24. The expression $\frac { 1 } { 50 } \left( \sqrt { \frac { 1 } { 50 } } + \sqrt { \frac { 2 } { 50 } } + \sqrt { \frac { 3 } { 50 } } + \cdots + \sqrt { \frac { 50 } { 50 } } \right)$ is a Riemann sum approximation for
(A) $\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 50 } } d x$
(B) $\int _ { 0 } ^ { 1 } \sqrt { x } d x$
(C) $\frac { 1 } { 50 } \int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 50 } } d x$
(D) $\frac { 1 } { 50 } \int _ { 0 } ^ { 1 } \sqrt { x } d x$
(E) $\frac { 1 } { 50 } \int _ { 0 } ^ { 50 } \sqrt { x } d x$ 25. $\int x \sin ( 2 x ) d x =$
(A) $- \frac { x } { 2 } \cos ( 2 x ) + \frac { 1 } { 4 } \sin ( 2 x ) + C$
(B) $- \frac { x } { 2 } \cos ( 2 x ) - \frac { 1 } { 4 } \sin ( 2 x ) + C$
(C) $\frac { x } { 2 } \cos ( 2 x ) - \frac { 1 } { 4 } \sin ( 2 x ) + C$
(D) $- 2 x \cos ( 2 x ) + \sin ( 2 x ) + C$
(E) $\quad - 2 x \cos ( 2 x ) - 4 \sin ( 2 x ) + C$
40 Minutes-Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number. 76. If $f ( x ) = \frac { e ^ { 2 x } } { 2 x }$, then $f ^ { \prime } ( x ) =$
(A) 1
(B) $\frac { e ^ { 2 x } ( 1 - 2 x ) } { 2 x ^ { 2 } }$
(C) $e ^ { 2 x }$
(D) $\frac { e ^ { 2 x } ( 2 x + 1 ) } { x ^ { 2 } }$
(E) $\frac { e ^ { 2 x } ( 2 x - 1 ) } { 2 x ^ { 2 } }$ 77. The graph of the function $y = x ^ { 3 } + 6 x ^ { 2 } + 7 x - 2 \cos x$ changes concavity at $x =$
(A) - 1.58
(B) - 1.63
(C) - 1.67
(D) - 1.89
(E) - 2.33
[Figure] 78. The graph of $f$ is shown in the figure above. If $\int _ { 1 } ^ { 3 } f ( x ) d x = 2.3$ and $F ^ { \prime } ( x ) = f ( x )$, then $F ( 3 ) - F ( 0 ) =$
(A) 0.3
(B) 1.3
(C) 3.3
(D) 4.3
(E) 5.3 79. Let $f$ be a function such that $\lim _ { h \rightarrow 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } = 5$. Which of the following must be true? I. $f$ is continuous at $x = 2$. II. $\quad f$ is differentiable at $x = 2$. III. The derivative of $f$ is continuous at $x = 2$.
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only 80. Let $f$ be the function given by $f ( x ) = 2 e ^ { 4 x ^ { 2 } }$. For what value of $x$ is the slope of the line tangent to the graph of $f$ at $( x , f ( x ) )$ equal to 3 ?
(A) 0.168
(B) 0.276
(C) 0.318
(D) 0.342
(E) 0.551 81. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection?
(A) 57.60
(B) 57.88
(C) 59.20
(D) 60.00
(E) 67.40 82. If $y = 2 x - 8$, what is the minimum value of the product $x y$ ?
(A) - 16
(B) - 8
(C) $\quad - 4$
(D) 0
(E) 2 83. What is the area of the region in the first quadrant enclosed by the graphs of $y = \cos x , y = x$, and the $y$-axis?
(A) 0.127
(B) 0.385
(C) 0.400
(D) 0.600
(E) 0.947 84. The base of a solid $S$ is the region enclosed by the graph of $y = \sqrt { \ln x }$, the line $x = e$, and the $x$-axis. If the cross sections of $S$ perpendicular to the $x$-axis are squares, then the volume of $S$ is
(A) $\frac { 1 } { 2 }$
(B) $\frac { 2 } { 3 }$
(C) 1
(D) 2
(E) $\frac { 1 } { 3 } \left( e ^ { 3 } - 1 \right)$ 85. If the derivative of $f$ is given by $f ^ { \prime } ( x ) = e ^ { x } - 3 x ^ { 2 }$, at which of the following values of $x$ does $f$ have a relative maximum value?
