   (A) $\frac { 1 } { \sqrt { x ^ { 2 } + 4 } }$
   (B) $\frac { 1 } { x ^ { 2 } + 4 }$
   (C) $\sqrt { x ^ { 2 } + 4 }$
   (D) $x ^ { 2 } + 4$
   (E) $x + 2$
8. If $x ^ { 2 } + x y + y ^ { 3 } = 0$, then, in terms of $x$ and $y , \frac { d y } { d x } =$
   (A) $- \frac { 2 x + y } { x + 3 y ^ { 2 } }$
   (B) $- \frac { x + 3 y ^ { 2 } } { 2 x + y }$
   (C) $\frac { - 2 x } { 1 + 3 y ^ { 2 } }$
   (D) $\frac { - 2 x } { x + 3 y ^ { 2 } }$
   (E) $- \frac { 2 x + y } { x + 3 y ^ { 2 } - 1 }$
9. The velocity of a particle moving on a line at time $t$ is $v = 3 t ^ { \frac { 1 } { 2 } } + 5 t ^ { \frac { 3 } { 2 } }$ meters per second. How many meters did the particle travel from $t = 0$ to $t = 4$ ?
   (A) 32
   (B) 40
   (C) 64
   (D) 80
   (E) 184
10. The domain of the function defined by $f ( x ) = \ln \left( x ^ { 2 } - 4 \right)$ is the set of all real numbers $x$ such that
    (A) $| x | < 2$
    (B) $| x | \leq 2$
    (C) $| x | > 2$
    (D) $| x | \geq 2$
    (E) $x$ is a real number
11. The function defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ for all real numbers $x$ has a relative maximum at $x =$
    (A) - 2
    (B) 0
    (C) 1
    (D) 2
    (E) 4
12. $\int _ { 0 } ^ { 1 } x e ^ { - x } d x =$
    (A) $1 - 2 e$
    (B) - 1
    (C) $1 - 2 e ^ { - 1 }$
    (D) 1
    (E) $2 e - 1$
13. If $y = \cos ^ { 2 } x - \sin ^ { 2 } x$, then $y ^ { \prime } =$
    (A) - 1
    (B) 0
    (C) $- 2 \sin ( 2 x )$
    (D) $\quad - 2 ( \cos x + \sin x )$
    (E) $\quad 2 ( \cos x - \sin x )$
14. If $f \left( x _ { 1 } \right) + f \left( x _ { 2 } \right) = f \left( x _ { 1 } + x _ { 2 } \right)$ for all real numbers $x _ { 1 }$ and $x _ { 2 }$, which of the following could define $f$ ?
    (A) $f ( x ) = x + 1$
    (B) $f ( x ) = 2 x$
    (C) $f ( x ) = \frac { 1 } { x }$
    (D) $f ( x ) = e ^ { x }$
    (E) $f ( x ) = x ^ { 2 }$
15. If $y = \arctan ( \cos x )$, then $\frac { d y } { d x } =$
    (A) $\frac { - \sin x } { 1 + \cos ^ { 2 } x }$
    (B) $- ( \operatorname { arcsec } ( \cos x ) ) ^ { 2 } \sin x$
    (C) $( \operatorname { arcsec } ( \cos x ) ) ^ { 2 }$
    (D) $\frac { 1 } { ( \arccos x ) ^ { 2 } + 1 }$
    (E) $\frac { 1 } { 1 + \cos ^ { 2 } x }$
16. If the domain of the function $f$ given by $f ( x ) = \frac { 1 } { 1 - x ^ { 2 } }$ is $\{ x : | x | > 1 \}$, what is the range of $f$ ?
    (A) $\quad \{ x : - \infty < x < - 1 \}$
    (B) $\{ x : - \infty < x < 0 \}$
    (C) $\{ x : - \infty < x < 1 \}$
    (D) $\quad \{ x : - 1 < x < \infty \}$
    (E) $\{ x : 0 < x < \infty \}$
17. $\int _ { 1 } ^ { 2 } \frac { x ^ { 2 } - 1 } { x + 1 } d x =$
    (A) $\frac { 1 } { 2 }$
    (B) 1
    (C) 2
    (D) $\frac { 5 } { 2 }$
    (E) $\quad \ln 3$
18. $\frac { d } { d x } \left( \frac { 1 } { x ^ { 3 } } - \frac { 1 } { x } + x ^ { 2 } \right)$ at $x = - 1$ is
    (A) $\quad - 6$
    (B) - 4
    (C) 0
    (D) 2
    (E) 6
19. If $\int _ { - 2 } ^ { 2 } \left( x ^ { 7 } + k \right) d x = 16$, then $k =$
    (A) - 12
    (B) - 4
    (C) 0
    (D) 4
    (E) 12
20. If $f ( x ) = e ^ { x }$, which of the following is equal to $f ^ { \prime } ( e )$ ?
    (A) $\lim _ { h \rightarrow 0 } \frac { e ^ { x + h } } { h }$
    (B) $\lim _ { h \rightarrow 0 } \frac { e ^ { x + h } - e ^ { e } } { h }$
    (C) $\lim _ { h \rightarrow 0 } \frac { e ^ { e + h } - e } { h }$
    (D) $\lim _ { h \rightarrow 0 } \frac { e ^ { x + h } - 1 } { h }$
    (E) $\lim _ { h \rightarrow 0 } \frac { e ^ { e + h } - e ^ { e } } { h }$

1985 AP Calculus AB: Section I

1. The graph of $y ^ { 2 } = x ^ { 2 } + 9$ is symmetric to which of the following? I. The $x$-axis II. The $y$-axis III. The origin
   (A) I only
   (B) II only
   (C) III only
   (D) I and II only
   (E) I, II, and III
2. $\int _ { 0 } ^ { 3 } | x - 1 | d x =$
   (A) 0
   (B) $\frac { 3 } { 2 }$
   (C) 2
   (D) $\frac { 5 } { 2 }$
   (E) 6
3. If the position of a particle on the $x$-axis at time $t$ is $- 5 t ^ { 2 }$, then the average velocity of the particle for $0 \leq t \leq 3$ is
   (A) - 45
   (B) - 30
   (C) - 15
   (D) - 10
   (E) - 5
4. Which of the following functions are continuous for all real numbers $x$ ? I. $y = x ^ { \frac { 2 } { 3 } }$ II. $y = e ^ { x }$ III. $y = \tan x$
   (A) None
   (B) I only
   (C) II only
   (D) I and II
   (E) I and III
5. $\int \tan ( 2 x ) d x =$
   (A) $\quad - 2 \ln | \cos ( 2 x ) | + C$
   (B) $\quad - \frac { 1 } { 2 } \ln | \cos ( 2 x ) | + C$
   (C) $\frac { 1 } { 2 } \ln | \cos ( 2 x ) | + C$
   (D) $\quad 2 \ln | \cos ( 2 x ) | + C$
   (E) $\frac { 1 } { 2 } \sec ( 2 x ) \tan ( 2 x ) + C$

1985 AP Calculus AB: Section I

1. The volume of a cone of radius $r$ and height $h$ is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$. If the radius and the height both increase at a constant rate of $\frac { 1 } { 2 }$ centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?
   (A) $\frac { 1 } { 2 } \pi$
   (B) $10 \pi$
   (C) $24 \pi$
   (D) $54 \pi$
   (E) $108 \pi$
2. $\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ( 3 x ) d x =$
   (A) - 2
   (B) $- \frac { 2 } { 3 }$
   (C) 0
   (D) $\frac { 2 } { 3 }$
   (E) 2 [Figure]
3. The graph of the derivative of $f$ is shown in the figure above. Which of the following could be the graph of $f$ ?
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]
   (E) [Figure]

