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.
- $\int _ { 1 } ^ { 2 } x ^ { - 3 } d x =$
(A) $- \frac { 7 } { 8 }$
(B) $- \frac { 3 } { 4 }$
(C) $\frac { 15 } { 64 }$
(D) $\frac { 3 } { 8 }$
(E) $\frac { 15 } { 16 }$ - 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 - 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) $\frac { 3 } { 2 x }$ - 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) $\quad - \frac { 1 } { 2 } \sin ( 2 x ) + C$ - $\lim _ { n \rightarrow \infty } \frac { 4 n ^ { 2 } } { n ^ { 2 } + 10,000 n }$ is
(A) 0
(B) $\frac { 1 } { 2,500 }$
(C) 1
(D) 4
(E) nonexistent
1985 AP Calculus AB: Section I
- If $f ( x ) = x$, then $f ^ { \prime } ( 5 ) =$
(A) 0
(B) $\frac { 1 } { 5 }$
(C) 1
(D) 5
(E) $\frac { 25 } { 2 }$ - 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) $\quad \int _ { 1 } ^ { 4 } \frac { 1 } { t } d t$ - 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 - 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) $2 k$ - 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) $\quad x ^ { 2 } ( \ln 10 ) 10 ^ { \left( x ^ { 2 } - 1 \right) }$ - 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 - 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) $x + 2$ - 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) $- \frac { 2 x + y } { x + 3 y ^ { 2 } - 1 }$ - 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 - 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 - 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 - $\int _ { 0 } ^ { 1 } x e ^ { - x } d x =$
(A) $1 - 2 e$
(B) - 1
(C) $1 - 2 e ^ { - 1 }$
(D) 1
(E) $2 e - 1$ - 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) $\quad 2 ( \cos x - \sin x )$ - 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) $f ( x ) = x ^ { 2 }$ - 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) $\frac { 1 } { 1 + \cos ^ { 2 } x }$ - 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) $\{ x : 0 < x < \infty \}$ - $\int _ { 1 } ^ { 2 } \frac { x ^ { 2 } - 1 } { x + 1 } d x =$
(A) $\frac { 1 } { 2 }$
(B) 1
(C) 2
(D) $\frac { 5 } { 2 }$
(E) $\quad \ln 3$ - $\frac { d } { d x } \left( \frac { 1 } { x ^ { 3 } } - \frac { 1 } { x } + x ^ { 2 } \right)$ at $x = - 1$ is
(A) $\quad - 6$
(B) - 4
(C) 0
(D) 2
(E) 6 - If $\int _ { - 2 } ^ { 2 } \left( x ^ { 7 } + k \right) d x = 16$, then $k =$
(A) - 12
(B) - 4
(C) 0
(D) 4
(E) 12 - 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
- 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 - $\int _ { 0 } ^ { 3 } | x - 1 | d x =$
(A) 0
(B) $\frac { 3 } { 2 }$
(C) 2
(D) $\frac { 5 } { 2 }$
(E) 6 - 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 - 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 - $\int \tan ( 2 x ) d x =$
(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
- 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) $108 \pi$ - $\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ( 3 x ) d x =$
(A) - 2
(B) $- \frac { 2 } { 3 }$
(C) 0
(D) $\frac { 2 } { 3 }$
(E) 2 [Figure] - 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
- 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] - 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) $y = \sin ( 2 x )$ - 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 - $\lim _ { x \rightarrow 0 } ( x \csc x )$ is
(A) $- \infty$
(B) - 1
(C) 0
(D) 1
(E) $\infty$
1985 AP Calculus AB: Section I
- 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 - 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 $e$. - 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 - 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
- $\frac { d } { d x } \int _ { 2 } ^ { x } \sqrt { 1 + t ^ { 2 } } d t =$
(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) $\frac { 1 } { 2 \sqrt { 1 + x ^ { 2 } } } - \frac { 1 } { 2 \sqrt { 5 } }$ - 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) $y = 4 x + 1$ - 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 - 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.