(A) - 0.46
(B) 0.20
(C) 0.91
(D) 0.95
(E) 3.73 86. Let $f ( x ) = \sqrt { x }$. If the rate of change of $f$ at $x = c$ is twice its rate of change at $x = 1$, then $c =$
(A) $\frac { 1 } { 4 }$
(B) 1
(C) 4
(D) $\frac { 1 } { \sqrt { 2 } }$
(E) $\frac { 1 } { 2 \sqrt { 2 } }$ 87. At time $t \geq 0$, the acceleration of a particle moving on the $x$-axis is $a ( t ) = t + \sin t$. At $t = 0$, the velocity of the particle is - 2 . For what value $t$ will the velocity of the particle be zero?
(A) 1.02
(B) 1.48
(C) 1.85
(D) 2.81
(E) 3.14
[Figure] 88. Let $f ( x ) = \int _ { a } ^ { x } h ( t ) d t$, where $h$ has the graph shown above. Which of the following could be the graph of $f$ ?
(A)
[Figure] [Figure](C)
[Figure] [Figure](E)
[Figure]| $x$ | 0 | 0.5 | 1.0 | 1.5 | 2.0 |
| $f ( x )$ | 3 | 3 | 5 | 8 | 13 |
- A table of values for a continuous function $f$ is shown above. If four equal subintervals of $[ 0,2 ]$ are used, which of the following is the trapezoidal approximation of $\int _ { 0 } ^ { 2 } f ( x ) d x ?$
(A) 8
(B) 12
(C) 16
(D) 24
(E) 32 - Which of the following are antiderivatives of $f ( x ) = \sin x \cos x$ ? I. $F ( x ) = \frac { \sin ^ { 2 } x } { 2 }$ II. $F ( x ) = \frac { \cos ^ { 2 } x } { 2 }$ III. $F ( x ) = \frac { - \cos ( 2 x ) } { 4 }$
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
50 Minutes-No Calculator
Note: 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.
- $\int _ { 0 } ^ { 1 } \sqrt { x } ( x + 1 ) d x =$
(A) 0
(B) 1
(C) $\frac { 16 } { 15 }$
(D) $\frac { 7 } { 5 }$
(E) 2 - If $x = e ^ { 2 t }$ and $y = \sin ( 2 t )$, then $\frac { d y } { d x } =$
(A) $\quad 4 e ^ { 2 t } \cos ( 2 \mathrm { t } )$
(B) $\frac { e ^ { 2 t } } { \cos ( 2 \mathrm { t } ) }$
(C) $\frac { \sin ( 2 \mathrm { t } ) } { 2 e ^ { 2 t } }$
(D) $\frac { \cos ( 2 \mathrm { t } ) } { 2 e ^ { 2 t } }$
(E) $\frac { \cos ( 2 \mathrm { t } ) } { e ^ { 2 t } }$ - The function $f$ given by $f ( x ) = 3 x ^ { 5 } - 4 x ^ { 3 } - 3 x$ has a relative maximum at $x =$
(A) - 1
(B) $- \frac { \sqrt { 5 } } { 5 }$
(C) 0
(D) $\frac { \sqrt { 5 } } { 5 }$
(E) 1 - $\frac { d } { d x } \left( x e ^ { \ln x ^ { 2 } } \right) =$
(A) $1 + 2 x$
(B) $x + x ^ { 2 }$
(C) $3 x ^ { 2 }$
(D) $x ^ { 3 }$
(E) $x ^ { 2 } + x ^ { 3 }$ - If $f ( x ) = ( x - 1 ) ^ { \frac { 3 } { 2 } } + \frac { e ^ { x - 2 } } { 2 }$, then $f ^ { \prime } ( 2 ) =$
(A) 1
(B) $\frac { 3 } { 2 }$
(C) 2
(D) $\frac { 7 } { 2 }$
(E) $\frac { 3 + e } { 2 }$ - The line normal to the curve $y = \sqrt { 16 - x }$ at the point $( 0,4 )$ has slope
(A) 8
(B) 4
(C) $\frac { 1 } { 8 }$
(D) $- \frac { 1 } { 8 }$
(E) - 8
Questions 7-9 refer to the graph and the information below.