1985 AP Calculus AB: Section I

1. The area of the region in the first quadrant that is enclosed by the graphs of $y = x ^ { 3 } + 8$ and $y = x + 8$ is
   (A) $\frac { 1 } { 4 }$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 3 } { 4 }$
   (D) 1
   (E) $\frac { 65 } { 4 }$ [Figure]
2. The figure above shows the graph of a sine function for one complete period. Which of the following is an equation for the graph?
   (A) $y = 2 \sin \left( \frac { \pi } { 2 } x \right)$
   (B) $y = \sin ( \pi x )$
   (C) $y = 2 \sin ( 2 x )$
   (D) $y = 2 \sin ( \pi x )$
   (E) $y = \sin ( 2 x )$
3. If $f$ is a continuous function defined for all real numbers $x$ and if the maximum value of $f ( x )$ is 5 and the minimum value of $f ( x )$ is - 7 , then which of the following must be true? I. The maximum value of $f ( | x | )$ is 5 . II. The maximum value of $| f ( x ) |$ is 7 . III. The minimum value of $f ( | x | )$ is 0 .
   (A) I only
   (B) II only
   (C) I and II only
   (D) II and III only
   (E) I, II, and III
4. $\lim _ { x \rightarrow 0 } ( x \csc x )$ is
   (A) $- \infty$
   (B) - 1
   (C) 0
   (D) 1
   (E) $\infty$

1985 AP Calculus AB: Section I

1. Let $f$ and $g$ have continuous first and second derivatives everywhere. If $f ( x ) \leq g ( x )$ for all real $x$, which of the following must be true? I. $f ^ { \prime } ( x ) \leq g ^ { \prime } ( x )$ for all real $x$ II. $f ^ { \prime \prime } ( x ) \leq g ^ { \prime \prime } ( x )$ for all real $x$ III. $\quad \int _ { 0 } ^ { 1 } f ( x ) d x \leq \int _ { 0 } ^ { 1 } g ( x ) d x$
   (A) None
   (B) I only
   (C) III only
   (D) I and II only
   (E) I, II, and III
2. If $f ( x ) = \frac { \ln x } { x }$, for all $x > 0$, which of the following is true?
   (A) $f$ is increasing for all $x$ greater than 0 .
   (B) $\quad f$ is increasing for all $x$ greater than 1 .
   (C) $f$ is decreasing for all $x$ between 0 and 1 .
   (D) $f$ is decreasing for all $x$ between 1 and $e$.
   (E) $f$ is decreasing for all $x$ greater than $e$.
3. Let $f$ be a continuous function on the closed interval $[ 0,2 ]$. If $2 \leq f ( x ) \leq 4$, then the greatest possible value of $\int _ { 0 } ^ { 2 } f ( x ) d x$ is
   (A) 0
   (B) 2
   (C) 4
   (D) 8
   (E) 16
4. If $\lim _ { x \rightarrow a } f ( x ) = L$, where $L$ is a real number, which of the following must be true?
   (A) $f ^ { \prime } ( a )$ exists.
   (B) $f ( x )$ is continuous at $x = a$.
   (C) $f ( x )$ is defined at $x = a$.
   (D) $f ( a ) = L$
   (E) None of the above

1985 AP Calculus AB: Section I

1. $\frac { d } { d x } \int _ { 2 } ^ { x } \sqrt { 1 + t ^ { 2 } } d t =$
   (A) $\frac { x } { \sqrt { 1 + x ^ { 2 } } }$
   (B) $\sqrt { 1 + x ^ { 2 } } - 5$
   (C) $\sqrt { 1 + x ^ { 2 } }$
   (D) $\frac { x } { \sqrt { 1 + x ^ { 2 } } } - \frac { 1 } { \sqrt { 5 } }$
   (E) $\frac { 1 } { 2 \sqrt { 1 + x ^ { 2 } } } - \frac { 1 } { 2 \sqrt { 5 } }$
2. An equation of the line tangent to $y = x ^ { 3 } + 3 x ^ { 2 } + 2$ at its point of inflection is
   (A) $y = - 6 x - 6$
   (B) $y = - 3 x + 1$
   (C) $y = 2 x + 10$
   (D) $y = 3 x - 1$
   (E) $y = 4 x + 1$
3. The average value of $f ( x ) = x ^ { 2 } \sqrt { x ^ { 3 } + 1 }$ on the closed interval $[ 0,2 ]$ is
   (A) $\frac { 26 } { 9 }$
   (B) $\frac { 13 } { 3 }$
   (C) $\frac { 26 } { 3 }$
   (D) 13
   (E) 26
4. The region enclosed by the graph of $y = x ^ { 2 }$, the line $x = 2$, and the $x$-axis is revolved about the $y$-axis. The volume of the solid generated is
   (A) $8 \pi$
   (B) $\frac { 32 } { 5 } \pi$
   (C) $\frac { 16 } { 3 } \pi$
   (D) $4 \pi$
   (E) $\frac { 8 } { 3 } \pi$

1985 AP Calculus BC: Section I

90 Minutes-No Calculator

Notes: (1) In this examination, $\ln x$ denotes the natural logarithm of $x$ (that is, logarithm to the base $e$ ).
(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 between the graph of $y = 4 x ^ { 3 } + 2$ and the $x$-axis from $x = 1$ to $x = 2$ is
   (A) 36
   (B) 23
   (C) 20
   (D) 17
   (E) 9
2. At what values of $x$ does $f ( x ) = 3 x ^ { 5 } - 5 x ^ { 3 } + 15$ have a relative maximum?
   (A) -1 only
   (B) 0 only
   (C) 1 only
   (D) -1 and 1 only
   (E) -1, 0 and 1
3. $\int _ { 1 } ^ { 2 } \frac { x + 1 } { x ^ { 2 } + 2 x } d x =$
   (A) $\quad \ln 8 - \ln 3$
   (B) $\frac { \ln 8 - \ln 3 } { 2 }$
   (C) $\quad \ln 8$
   (D) $\frac { 3 \ln 2 } { 2 }$
   (E) $\frac { 3 \ln 2 + 2 } { 2 }$
4. A particle moves in the $x y$-plane so that at any time $t$ its coordinates are $x = t ^ { 2 } - 1$ and $y = t ^ { 4 } - 2 t ^ { 3 }$. At $t = 1$, its acceleration vector is
   (A) $( 0 , - 1 )$
   (B) $( 0,12 )$
   (C) $( 2 , - 2 )$
   (D) $( 2,0 )$
   (E) $( 2,8 )$ [Figure]
5. The curves $y = f ( x )$ and $y = g ( x )$ shown in the figure above intersect at the point $( a , b )$. The area of the shaded region enclosed by these curves and the line $x = - 1$ is given by
   (A) $\quad \int _ { 0 } ^ { a } ( f ( x ) - g ( x ) ) d x + \int _ { - 1 } ^ { 0 } ( f ( x ) + g ( x ) ) d x$
   (B) $\quad \int _ { - 1 } ^ { b } g ( x ) d x + \int _ { b } ^ { c } f ( x ) d x$
   (C) $\quad \int _ { - 1 } ^ { c } ( f ( x ) - g ( x ) ) d x$
   (D) $\quad \int _ { - 1 } ^ { a } ( f ( x ) - g ( x ) ) d x$
   (E) $\quad \int _ { - 1 } ^ { a } ( | f ( x ) | - | g ( x ) | ) d x$
6. If $f ( x ) = \frac { x } { \tan x }$, then $f ^ { \prime } \left( \frac { \pi } { 4 } \right) =$
   (A) 2
   (B) $\frac { 1 } { 2 }$
   (C) $1 + \frac { \pi } { 2 }$
   (D) $\frac { \pi } { 2 } - 1$
   (E) $\quad 1 - \frac { \pi } { 2 }$