- 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 - 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 - $\int _ { 1 } ^ { 2 } \frac { x + 1 } { x ^ { 2 } + 2 x } d x =$
(A) $\quad \ln 8 - \ln 3$
(B) $\frac { \ln 8 - \ln 3 } { 2 }$
(C) $\quad \ln 8$
(D) $\frac { 3 \ln 2 } { 2 }$
(E) $\frac { 3 \ln 2 + 2 } { 2 }$ - 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] - 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) $\quad \int _ { - 1 } ^ { a } ( | f ( x ) | - | g ( x ) | ) d x$ - 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
- 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) $\quad 2 \sqrt { 25 - x ^ { 2 } } + C$ - 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) $f ( 2 ) = 0$ - 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 - 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
- $\frac { d } { d x } \ln \left( \frac { 1 } { 1 - x } \right) =$
(A) $\frac { 1 } { 1 - x }$
(B) $\frac { 1 } { x - 1 }$
(C) $1 - x$
(D) $\quad x - 1$
(E) $( 1 - x ) ^ { 2 }$ - $\int \frac { d x } { ( x - 1 ) ( x + 2 ) } =$
(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) $\quad \ln \left| ( x - 1 ) ( x + 2 ) ^ { 2 } \right| + C$ - 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 - 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 - 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
- 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) $f ( x ) = x ^ { 3 }$ - 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) $\frac { 1 } { x }$ - $\int \sin ( 2 x + 3 ) d x =$
(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) $\quad \cos ( 2 x + 3 ) + C$ - 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] - 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
- 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) $\quad \cos x$ - 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) $12 \sqrt { 3 }$ - $\lim _ { h \rightarrow 0 } \frac { \int _ { 1 } ^ { 1 + h } \sqrt { x ^ { 5 } + 8 } d x } { h }$ is
(A) 0
(B) 1
(C) 3
(D) $2 \sqrt { 2 }$
(E) nonexistent - 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) $\frac { \pi } { 4 }$ - 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 $\frac { 1 } { 2 }$ - 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] - 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 - 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) $e ^ { e ^ { x } }$ - $\lim _ { x \rightarrow \frac { \pi } { 4 } } \frac { \sin \left( x - \frac { \pi } { 4 } \right) } { x - \frac { \pi } { 4 } }$ is
(A) 0
(B) $\frac { 1 } { \sqrt { 2 } }$
(C) $\frac { \pi } { 4 }$
(D) 1
(E) nonexistent - 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 - 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
- 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) $x + 4 y = - 25$ - 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) $\quad \ln 2$ - 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) $2 \pi \int _ { 0 } ^ { 1 } y \sqrt { 1 + y ^ { 6 } } d y$ - 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
- $\int _ { - 1 } ^ { 1 } \frac { 3 } { x ^ { 2 } } d x$ is
(A) - 6
(B) - 3
(C) 0
(D) 6
(E) nonexistent - 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) $y = C _ { 1 } e ^ { x } + C _ { 2 } x e ^ { - x }$ - $\lim _ { x \rightarrow \infty } \left( 1 + 5 e ^ { x } \right) ^ { \frac { 1 } { x } }$ is
(A) 0
(B) 1
(C) $e$
(D) $e ^ { 5 }$
(E) nonexistent - 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) $1 - e ^ { - 3 }$ - 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) $\quad \int _ { 2 } ^ { 4 } \frac { 1 - u ^ { 2 } } { 2 u } d u$ - 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 - 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) $\frac { 9 } { 2 }$ - 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] - 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) $y ^ { 2 } = x ^ { 3 } + 8$ - 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.
- 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) $\quad 2 x + e$ - 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 $x \neq 3 \}$ - 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) $\frac { e ^ { 3 } } { 3 }$ - 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) $x > 2$ - $\int \sec ^ { 2 } x d x =$
(A) $\quad \tan x + C$
(B) $\csc ^ { 2 } x + C$
(C) $\cos ^ { 2 } x + C$
(D) $\frac { \sec ^ { 3 } x } { 3 } + C$
(E) $2 \sec ^ { 2 } x \tan x + C$ - 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) $\frac { 1 + \ln x } { x ^ { 2 } }$ - $\int \frac { x d x } { \sqrt { 3 x ^ { 2 } + 5 } } =$
(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] - 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 - 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 - 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 - 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) $\quad y = 7 x - 8$ - 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) $\sqrt { 3 }$ - 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) $f ^ { \prime \prime } ( c ) - f ^ { \prime \prime } ( 0 )$ - $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos \theta } { \sqrt { 1 + \sin \theta } } d \theta =$
(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
- 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) $\sqrt { 2 }$ - 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 - $\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 2 } d x =$
(A) $- \frac { 7 } { 3 }$
(B) $- \frac { 7 } { 9 }$
(C) $\frac { 1 } { 9 }$
(D) 1
(E) 3 - 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) $- \frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right)$ - $\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } d x =$
(A) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
(B) $\frac { 1 } { 2 } \ln 2$
(C) $\ln 2$
(D) $2 \ln 2$
(E) $\frac { 1 } { 2 } \ln 5$ - 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
- 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) $\frac { 14 } { 3 }$ - 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) $e ^ { 2 }$ - 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 - $\frac { d } { d x } \left( x ^ { \ln x } \right) =$
(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) $\quad 2 ( \ln x ) \left( x ^ { \ln x } \right)$ - 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) $e ^ { x }$ - $\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x =$
(A) $- \frac { \pi } { 2 }$
(B) - 1
(C) $1 - \frac { \pi } { 2 }$
(D) 1
(E) $\frac { \pi } { 2 } - 1$
1988 AP Calculus AB: Section I
- 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. - $\int _ { 1 } ^ { 4 } | x - 3 | d x =$
(A) $- \frac { 3 } { 2 }$
(B) $\frac { 3 } { 2 }$
(C) $\frac { 5 } { 2 }$
(D) $\frac { 9 } { 2 }$
(E) 5 - 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 - 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
- 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) $x$ - 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) $f ( x ) = \tan ^ { 2 } x$ - 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] - 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
- $4 \cos \left( x + \frac { \pi } { 3 } \right) =$
(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) $\quad 4 \cos x + 2$ - 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) $\frac { 9 } { 2 }$ - 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 - 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) $\frac { ( \ln x ) ^ { 2 } } { 2 } + C$ - 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 - 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 - 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 - 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) $y = e ^ { - x }$ - 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
- 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 - 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.
- 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 - $\int _ { 0 } ^ { 1 } x \left( x ^ { 2 } + 2 \right) ^ { 2 } d x =$
(A) $\frac { 19 } { 2 }$
(B) $\frac { 19 } { 3 }$
(C) $\frac { 9 } { 2 }$
(D) $\frac { 19 } { 6 }$
(E) $\frac { 1 } { 6 }$ - 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) $\frac { 2 } { x ^ { 2 } }$ - 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) $\frac { u v ^ { \prime } w + u ^ { \prime } v w - u v w ^ { \prime } } { w ^ { 2 } }$ - 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