[Figure]The function $f$ is defined on the closed interval $[ 0,8 ]$. The graph of its derivative $f ^ { \prime }$ is shown above. 7. The point $( 3,5 )$ is on the graph of $y = f ( x )$. An equation of the line tangent to the graph of $f$ at $( 3,5 )$ is
(A) $y = 2$
(B) $y = 5$
(C) $y - 5 = 2 ( x - 3 )$
(D) $y + 5 = 2 ( x - 3 )$
(E) $y + 5 = 2 ( x + 3 )$ 8. How many points of inflection does the graph of $f$ have?
(A) Two
(B) Three
(C) Four
(D) Five
(E) Six 9. At what value of $x$ does the absolute minimum of $f$ occur?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8 10. If $y = x y + x ^ { 2 } + 1$, then when $x = - 1 , \frac { d y } { d x }$ is
(A) $\frac { 1 } { 2 }$
(B) $- \frac { 1 } { 2 }$
(C) $\quad - 1$
(D) - 2
(E) nonexistent 11. $\int _ { 1 } ^ { \infty } \frac { x } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x$ is
(A) $- \frac { 1 } { 2 }$
(B) $- \frac { 1 } { 4 }$
(C) $\frac { 1 } { 4 }$
(D) $\frac { 1 } { 2 }$
(E) divergent
[Figure] 12. The graph of $f ^ { \prime }$, the derivative of $f$, is shown in the figure above. Which of the following describes all relative extrema of $f$ on the open interval $( a , b )$ ?
(A) One relative maximum and two relative minima
(B) Two relative maxima and one relative minimum
(C) Three relative maxima and one relative minimum
(D) One relative maximum and three relative minima
(E) Three relative maxima and two relative minima 13. A particle moves along the $x$-axis so that its acceleration at any time $t$ is $a ( t ) = 2 t - 7$. If the initial velocity of the particle is 6 , at what time $t$ during the interval $0 \leq t \leq 4$ is the particle farthest to the right?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4 14. The sum of the infinite geometric series $\frac { 3 } { 2 } + \frac { 9 } { 16 } + \frac { 27 } { 128 } + \frac { 81 } { 1,024 } + \ldots$ is
(A) 1.60
(B) 2.35
(C) 2.40
(D) 2.45
(E) 2.50 15. The length of the path described by the parametric equations $x = \cos ^ { 3 } t$ and $y = \sin ^ { 3 } t$, for $0 \leq t \leq \frac { \pi } { 2 }$, is given by
(A) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 3 \cos ^ { 2 } t + 3 \sin ^ { 2 } t } d t$
(B) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { - 3 \cos ^ { 2 } t \sin t + 3 \sin ^ { 2 } t \cos t } d t$
(C) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 9 \cos ^ { 4 } t + 9 \sin ^ { 4 } t } d t$
(D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 9 \cos ^ { 4 } t \sin ^ { 2 } t + 9 \sin ^ { 4 } t \cos ^ { 2 } t } d t$
(E) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { \cos ^ { 6 } t + \sin ^ { 6 } t } d t$ 16. $\lim _ { h \rightarrow 0 } \frac { e ^ { h } - 1 } { 2 h }$ is
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) $e$
(E) nonexistent 17. Let $f$ be the function given by $f ( x ) = \ln ( 3 - x )$. The third-degree Taylor polynomial for $f$ about $x = 2$ is
(A) $\quad - ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 2 } - \frac { ( x - 2 ) ^ { 3 } } { 3 }$
(B) $\quad - ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 2 } - \frac { ( x - 2 ) ^ { 3 } } { 3 }$
(C) $( x - 2 ) + ( x - 2 ) ^ { 2 } + ( x - 2 ) ^ { 3 }$
(D) $\quad ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 2 } + \frac { ( x - 2 ) ^ { 3 } } { 3 }$
(E) $\quad ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 2 } + \frac { ( x - 2 ) ^ { 3 } } { 3 }$ 18. For what values of $t$ does the curve given by the parametric equations $x = t ^ { 3 } - t ^ { 2 } - 1$ and $y = t ^ { 4 } + 2 t ^ { 2 } - 8 t$ have a vertical tangent?