1985 AP Calculus BC: Section I

1. Which of the following is equal to $\int \frac { 1 } { \sqrt { 25 - x ^ { 2 } } } d x$ ?
   (A) $\arcsin \frac { x } { 5 } + C$
   (B) $\quad \arcsin x + C$
   (C) $\frac { 1 } { 5 } \arcsin \frac { x } { 5 } + C$
   (D) $\sqrt { 25 - x ^ { 2 } } + C$
   (E) $\quad 2 \sqrt { 25 - x ^ { 2 } } + C$
2. If $f$ is a function such that $\lim _ { x \rightarrow 2 } \frac { f ( x ) - f ( 2 ) } { x - 2 } = 0$, which of the following must be true?
   (A) The limit of $f ( x )$ as $x$ approaches 2 does not exist.
   (B) $f$ is not defined at $x = 2$.
   (C) The derivative of $f$ at $x = 2$ is 0 .
   (D) $f$ is continuous at $x = 0$.
   (E) $f ( 2 ) = 0$
3. If $x y ^ { 2 } + 2 x y = 8$, then, at the point $( 1,2 ) , y ^ { \prime }$ is
   (A) $- \frac { 5 } { 2 }$
   (B) $- \frac { 4 } { 3 }$
   (C) - 1
   (D) $- \frac { 1 } { 2 }$
   (E) 0
4. For $- 1 < x < 1$ if $f ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } x ^ { 2 n - 1 } } { 2 n - 1 }$, then $f ^ { \prime } ( x ) =$
   (A) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } x ^ { 2 n - 2 }$
   (B) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } x ^ { 2 n - 2 }$
   (C) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { 2 n } x ^ { 2 n }$
   (D) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } x ^ { 2 n }$
   (E) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } x ^ { 2 n }$

1985 AP Calculus BC: Section I

1. $\frac { d } { d x } \ln \left( \frac { 1 } { 1 - x } \right) =$
   (A) $\frac { 1 } { 1 - x }$
   (B) $\frac { 1 } { x - 1 }$
   (C) $1 - x$
   (D) $\quad x - 1$
   (E) $( 1 - x ) ^ { 2 }$
2. $\int \frac { d x } { ( x - 1 ) ( x + 2 ) } =$
   (A) $\frac { 1 } { 3 } \ln \left| \frac { x - 1 } { x + 2 } \right| + C$
   (B) $\frac { 1 } { 3 } \ln \left| \frac { x + 2 } { x - 1 } \right| + C$
   (C) $\frac { 1 } { 3 } \ln | ( x - 1 ) ( x + 2 ) | + C$
   (D) $( \ln | x - 1 | ) ( \ln | x + 2 | ) + C$
   (E) $\quad \ln \left| ( x - 1 ) ( x + 2 ) ^ { 2 } \right| + C$
3. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$. What are all values of $c$ that satisfy the conclusion of the Mean Value Theorem of differential calculus on the closed interval $[ 0,3 ]$ ?
   (A) 0 only
   (B) 2 only
   (C) 3 only
   (D) 0 and 3
   (E) 2 and 3
4. Which of the following series are convergent? I. $\quad 1 + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } + \ldots$ II. $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } + \ldots$ III. $\quad 1 - \frac { 1 } { 3 } + \frac { 1 } { 3 ^ { 2 } } - \ldots + \frac { ( - 1 ) ^ { n + 1 } } { 3 ^ { n - 1 } } + \ldots$
   (A) I only
   (B) III only
   (C) I and III only
   (D) II and III only
   (E) I, II, and III
5. If the velocity of a particle moving along the $x$-axis is $v ( t ) = 2 t - 4$ and if at $t = 0$ its position is 4 , then at any time $t$ its position $x ( t )$ is
   (A) $t ^ { 2 } - 4 t$
   (B) $t ^ { 2 } - 4 t - 4$
   (C) $t ^ { 2 } - 4 t + 4$
   (D) $2 t ^ { 2 } - 4 t$
   (E) $2 t ^ { 2 } - 4 t + 4$

1985 AP Calculus BC: Section I

1. Which of the following functions shows that the statement "If a function is continuous at $x = 0$, then it is differentiable at $x = 0$ " is false?
   (A) $f ( x ) = x ^ { - \frac { 4 } { 3 } }$
   (B) $f ( x ) = x ^ { - \frac { 1 } { 3 } }$
   (C) $f ( x ) = x ^ { \frac { 1 } { 3 } }$
   (D) $f ( x ) = x ^ { \frac { 4 } { 3 } }$
   (E) $f ( x ) = x ^ { 3 }$
2. If $f ( x ) = x \ln \left( x ^ { 2 } \right)$, then $f ^ { \prime } ( x ) =$
   (A) $\quad \ln \left( x ^ { 2 } \right) + 1$
   (B) $\quad \ln \left( x ^ { 2 } \right) + 2$
   (C) $\quad \ln \left( x ^ { 2 } \right) + \frac { 1 } { x }$
   (D) $\frac { 1 } { x ^ { 2 } }$
   (E) $\frac { 1 } { x }$
3. $\int \sin ( 2 x + 3 ) d x =$
   (A) $- 2 \cos ( 2 x + 3 ) + C$
   (B) $- \cos ( 2 x + 3 ) + C$
   (C) $- \frac { 1 } { 2 } \cos ( 2 x + 3 ) + C$
   (D) $\frac { 1 } { 2 } \cos ( 2 x + 3 ) + C$
   (E) $\quad \cos ( 2 x + 3 ) + C$
4. If $f$ and $g$ are twice differentiable functions such that $g ( x ) = e ^ { f ( x ) }$ and $g ^ { \prime \prime } ( x ) = h ( x ) e ^ { f ( x ) }$, then $h ( x ) =$
   (A) $f ^ { \prime } ( x ) + f ^ { \prime \prime } ( x )$
   (B) $f ^ { \prime } ( x ) + \left( f ^ { \prime \prime } ( x ) \right) ^ { 2 }$
   (C) $\left( f ^ { \prime } ( x ) + f ^ { \prime \prime } ( x ) \right) ^ { 2 }$
   (D) $\left( f ^ { \prime } ( x ) \right) ^ { 2 } + f ^ { \prime \prime } ( x )$
   (E) $2 f ^ { \prime } ( x ) + f ^ { \prime \prime } ( x )$ [Figure]
5. The graph of $y = f ( x )$ on the closed interval [2,7] is shown above. How many points of inflection does this graph have on this interval?
   (A) One
   (B) Two
   (C) Three
   (D) Four
   (E) Five

1985 AP Calculus BC: Section I

1. If $\int f ( x ) \sin x d x = - f ( x ) \cos x + \int 3 x ^ { 2 } \cos x d x$, then $f ( x )$ could be
   (A) $3 x ^ { 2 }$
   (B) $x ^ { 3 }$
   (C) $- x ^ { 3 }$
   (D) $\quad \sin x$
   (E) $\quad \cos x$
2. The area of a circular region is increasing at a rate of $96 \pi$ square meters per second. When the area of the region is $64 \pi$ square meters, how fast, in meters per second, is the radius of the region increasing?
   (A) 6
   (B) 8
   (C) 16
   (D) $4 \sqrt { 3 }$
   (E) $12 \sqrt { 3 }$
3. $\lim _ { h \rightarrow 0 } \frac { \int _ { 1 } ^ { 1 + h } \sqrt { x ^ { 5 } + 8 } d x } { h }$ is
   (A) 0
   (B) 1
   (C) 3
   (D) $2 \sqrt { 2 }$
   (E) nonexistent
4. The area of the region enclosed by the polar curve $r = \sin ( 2 \theta )$ for $0 \leq \theta \leq \frac { \pi } { 2 }$ is
   (A) 0
   (B) $\frac { 1 } { 2 }$
   (C) 1
   (D) $\frac { \pi } { 8 }$
   (E) $\frac { \pi } { 4 }$
5. A particle moves along the $x$-axis so that at any time $t$ its position is given by $x ( t ) = t e ^ { - 2 t }$. For what values of $t$ is the particle at rest?
   (A) No values
   (B) 0 only
   (C) $\frac { 1 } { 2 }$ only
   (D) 1 only
   (E) 0 and $\frac { 1 } { 2 }$
6. For $0 < x < \frac { \pi } { 2 }$, if $y = ( \sin x ) ^ { x }$, then $\frac { d y } { d x }$ is
   (A) $\quad x \ln ( \sin x )$
   (B) $( \sin x ) ^ { x } \cot x$
   (C) $\quad x ( \sin x ) ^ { x - 1 } ( \cos x )$
   (D) $( \sin x ) ^ { x } ( x \cos x + \sin x )$
   (E) $\quad ( \sin x ) ^ { x } ( x \cot x + \ln ( \sin x ) )$ [Figure]
7. If $f$ is the continuous, strictly increasing function on the interval $a \leq x \leq b$ as shown above, which of the following must be true? I. $\quad \int _ { a } ^ { b } f ( x ) d x < f ( b ) ( b - a )$ II. $\quad \int _ { a } ^ { b } f ( x ) d x > f ( a ) ( b - a )$ III. $\quad \int _ { a } ^ { b } f ( x ) d x = f ( c ) ( b - a )$ for some number $c$ such that $a < c < b$
   (A) I only
   (B) II only
   (C) III only
   (D) I and III only
   (E) I, II, and III
8. An antiderivative of $f ( x ) = e ^ { x + e ^ { x } }$ is
   (A) $\frac { e ^ { x + e ^ { x } } } { 1 + e ^ { x } }$
   (B) $\left( 1 + e ^ { x } \right) e ^ { x + e ^ { x } }$
   (C) $e ^ { 1 + e ^ { x } }$
   (D) $e ^ { x + e ^ { x } }$
   (E) $e ^ { e ^ { x } }$
9. $\lim _ { x \rightarrow \frac { \pi } { 4 } } \frac { \sin \left( x - \frac { \pi } { 4 } \right) } { x - \frac { \pi } { 4 } }$ is
   (A) 0
   (B) $\frac { 1 } { \sqrt { 2 } }$
   (C) $\frac { \pi } { 4 }$
   (D) 1
   (E) nonexistent
10. If $x = t ^ { 3 } - t$ and $y = \sqrt { 3 t + 1 }$, then $\frac { d y } { d x }$ at $t = 1$ is
    (A) $\frac { 1 } { 8 }$
    (B) $\frac { 3 } { 8 }$
    (C) $\frac { 3 } { 4 }$
    (D) $\frac { 8 } { 3 }$
    (E) 8
11. What are all values of $x$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { n }$ converges?
    (A) $- 1 \leq x < 1$
    (B) $- 1 \leq x \leq 1$
    (C) $0 < x < 2$
    (D) $0 \leq x < 2$
    (E) $0 \leq x \leq 2$

1985 AP Calculus BC: Section I

1. An equation of the line normal to the graph of $y = x ^ { 3 } + 3 x ^ { 2 } + 7 x - 1$ at the point where $x = - 1$ is
   (A) $4 x + y = - 10$
   (B) $x - 4 y = 23$
   (C) $4 x - y = 2$
   (D) $x + 4 y = 25$
   (E) $x + 4 y = - 25$
2. If $\frac { d y } { d t } = - 2 y$ and if $y = 1$ when $t = 0$, what is the value of $t$ for which $y = \frac { 1 } { 2 }$ ?
   (A) $- \frac { \ln 2 } { 2 }$
   (B) $- \frac { 1 } { 4 }$
   (C) $\frac { \ln 2 } { 2 }$
   (D) $\frac { \sqrt { 2 } } { 2 }$
   (E) $\quad \ln 2$
3. Which of the following gives the area of the surface generated by revolving about the $y$-axis the arc of $x = y ^ { 3 }$ from $y = 0$ to $y = 1$ ?
   (A) $2 \pi \int _ { 0 } ^ { 1 } y ^ { 3 } \sqrt { 1 + 9 y ^ { 4 } } d y$
   (B) $2 \pi \int _ { 0 } ^ { 1 } y ^ { 3 } \sqrt { 1 + y ^ { 6 } } d y$
   (C) $2 \pi \int _ { 0 } ^ { 1 } y ^ { 3 } \sqrt { 1 + 3 y ^ { 2 } } d y$
   (D) $2 \pi \int _ { 0 } ^ { 1 } y \sqrt { 1 + 9 y ^ { 4 } } d y$
   (E) $2 \pi \int _ { 0 } ^ { 1 } y \sqrt { 1 + y ^ { 6 } } d y$
4. The region in the first quadrant between the $x$-axis and the graph of $y = 6 x - x ^ { 2 }$ is rotated around the $y$-axis. The volume of the resulting solid of revolution is given by
   (A) $\int _ { 0 } ^ { 6 } \pi \left( 6 x - x ^ { 2 } \right) ^ { 2 } d x$
   (B) $\int _ { 0 } ^ { 6 } 2 \pi x \left( 6 x - x ^ { 2 } \right) d x$
   (C) $\int _ { 0 } ^ { 6 } \pi x \left( 6 x - x ^ { 2 } \right) ^ { 2 } d x$
   (D) $\int _ { 0 } ^ { 6 } \pi ( 3 + \sqrt { 9 - y } ) ^ { 2 } d y$
   (E) $\int _ { 0 } ^ { 9 } \pi ( 3 + \sqrt { 9 - y } ) ^ { 2 } d y$

1985 AP Calculus BC: Section I

1. $\int _ { - 1 } ^ { 1 } \frac { 3 } { x ^ { 2 } } d x$ is
   (A) - 6
   (B) - 3
   (C) 0
   (D) 6
   (E) nonexistent
2. The general solution for the equation $\frac { d y } { d x } + y = x e ^ { - x }$ is
   (A) $y = \frac { x ^ { 2 } } { 2 } e ^ { - x } + C e ^ { - x }$
   (B) $y = \frac { x ^ { 2 } } { 2 } e ^ { - x } + e ^ { - x } + C$
   (C) $y = - e ^ { - x } + \frac { C } { 1 + x }$
   (D) $y = x e ^ { - x } + C e ^ { - x }$
   (E) $y = C _ { 1 } e ^ { x } + C _ { 2 } x e ^ { - x }$
3. $\lim _ { x \rightarrow \infty } \left( 1 + 5 e ^ { x } \right) ^ { \frac { 1 } { x } }$ is
   (A) 0
   (B) 1
   (C) $e$
   (D) $e ^ { 5 }$
   (E) nonexistent
4. The base of a solid is the region enclosed by the graph of $y = e ^ { - x }$, the coordinate axes, and the line $x = 3$. If all plane cross sections perpendicular to the $x$-axis are squares, then its volume is
   (A) $\frac { \left( 1 - e ^ { - 6 } \right) } { 2 }$
   (B) $\frac { 1 } { 2 } e ^ { - 6 }$
   (C) $e ^ { - 6 }$
   (D) $e ^ { - 3 }$
   (E) $1 - e ^ { - 3 }$
5. If the substitution $u = \frac { x } { 2 }$ is made, the integral $\int _ { 2 } ^ { 4 } \frac { 1 - \left( \frac { x } { 2 } \right) ^ { 2 } } { x } d x =$
   (A) $\quad \int _ { 1 } ^ { 2 } \frac { 1 - u ^ { 2 } } { u } d u$
   (B) $\quad \int _ { 2 } ^ { 4 } \frac { 1 - u ^ { 2 } } { u } d u$
   (C) $\quad \int _ { 1 } ^ { 2 } \frac { 1 - u ^ { 2 } } { 2 u } d u$
   (D) $\quad \int _ { 1 } ^ { 2 } \frac { 1 - u ^ { 2 } } { 4 u } d u$
   (E) $\quad \int _ { 2 } ^ { 4 } \frac { 1 - u ^ { 2 } } { 2 u } d u$
6. What is the length of the arc of $y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } }$ from $x = 0$ to $x = 3$ ?
   (A) $\frac { 8 } { 3 }$
   (B) 4
   (C) $\frac { 14 } { 3 }$
   (D) $\frac { 16 } { 3 }$
   (E) 7
7. The coefficient of $x ^ { 3 }$ in the Taylor series for $e ^ { 3 x }$ about $x = 0$ is
   (A) $\frac { 1 } { 6 }$
   (B) $\frac { 1 } { 3 }$
   (C) $\frac { 1 } { 2 }$
   (D) $\frac { 3 } { 2 }$
   (E) $\frac { 9 } { 2 }$
8. Let $f$ be a function that is continuous on the closed interval $[ - 2,3 ]$ such that $f ^ { \prime } ( 0 )$ does not exist, $f ^ { \prime } ( 2 ) = 0$, and $f ^ { \prime \prime } ( x ) < 0$ for all $x$ except $x = 0$. Which of the following could be the graph of $f$ ?