(A) 0 only
(B) 1 only
(C) 0 and $\frac { 2 } { 3 }$ only
(D) $0 , \frac { 2 } { 3 }$, and 1
(E) No value
[Figure] 19. The graph of $y = f ( x )$ is shown in the figure above. If $A _ { 1 }$ and $A _ { 2 }$ are positive numbers that represent the areas of the shaded regions, then in terms of $A _ { 1 }$ and $A _ { 2 }$, $\int _ { - 4 } ^ { 4 } f ( x ) d x - 2 \int _ { - 1 } ^ { 4 } f ( x ) d x =$
(A) $A _ { 1 }$
(B) $A _ { 1 } - A _ { 2 }$
(C) $2 A _ { 1 } - A _ { 2 }$
(D) $A _ { 1 } + A _ { 2 }$
(E) $A _ { 1 } + 2 A _ { 2 }$ 20. What are all values of $x$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 2 ) ^ { n } } { n \cdot 3 ^ { n } }$ converges?
(A) $- 3 \leq x \leq 3$
(B) $- 3 < x < 3$
(C) $- 1 < x \leq 5$
(D) $- 1 \leq x \leq 5$
(E) $- 1 \leq x < 5$ 21. Which of the following is equal to the area of the region inside the polar curve $r = 2 \cos \theta$ and outside the polar curve $r = \cos \theta$ ?
(A) $3 \int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } \theta d \theta$
(B) $3 \int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta d \theta$
(C) $\frac { 3 } { 2 } \int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } \theta d \theta$
(D) $3 \int _ { 0 } ^ { \frac { \pi } { 2 } } \cos \theta d \theta$
(E) $3 \int _ { 0 } ^ { \pi } \cos \theta d \theta$
[Figure] 22. The graph of $f$ is shown in the figure above. If $g ( x ) = \int _ { a } ^ { x } f ( t ) d t$, for what value of $x$ does $g ( x )$ have a maximum?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) It cannot be determined from the information given. 23. In the triangle shown above, if $\theta$ increases at a constant rate of 3 radians per minute, at what rate is $x$ increasing in units per minute when $x$ equals 3 units?
(A) 3
(B) $\frac { 15 } { 4 }$
(C) 4
(D) 9
(E) 12 24. The Taylor series for $\sin x$ about $x = 0$ is $x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \ldots$. If $f$ is a function such that $f ^ { \prime } ( x ) = \sin \left( x ^ { 2 } \right)$, then the coefficient of $x ^ { 7 }$ in the Taylor series for $f ( x )$ about $x = 0$ is
(A) $\frac { 1 } { 7 ! }$
(B) $\frac { 1 } { 7 }$
(C) 0
(D) $- \frac { 1 } { 42 }$
(E) $- \frac { 1 } { 7 ! }$ 25. The closed interval $[ a , b ]$ is partitioned into $n$ equal subintervals, each of width $\Delta x$, by the numbers $x _ { 0 } , x _ { 1 } , \ldots , x _ { n }$ where $a = x _ { 0 } < x _ { 1 } < x _ { 2 } < \cdots < x _ { n - 1 } < x _ { n } = b$. What is $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \sqrt { x _ { i } } \Delta x$ ?