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
(E) [Figure]

1. At each point $( x , y )$ on a certain curve, the slope of the curve is $3 x ^ { 2 } y$. If the curve contains the point $( 0,8 )$, then its equation is
   (A) $y = 8 e ^ { x ^ { 3 } }$
   (B) $y = x ^ { 3 } + 8$
   (C) $y = e ^ { x ^ { 3 } } + 7$
   (D) $y = \ln ( x + 1 ) + 8$
   (E) $y ^ { 2 } = x ^ { 3 } + 8$
2. If $n$ is a positive integer, then $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \left[ \left( \frac { 1 } { n } \right) ^ { 2 } + \left( \frac { 2 } { n } \right) ^ { 2 } + \ldots + \left( \frac { 3 n } { n } \right) ^ { 2 } \right]$ can be expressed as
   (A) $\int _ { 0 } ^ { 1 } \frac { 1 } { x ^ { 2 } } d x$
   (B) $3 \int _ { 0 } ^ { 1 } \left( \frac { 1 } { x } \right) ^ { 2 } d x$
   (C) $\int _ { 0 } ^ { 3 } \left( \frac { 1 } { x } \right) ^ { 2 } d x$
   (D) $\int _ { 0 } ^ { 3 } x ^ { 2 } d x$
   (E) $3 \int _ { 0 } ^ { 3 } x ^ { 2 } d x$

1988 AP Calculus AB: Section I

90 Minutes-No Calculator

Notes: (1) In this examination, $\ln x$ denotes the natural logarithm of $x$ (that is, logarithm to the base $e$ ).
(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 $y = x ^ { 2 } e ^ { x }$, then $\frac { d y } { d x } =$
   (A) $\quad 2 x e ^ { x }$
   (B) $\quad x \left( x + 2 e ^ { x } \right)$
   (C) $x e ^ { x } ( x + 2 )$
   (D) $2 x + e ^ { x }$
   (E) $\quad 2 x + e$
2. What is the domain of the function $f$ given by $f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x - 3 }$ ?
   (A) $\quad \{ x : x \neq 3 \}$
   (B) $\quad \{ x : | x | \leq 2 \}$
   (C) $\{ x : | x | \geq 2 \}$
   (D) $\quad \{ x : | x | \geq 2$ and $x \neq 3 \}$
   (E) $\quad \{ x : x \geq 2$ and $x \neq 3 \}$
3. A particle with velocity at any time $t$ given by $v ( t ) = e ^ { t }$ moves in a straight line. How far does the particle move from $t = 0$ to $t = 2$ ?
   (A) $e ^ { 2 } - 1$
   (B) $e - 1$
   (C) $2 e$
   (D) $e ^ { 2 }$
   (E) $\frac { e ^ { 3 } } { 3 }$
4. The graph of $y = \frac { - 5 } { x - 2 }$ is concave downward for all values of $x$ such that
   (A) $x < 0$
   (B) $x < 2$
   (C) $x < 5$
   (D) $x > 0$
   (E) $x > 2$
5. $\int \sec ^ { 2 } x d x =$
   (A) $\quad \tan x + C$
   (B) $\csc ^ { 2 } x + C$
   (C) $\cos ^ { 2 } x + C$
   (D) $\frac { \sec ^ { 3 } x } { 3 } + C$
   (E) $2 \sec ^ { 2 } x \tan x + C$
6. If $y = \frac { \ln x } { x }$, then $\frac { d y } { d x } =$
   (A) $\frac { 1 } { x }$
   (B) $\frac { 1 } { x ^ { 2 } }$
   (C) $\frac { \ln x - 1 } { x ^ { 2 } }$
   (D) $\frac { 1 - \ln x } { x ^ { 2 } }$
   (E) $\frac { 1 + \ln x } { x ^ { 2 } }$
7. $\int \frac { x d x } { \sqrt { 3 x ^ { 2 } + 5 } } =$
   (A) $\frac { 1 } { 9 } \left( 3 x ^ { 2 } + 5 \right) ^ { \frac { 3 } { 2 } } + C$
   (B) $\frac { 1 } { 4 } \left( 3 x ^ { 2 } + 5 \right) ^ { \frac { 3 } { 2 } } + C$
   (C) $\frac { 1 } { 12 } \left( 3 x ^ { 2 } + 5 \right) ^ { \frac { 1 } { 2 } } + C$
   (D) $\frac { 1 } { 3 } \left( 3 x ^ { 2 } + 5 \right) ^ { \frac { 1 } { 2 } } + C$
   (E) $\frac { 3 } { 2 } \left( 3 x ^ { 2 } + 5 \right) ^ { \frac { 1 } { 2 } } + C$ [Figure]
8. The graph of $y = f ( x )$ is shown in the figure above. On which of the following intervals are $\frac { d y } { d x } > 0$ and $\frac { d ^ { 2 } y } { d x ^ { 2 } } < 0$ ? I. $a < x < b$ II. $b < x < c$ III. $c < x < d$
   (A) I only
   (B) II only
   (C) III only
   (D) I and II
   (E) II and III
9. If $x + 2 x y - y ^ { 2 } = 2$, then at the point $( 1,1 ) , \frac { d y } { d x }$ is
   (A) $\frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 }$
   (C) 0
   (D) $- \frac { 3 } { 2 }$
   (E) nonexistent
10. If $\int _ { 0 } ^ { k } \left( 2 k x - x ^ { 2 } \right) d x = 18$, then $k =$
    (A) $\quad - 9$
    (B) - 3
    (C) 3
    (D) 9
    (E) 18
11. An equation of the line tangent to the graph of $f ( x ) = x ( 1 - 2 x ) ^ { 3 }$ at the point $( 1 , - 1 )$ is
    (A) $y = - 7 x + 6$
    (B) $y = - 6 x + 5$
    (C) $y = - 2 x + 1$
    (D) $y = 2 x - 3$
    (E) $\quad y = 7 x - 8$
12. If $f ( x ) = \sin x$, then $f ^ { \prime } \left( \frac { \pi } { 3 } \right) =$
    (A) $- \frac { 1 } { 2 }$
    (B) $\frac { 1 } { 2 }$
    (C) $\frac { \sqrt { 2 } } { 2 }$
    (D) $\frac { \sqrt { 3 } } { 2 }$
    (E) $\sqrt { 3 }$
13. If the function $f$ has a continuous derivative on $[ 0 , c ]$, then $\int _ { 0 } ^ { c } f ^ { \prime } ( x ) d x =$
    (A) $f ( c ) - f ( 0 )$
    (B) $| f ( c ) - f ( 0 ) |$
    (C) $f ( c )$
    (D) $f ( x ) + c$
    (E) $f ^ { \prime \prime } ( c ) - f ^ { \prime \prime } ( 0 )$
14. $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos \theta } { \sqrt { 1 + \sin \theta } } d \theta =$
    (A) $- 2 ( \sqrt { 2 } - 1 )$
    (B) $- 2 \sqrt { 2 }$
    (C) $2 \sqrt { 2 }$
    (D) $2 ( \sqrt { 2 } - 1 )$
    (E) $2 ( \sqrt { 2 } + 1 )$