(A) $\frac { 2 } { 3 } \left( b ^ { \frac { 3 } { 2 } } - a ^ { \frac { 3 } { 2 } } \right)$
(B) $b ^ { \frac { 3 } { 2 } } - a ^ { \frac { 3 } { 2 } }$
(C) $\frac { 3 } { 2 } \left( b ^ { \frac { 3 } { 2 } } - a ^ { \frac { 3 } { 2 } } \right)$
(D) $b ^ { \frac { 1 } { 2 } } - a ^ { \frac { 1 } { 2 } }$
(E) $\quad 2 \left( b ^ { \frac { 1 } { 2 } } - a ^ { \frac { 1 } { 2 } } \right)$
40 Minutes-Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number. 76. Which of the following sequences converge? I. $\left\{ \frac { 5 n } { 2 n - 1 } \right\}$ II. $\left\{ \frac { e ^ { n } } { n } \right\}$ III. $\left\{ \frac { e ^ { n } } { 1 + e ^ { n } } \right\}$
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III 77. When the region enclosed by the graphs of $y = x$ and $y = 4 x - x ^ { 2 }$ is revolved about the $y$-axis, the volume of the solid generated is given by
(A) $\pi \int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) d x$
(B) $\pi \int _ { 0 } ^ { 3 } \left( x ^ { 2 } - \left( 4 x - x ^ { 2 } \right) ^ { 2 } \right) d x$
(C) $\pi \int _ { 0 } ^ { 3 } \left( 3 x - x ^ { 2 } \right) ^ { 2 } d x$
(D) $2 \pi \int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) d x$
(E) $\quad 2 \pi \int _ { 0 } ^ { 3 } \left( 3 x ^ { 2 } - x ^ { 3 } \right) d x$ 78. $\lim _ { h \rightarrow 0 } \frac { \ln ( e + h ) - 1 } { h }$ is
(A) $f ^ { \prime } ( e )$, where $f ( x ) = \ln x$
(B) $f ^ { \prime } ( e )$, where $f ( x ) = \frac { \ln x } { x }$
(C) $f ^ { \prime } ( 1 )$, where $f ( x ) = \ln x$
(D) $f ^ { \prime } ( 1 )$, where $f ( x ) = \ln ( x + e )$
(E) $\quad f ^ { \prime } ( 0 )$, where $f ( x ) = \ln x$ 79. The position of an object attached to a spring is given by $y ( t ) = \frac { 1 } { 6 } \cos ( 5 t ) - \frac { 1 } { 4 } \sin ( 5 t )$, where $t$ is time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to 0 ?
(A) Zero
(B) Three
(C) Five
(D) Six
(E) Seven 80. Let $f$ be the function given by $f ( x ) = \cos ( 2 x ) + \ln ( 3 x )$. What is the least value of $x$ at which the graph of $f$ changes concavity?
(A) 0.56
(B) 0.93
(C) 1.18
(D) 2.38
(E) 2.44 81. Let $f$ be a continuous function on the closed interval $[ - 3,6 ]$. If $f ( - 3 ) = - 1$ and $f ( 6 ) = 3$, then the Intermediate Value Theorem guarantees that
(A) $f ( 0 ) = 0$
(B) $f ^ { \prime } ( c ) = \frac { 4 } { 9 }$ for at least one $c$ between - 3 and 6
(C) $- 1 \leq f ( x ) \leq 3$ for all $x$ between - 3 and 6
(D) $f ( c ) = 1$ for at least one $c$ between - 3 and 6
(E) $\quad f ( c ) = 0$ for at least one $c$ between - 1 and 3 82. If $0 \leq x \leq 4$, of the following, which is the greatest value of $x$ such that $\int _ { 0 } ^ { x } \left( t ^ { 2 } - 2 t \right) d t \geq \int _ { 2 } ^ { x } t d t$ ?
(A) 1.35
(B) 1.38
(C) 1.41
(D) 1.48
(E) 1.59 83. If $\frac { d y } { d x } = ( 1 + \ln x ) y$ and if $y = 1$ when $x = 1$, then $y =$
(A) $e ^ { \frac { x ^ { 2 } - 1 } { x ^ { 2 } } }$
(B) $1 + \ln x$
(C) $\ln x$
(D) $e ^ { 2 x + x \ln x - 2 }$
(E) $e ^ { x \ln x }$ 84. $\int x ^ { 2 } \sin x d x =$
(A) $- x ^ { 2 } \cos x - 2 x \sin x - 2 \cos x + C$
(B) $- x ^ { 2 } \cos x + 2 x \sin x - 2 \cos x + C$
(C) $- x ^ { 2 } \cos x + 2 x \sin x + 2 \cos x + C$
(D) $- \frac { x ^ { 3 } } { 3 } \cos x + C$
(E) $\quad 2 x \cos x + C$ 85. Let $f$ be a twice differentiable function such that $f ( 1 ) = 2$ and $f ( 3 ) = 7$. Which of the following must be true for the function $f$ on the interval $1 \leq x \leq 3$ ? I. The average rate of change of $f$ is $\frac { 5 } { 2 }$. II. The average value of $f$ is $\frac { 9 } { 2 }$. III. The average value of $f ^ { \prime }$ is $\frac { 5 } { 2 }$.
(A) None
(B) I only
(C) III only
(D) I and III only
(E) II and III only 86. $\int \frac { d x } { ( x - 1 ) ( x + 3 ) } =$
(A) $\frac { 1 } { 4 } \ln \left| \frac { x - 1 } { x + 3 } \right| + C$
(B) $\frac { 1 } { 4 } \ln \left| \frac { x + 3 } { x - 1 } \right| + C$
(C) $\frac { 1 } { 2 } \ln | ( x - 1 ) ( x + 3 ) | + C$
(D) $\frac { 1 } { 2 } \ln \left| \frac { 2 x + 2 } { ( x - 1 ) ( x + 3 ) } \right| + C$
(E) $\quad \ln | ( x - 1 ) ( x + 3 ) | + C$ 87. The base of a solid is the region in the first quadrant enclosed by the graph of $y = 2 - x ^ { 2 }$ and the coordinate axes. If every cross section of the solid perpendicular to the $y$-axis is a square, the volume of the solid is given by
(A) $\pi \int _ { 0 } ^ { 2 } ( 2 - y ) ^ { 2 } d y$
(B) $\int _ { 0 } ^ { 2 } ( 2 - y ) d y$
(C) $\pi \int _ { 0 } ^ { \sqrt { 2 } } \left( 2 - x ^ { 2 } \right) ^ { 2 } d x$
(D) $\int _ { 0 } ^ { \sqrt { 2 } } \left( 2 - x ^ { 2 } \right) ^ { 2 } d x$
(E) $\int _ { 0 } ^ { \sqrt { 2 } } \left( 2 - x ^ { 2 } \right) d x$ 88. Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sin t d t$. At how many points in the closed interval $[ 0 , \sqrt { \pi } ]$ does the instantaneous rate of change of $f$ equal the average rate of change of $f$ on that interval?
(A) Zero
(B) One
(C) Two
(D) Three
(E) Four 89. If $f$ is the antiderivative of $\frac { x ^ { 2 } } { 1 + x ^ { 5 } }$ such that $f ( 1 ) = 0$, then $f ( 4 ) =$
(A) - 0.012
(B) 0
(C) 0.016
(D) 0.376
(E) 0.629 90. A force of 10 pounds is required to stretch a spring 4 inches beyond its natural length. Assuming Hooke's law applies, how much work is done in stretching the spring from its natural length to 6 inches beyond its natural length?
(A) 60.0 inch-pounds
(B) 45.0 inch-pounds
(C) 40.0 inch-pounds
(D) 15.0 inch-pounds
(E) 7.2 inch-pounds
55 Minutes-No Calculator
Note: 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.
- What is the $x$-coordinate of the point of inflection on the graph of $y = \frac { 1 } { 3 } x ^ { 3 } + 5 x ^ { 2 } + 24$ ?
(A) 5
(B) 0
(C) $- \frac { 10 } { 3 }$
(D) - 5
(E) - 10 [Figure] - The graph of a piecewise-linear function $f$, for $- 1 \leq x \leq 4$, is shown above. What is the value of $\int _ { - 1 } ^ { 4 } f ( x ) d x$ ?