1988 AP Calculus AB: Section I

1. If $f ( x ) = \sqrt { 2 x }$, then $f ^ { \prime } ( 2 ) =$
   (A) $\frac { 1 } { 4 }$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { \sqrt { 2 } } { 2 }$
   (D) 1
   (E) $\sqrt { 2 }$
2. A particle moves along the $x$-axis so that at any time $t \geq 0$ its position is given by $x ( t ) = t ^ { 3 } - 3 t ^ { 2 } - 9 t + 1$. For what values of $t$ is the particle at rest?
   (A) No values
   (B) 1 only
   (C) 3 only
   (D) 5 only
   (E) 1 and 3
3. $\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 2 } d x =$
   (A) $- \frac { 7 } { 3 }$
   (B) $- \frac { 7 } { 9 }$
   (C) $\frac { 1 } { 9 }$
   (D) 1
   (E) 3
4. If $y = 2 \cos \left( \frac { x } { 2 } \right)$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } =$
   (A) $- 8 \cos \left( \frac { x } { 2 } \right)$
   (B) $- 2 \cos \left( \frac { x } { 2 } \right)$
   (C) $- \sin \left( \frac { x } { 2 } \right)$
   (D) $- \cos \left( \frac { x } { 2 } \right)$
   (E) $- \frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right)$
5. $\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } d x =$
   (A) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 } \ln 2$
   (C) $\ln 2$
   (D) $2 \ln 2$
   (E) $\frac { 1 } { 2 } \ln 5$
6. Let $f$ be a polynomial function with degree greater than 2 . If $a \neq b$ and $f ( a ) = f ( b ) = 1$, which of the following must be true for at least one value of $x$ between $a$ and $b$ ? I. $f ( x ) = 0$ II. $f ^ { \prime } ( x ) = 0$ III. $f ^ { \prime \prime } ( x ) = 0$
   (A) None
   (B) I only
   (C) II only
   (D) I and II only
   (E) I, II, and III

1988 AP Calculus AB: Section I

1. The area of the region enclosed by the graphs of $y = x$ and $y = x ^ { 2 } - 3 x + 3$ is
   (A) $\frac { 2 } { 3 }$
   (B) 1
   (C) $\frac { 4 } { 3 }$
   (D) 2
   (E) $\frac { 14 } { 3 }$
2. If $\ln x - \ln \left( \frac { 1 } { x } \right) = 2$, then $x =$
   (A) $\frac { 1 } { e ^ { 2 } }$
   (B) $\frac { 1 } { e }$
   (C) $e$
   (D) $2 e$
   (E) $e ^ { 2 }$
3. If $f ^ { \prime } ( x ) = \cos x$ and $g ^ { \prime } ( x ) = 1$ for all $x$, and if $f ( 0 ) = g ( 0 ) = 0$, then $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { g ( x ) }$ is
   (A) $\frac { \pi } { 2 }$
   (B) 1
   (C) 0
   (D) - 1
   (E) nonexistent
4. $\frac { d } { d x } \left( x ^ { \ln x } \right) =$
   (A) $x ^ { \ln x }$
   (B) $( \ln x ) ^ { x }$
   (C) $\frac { 2 } { x } ( \ln x ) \left( x ^ { \ln x } \right)$
   (D) $\quad ( \ln x ) \left( x ^ { \ln x - 1 } \right)$
   (E) $\quad 2 ( \ln x ) \left( x ^ { \ln x } \right)$
5. For all $x > 1$, if $f ( x ) = \int _ { 1 } ^ { x } \frac { 1 } { t } d t$, then $f ^ { \prime } ( x ) =$
   (A) 1
   (B) $\frac { 1 } { x }$
   (C) $\quad \ln x - 1$
   (D) $\quad \ln x$
   (E) $e ^ { x }$
6. $\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x =$
   (A) $- \frac { \pi } { 2 }$
   (B) - 1
   (C) $1 - \frac { \pi } { 2 }$
   (D) 1
   (E) $\frac { \pi } { 2 } - 1$

1988 AP Calculus AB: Section I

1. At $x = 3$, the function given by $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } , & x < 3 \\ 6 x - 9 , & x \geq 3 \end{array} \right.$ is
   (A) undefined.
   (B) continuous but not differentiable.
   (C) differentiable but not continuous.
   (D) neither continuous nor differentiable.
   (E) both continuous and differentiable.
2. $\int _ { 1 } ^ { 4 } | x - 3 | d x =$
   (A) $- \frac { 3 } { 2 }$
   (B) $\frac { 3 } { 2 }$
   (C) $\frac { 5 } { 2 }$
   (D) $\frac { 9 } { 2 }$
   (E) 5
3. The $\lim _ { h \rightarrow 0 } \frac { \tan 3 ( x + h ) - \tan 3 x } { h }$ is
   (A) 0
   (B) $3 \sec ^ { 2 } ( 3 x )$
   (C) $\sec ^ { 2 } ( 3 x )$
   (D) $3 \cot ( 3 x )$
   (E) nonexistent
4. A region in the first quadrant is enclosed by the graphs of $y = e ^ { 2 x } , x = 1$, and the coordinate axes. If the region is rotated about the $y$-axis, the volume of the solid that is generated is represented by which of the following integrals?
   (A) $\quad 2 \pi \int _ { 0 } ^ { 1 } x e ^ { 2 x } d x$
   (B) $2 \pi \int _ { 0 } ^ { 1 } e ^ { 2 x } d x$
   (C) $\pi \int _ { 0 } ^ { 1 } e ^ { 4 x } d x$
   (D) $\pi \int _ { 0 } ^ { e } y \ln y d y$
   (E) $\frac { \pi } { 4 } \int _ { 0 } ^ { e } \ln ^ { 2 } y d y$

1988 AP Calculus AB: Section I

1. If $f ( x ) = \frac { x } { x + 1 }$, then the inverse function, $f ^ { - 1 }$, is given by $f ^ { - 1 } ( x ) =$
   (A) $\frac { x - 1 } { x }$
   (B) $\frac { x + 1 } { x }$
   (C) $\frac { x } { 1 - x }$
   (D) $\frac { x } { x + 1 }$
   (E) $x$
2. Which of the following does NOT have a period of $\pi$ ?
   (A) $f ( x ) = \sin \left( \frac { 1 } { 2 } x \right)$
   (B) $\quad f ( x ) = | \sin x |$
   (C) $f ( x ) = \sin ^ { 2 } x$
   (D) $f ( x ) = \tan x$
   (E) $f ( x ) = \tan ^ { 2 } x$
3. The absolute maximum value of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 12$ on the closed interval $[ - 2,4 ]$ occurs at $x =$
   (A) 4
   (B) 2
   (C) 1
   (D) 0
   (E) - 2 [Figure]
4. The area of the shaded region in the figure above is represented by which of the following integrals?
   (A) $\int _ { a } ^ { c } ( | f ( x ) | - | g ( x ) | ) d x$
   (B) $\int _ { b } ^ { c } f ( x ) d x - \int _ { a } ^ { c } g ( x ) d x$
   (C) $\int _ { a } ^ { c } ( g ( x ) - f ( x ) ) d x$
   (D) $\int _ { a } ^ { c } ( f ( x ) - g ( x ) ) d x$
   (E) $\int _ { a } ^ { b } ( g ( x ) - f ( x ) ) d x + \int _ { b } ^ { c } ( f ( x ) - g ( x ) ) d x$

1988 AP Calculus AB: Section I

1. $4 \cos \left( x + \frac { \pi } { 3 } \right) =$
   (A) $2 \sqrt { 3 } \cos x - 2 \sin x$
   (B) $2 \cos x - 2 \sqrt { 3 } \sin x$
   (C) $2 \cos x + 2 \sqrt { 3 } \sin x$
   (D) $2 \sqrt { 3 } \cos x + 2 \sin x$
   (E) $\quad 4 \cos x + 2$
2. What is the average value of $y$ for the part of the curve $y = 3 x - x ^ { 2 }$ which is in the first quadrant?
   (A) - 6
   (B) - 2
   (C) $\frac { 3 } { 2 }$
   (D) $\frac { 9 } { 4 }$
   (E) $\frac { 9 } { 2 }$
3. If $f ( x ) = e ^ { x } \sin x$, then the number of zeros of $f$ on the closed interval $[ 0,2 \pi ]$ is
   (A) 0
   (B) 1
   (C) 2
   (D) 3
   (E) 4
4. For $x > 0 , \int \left( \frac { 1 } { x } \int _ { 1 } ^ { x } \frac { d u } { u } \right) d x =$
   (A) $\frac { 1 } { x ^ { 3 } } + C$
   (B) $\frac { 8 } { x ^ { 4 } } - \frac { 2 } { x ^ { 2 } } + C$
   (C) $\quad \ln ( \ln x ) + C$
   (D) $\frac { \ln \left( x ^ { 2 } \right) } { 2 } + C$
   (E) $\frac { ( \ln x ) ^ { 2 } } { 2 } + C$
5. If $\int _ { 1 } ^ { 10 } f ( x ) d x = 4$ and $\int _ { 10 } ^ { 3 } f ( x ) d x = 7$, then $\int _ { 1 } ^ { 3 } f ( x ) d x =$
   (A) - 3
   (B) 0
   (C) 3
   (D) 10
   (E) 11
6. The sides of the rectangle above increase in such a way that $\frac { d z } { d t } = 1$ and $\frac { d x } { d t } = 3 \frac { d y } { d t }$. At the instant when $x = 4$ and $y = 3$, what is the value of $\frac { d x } { d t }$ ?
   (A) $\frac { 1 } { 3 }$
   (B) 1
   (C) 2
   (D) $\sqrt { 5 }$
   (E) 5
7. If $\lim _ { x \rightarrow 3 } f ( x ) = 7$, which of the following must be true? I. $f$ is continuous at $x = 3$. II. $f$ is differentiable at $x = 3$. III. $f ( 3 ) = 7$
   (A) None
   (B) II only
   (C) III only
   (D) I and III only
   (E) I, II, and III
8. The graph of which of the following equations has $y = 1$ as an asymptote?
   (A) $y = \ln x$
   (B) $y = \sin x$
   (C) $y = \frac { x } { x + 1 }$
   (D) $y = \frac { x ^ { 2 } } { x - 1 }$
   (E) $y = e ^ { - x }$
9. The volume of the solid obtained by revolving the region enclosed by the ellipse $x ^ { 2 } + 9 y ^ { 2 } = 9$ about the $x$-axis is
   (A) $2 \pi$
   (B) $4 \pi$
   (C) $6 \pi$
   (D) $9 \pi$
   (E) $12 \pi$

1988 AP Calculus AB: Section I

1. Let $f$ and $g$ be odd functions. If $p , r$, and $s$ are nonzero functions defined as follows, which must be odd? I. $p ( x ) = f ( g ( x ) )$ II. $r ( x ) = f ( x ) + g ( x )$ III. $s ( x ) = f ( x ) g ( x )$
   (A) I only
   (B) II only
   (C) I and II only
   (D) II and III only
   (E) I, II, and III
2. The volume of a cylindrical tin can with a top and a bottom is to be $16 \pi$ cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?
   (A) $2 \sqrt [ 3 ] { 2 }$
   (B) $2 \sqrt { 2 }$
   (C) $2 \sqrt [ 3 ] { 4 }$
   (D) 4
   (E) 8

1988 AP Calculus BC: Section I

90 Minutes-No Calculator

Notes: (1) In this examination, $\ln x$ denotes the natural logarithm of $x$ (that is, logarithm to the base $e$ ).
(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 in the first quadrant enclosed by the graph of $y = x ( 1 - x )$ and the $x$-axis is
   (A) $\frac { 1 } { 6 }$
   (B) $\frac { 1 } { 3 }$
   (C) $\frac { 2 } { 3 }$
   (D) $\frac { 5 } { 6 }$
   (E) 1
2. $\int _ { 0 } ^ { 1 } x \left( x ^ { 2 } + 2 \right) ^ { 2 } d x =$
   (A) $\frac { 19 } { 2 }$
   (B) $\frac { 19 } { 3 }$
   (C) $\frac { 9 } { 2 }$
   (D) $\frac { 19 } { 6 }$
   (E) $\frac { 1 } { 6 }$
3. If $f ( x ) = \ln ( \sqrt { x } )$, then $f ^ { \prime \prime } ( x ) =$
   (A) $- \frac { 2 } { x ^ { 2 } }$
   (B) $- \frac { 1 } { 2 x ^ { 2 } }$
   (C) $- \frac { 1 } { 2 x }$
   (D) $- \frac { 1 } { 2 x ^ { \frac { 3 } { 2 } } }$
   (E) $\frac { 2 } { x ^ { 2 } }$
4. If $u , v$, and $w$ are nonzero differentiable functions, then the derivative of $\frac { u v } { w }$ is
   (A) $\frac { u v ^ { \prime } + u ^ { \prime } v } { w ^ { \prime } }$
   (B) $\frac { u ^ { \prime } v ^ { \prime } w - u v w ^ { \prime } } { w ^ { 2 } }$
   (C) $\frac { u v w ^ { \prime } - u v ^ { \prime } w - u ^ { \prime } v w } { w ^ { 2 } }$
   (D) $\frac { u ^ { \prime } v w + u v ^ { \prime } w + u v w ^ { \prime } } { w ^ { 2 } }$
   (E) $\frac { u v ^ { \prime } w + u ^ { \prime } v w - u v w ^ { \prime } } { w ^ { 2 } }$
5. Let $f$ be the function defined by the following.

$$f ( x ) = \left\{ \begin{aligned} \sin x , & x < 0 \\ x ^ { 2 } , & 0 \leq x < 1 \\ 2 - x , & 1 \leq x < 2 \\ x - 3 , & x \geq 2 \end{aligned} \right.$$
For what values of $x$ is $f$ NOT continuous?
(A) 0 only
(B) 1 only
(C) 2 only
(D) 0 and 2 only
(E) 0, 1, and 2

Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.

Let $f$ be the function defined for $\frac { \pi } { 6 } \leqq x \leq \frac { 5 \pi } { 6 }$ by $f ( x ) = x + \sin ^ { 2 } x$. (a) Find all values of $x$ for which $f ^ { \prime } ( x ) = 1$. (b) Find the $x$-coordinates of all minimum points of $f$. Justify your answer. (c) Find the $x$-coordinates of all inflection points of $f$. Justify your answer.

13. The graph of the function $f$ shown in the figure above has a vertical tangent at the point $( 2,0 )$ and horizontal tangents at the points $( 1 , - 1 )$ and $( 3,1 )$. For what values of $x , - 2 < x < 4$, is $f$ not differentiable?
(A) 0 only
(B) 0 and 2 only
(C) 1 and 3 only
(D) 0, 1, and 3 only
(E) 0, 1, 2, and 3

17. The graph of a twice-differentiable function $f$ is shown in the figure above. Which of the following is true?
(A) $f ( 1 ) < f ^ { \prime } ( 1 ) < f ^ { \prime \prime } ( 1 )$
(B) $f ( 1 ) < f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 )$
(C) $f ^ { \prime } ( 1 ) < f ( 1 ) < f ^ { \prime \prime } ( 1 )$
(D) $f ^ { \prime \prime } ( 1 ) < f ( 1 ) < f ^ { \prime } ( 1 )$
(E) $f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 ) < f ( 1 )$

19. If $f ^ { \prime \prime } ( x ) = x ( x + 1 ) ( x - 2 ) ^ { 2 }$, then the graph of $f$ has inflection points when $x =$
(A) - 1 only
(B) 2 only
(C) - 1 and 0 only
(D) -1 and 2 only
(E) $- 1,0$, and 2 only

22. The function $f$ is given by $f ( x ) = x ^ { 4 } + x ^ { 2 } - 2$. On which of the following intervals is $f$ increasing?
(A) $\left( - \frac { 1 } { \sqrt { 2 } } , \infty \right)$
(B) $\left( - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
(C) $( 0 , \infty )$
(D) $( - \infty , 0 )$
(E) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right)$ [Figure]

23. The graph of $f$ is shown in the figure above. Which of the following could be the graph of the derivative of $f$ ?
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
(E) [Figure]

Given the function defined by $\mathrm { y } = \mathrm { e } ^ { \sin \mathrm { x } }$ for all x such that $- \pi \leqq \mathrm { x } \leqq 2 \pi$. (a) Find the x - and y -coordinates of all maximum and minimum points on the given interval. Justify your answe (b) On the axes provided, sketch the graph of the function. (c) Write an equation for the axis of symmetry of the graph.

The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, for $- 7 \leq x \leq 7$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = - 3 , x = 2$, and $x = 5$, and a vertical tangent line at $x = 3$.
(a) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative minimum. Justify your answer.
(b) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative maximum. Justify your answer.
(c) Find all values of $x$, for $- 7 < x < 7$, at which $f ^ { \prime \prime } ( x ) < 0$.
(d) At what value of $x$, for $- 7 \leq x \leq 7$, does $f$ attain its absolute maximum? Justify your answer.

Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h^{\prime}(x) = \dfrac{x^{2} - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?

$$h ( x ) = \int _ { 1 } ^ { x } t ( t ) d t$$
a) $h ( 1 ) = \int ^ { \prime } f ( t ) d t = 0$
b) $$\begin{aligned} & h h ^ { \prime } ( x ) = \frac { d } { d x } \int _ { 0 } ^ { x } f ( t ) d t = f ( x ) \\ & h ^ { \prime } ( 4 ) = f ( 4 ) = 2 \end{aligned}$$
oncave up on $( 1,3 )$ and $( 6,7 )$ since $h ^ { \prime } ( x )$ is increa. those intervals
d) h(x) has no relotive minimum on [1,7] since $h ^ { \prime } ( x )$ does
change sign from - ve to tre
$\therefore$ Minimum must accur at either enopoint
$$\begin{aligned} & h ( 1 ) = 0 \\ & h ( 7 ) = \int _ { 1 } ^ { 7 } f ( t ) d t \end{aligned}$$
$h ( 7 ) > h ( 1 ) \quad$ since $h ( x )$ has a relative maximum $a t \quad x = 5$, and connot decrease to zero, $\int ^ { 5 } h ^ { \prime } ( x ) > \int _ { 5 } ^ { 7 } h ^ { \prime } ( x ) d x$

Let $f$ be a function that is differentiable for all real numbers. The table below gives the values of $f$ and its derivative $f ^ { \prime }$ for selected points $x$ in the closed interval $- 1.5 \leq x \leq 1.5$. The second derivative of $f$ has the property that $f ^ { \prime \prime } ( x ) > 0$ for $- 1.5 \leq x \leq 1.5$.

$x$	- 1.5	- 1.0	- 0.5	0	0.5	1.0	1.5
$f ( x )$	- 1	- 4	- 6	- 7	- 6	- 4	- 1
$f ^ { \prime } ( x )$	- 7	- 5	- 3	0	3	5	7

(a) Evaluate $\int _ { 0 } ^ { 1.5 } \left( 3 f ^ { \prime } ( x ) + 4 \right) d x$. Show the work that leads to your answer.
(b) Write an equation of the line tangent to the graph of $f$ at the point where $x = 1$. Use this line to approximate the value of $f ( 1.2 )$. Is this approximation greater than or less than the actual value of $f ( 1.2 )$? Give a reason for your answer.
(c) Find a positive real number $r$ having the property that there must exist a value $c$ with $0 < c < 0.5$ and $f ^ { \prime \prime } ( c ) = r$. Give a reason for your answer.
(d) Let $g$ be the function given by $g ( x ) = \begin{cases} 2 x ^ { 2 } - x - 7 & \text { for } x < 0 \\ 2 x ^ { 2 } + x - 7 & \text { for } x \geq 0 . \end{cases}$ The graph of $g$ passes through each of the points $( x , f ( x ) )$ given in the table above. Is it possible that $f$ and $g$ are the same function? Give a reason for your answer.

Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$.
(d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.

The figure above shows the graph of $f'$, the derivative of the function $f$, on the closed interval $-1 \leq x \leq 5$. The graph of $f'$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f(2) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $-1 \leq x \leq 5$? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $-1 \leq x \leq 5$? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g(x) = x f(x)$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.

4. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + p$, where $p$ is an arbitrary constant.
(a) Write an expression for $f ^ { \prime } ( x )$ and use it to find the relative maximum and minimum values of $f$ in terms of $p$. Show the analysis that leads to your conclusion.
(b) For what values of the constant $p$ does $f$ have 3 distinct real roots?
(c) Find the value of $p$ such that the average value of $f$ over the closed interval $[ - 1,2 ]$ is 1 . [Figure]

5. The graph of a function $f$ consists of a semicircle and two line segments as shown above. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 3 )$.
(b) Find all values of $x$ on the open interval $( - 2,5 )$ at which $g$ has a relative maximum. Justify your answer.
(c) Write an equation for the line tangent to the graph of $g$ at $x = 3$.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $( - 2,5 )$. Justify your answer.

Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table below, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.

$x$	0	$0 < x < 1$	1	$1 < x < 2$	2	$2 < x < 3$	3	$3 < x < 4$
$f ( x )$	- 1	Negative	0	Positive	2	Positive	0	Negative
$f ^ { \prime } ( x )$	4	Positive	0	Positive	DNE	Negative	- 3	Negative
$f ^ { \prime \prime } ( x )$	- 2	Negative	0	Positive	DNE	Negative	0	Positive

(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.

Let $f$ be the function defined by $f ( x ) = 2 x e ^ { - x }$ for all real numbers $x$. (a) Write an equation of the horizontal asymptote for the graph of $f$. (b) Find the $x$-coordinate of each critical point of $f$. For each such $x$, determine whether $f ( x )$ relative maximum, a relative minimum, or neither. (c) For what values of $x$ is the graph of $f$ concave down? (d) Using the results found in parts (a), (b), and (c), sketch the graph of $y = f ( x )$ in the $x y$ provided below. Note: The $x y$-plane is provided in the pink test booklet only.