(A) 1
(B) 2.5
(C) 4
(D) 5.5
(E) 8 - $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x =$
(A) $- \frac { 1 } { 2 }$
(B) $\frac { 7 } { 24 }$
(C) $\frac { 1 } { 2 }$
(D) 1
(E) $2 \ln 2$ - If $f$ is continuous for $a \leq x \leq b$ and differentiable for $a < x < b$, which of the following could be false?
(A) $f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a }$ for some $c$ such that $a < c < b$.
(B) $f ^ { \prime } ( c ) = 0$ for some $c$ such that $a < c < b$.
(C) $f$ has a minimum value on $a \leq x \leq b$.
(D) $f$ has a maximum value on $a \leq x \leq b$.
(E) $\int _ { a } ^ { b } f ( x ) d x$ exists. - $\int _ { 0 } ^ { x } \sin t d t =$
(A) $\sin x$
(B) $- \cos x$
(C) $\cos x$
(D) $\cos x - 1$
(E) $1 - \cos x$ - If $x ^ { 2 } + x y = 10$, then when $x = 2 , \frac { d y } { d x } =$
(A) $- \frac { 7 } { 2 }$
(B) - 2
(C) $\frac { 2 } { 7 }$
(D) $\frac { 3 } { 2 }$
(E) $\frac { 7 } { 2 }$ - $\int _ { 1 } ^ { e } \left( \frac { x ^ { 2 } - 1 } { x } \right) d x =$
(A) $e - \frac { 1 } { e }$
(B) $e ^ { 2 } - e$
(C) $\frac { e ^ { 2 } } { 2 } - e + \frac { 1 } { 2 }$
(D) $e ^ { 2 } - 2$
(E) $\frac { e ^ { 2 } } { 2 } - \frac { 3 } { 2 }$ - Let $f$ and $g$ be differentiable functions with the following properties:
(i) $g ( x ) > 0$ for all $x$
(ii) $\quad f ( 0 ) = 1$
If $h ( x ) = f ( x ) g ( x )$ and $h ^ { \prime } ( x ) = f ( x ) g ^ { \prime } ( x )$, then $f ( x ) =$
(A) $f ^ { \prime } ( x )$
(B) $g ( x )$
(C) $e ^ { x }$
(D) 0
(E) 1
[Figure] 9. The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?
(A) 500
(B) 600
(C) 2,400
(D) 3,000
(E) 4,800 10. What is the instantaneous rate of change at $x = 2$ of the function $f$ given by $f ( x ) = \frac { x ^ { 2 } - 2 } { x - 1 }$ ?
(A) - 2
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 2 }$
(D) 2
(E) 6 11. If $f$ is a linear function and $0 < a < b$, then $\int _ { a } ^ { b } f ^ { \prime \prime } ( x ) d x =$
(A) 0
(B) 1
(C) $\frac { a b } { 2 }$
(D) $b - a$
(E) $\frac { b ^ { 2 } - a ^ { 2 } } { 2 }$ 12. If $f ( x ) = \left\{ \begin{array} { r r } \ln x & \text { for } 0 < x \leq 2 \\ x ^ { 2 } \ln 2 & \text { for } 2 < x \leq 4 , \end{array} \right.$ then $\lim _ { x \rightarrow 2 } f ( x )$ is
(A) $\ln 2$
(B) $\quad \ln 8$
(C) $\quad \ln 16$
(D) 4
(E) nonexistent
[Figure] 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 14. A particle moves along the $x$-axis so that its position at time $t$ is given by $x ( t ) = t ^ { 2 } - 6 t + 5$. For what value of $t$ is the velocity of the particle zero?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5 15. If $F ( x ) = \int _ { 0 } ^ { x } \sqrt { t ^ { 3 } + 1 } d t$, then $F ^ { \prime } ( 2 ) =$
(A) - 3
(B) - 2
(C) 2
(D) 3
(E) 18 16. If $f ( x ) = \sin \left( e ^ { - x } \right)$, then $f ^ { \prime } ( x ) =$
(A) $\quad - \cos \left( e ^ { - x } \right)$
(B) $\quad \cos \left( e ^ { - x } \right) + e ^ { - x }$
(C) $\quad \cos \left( e ^ { - x } \right) - e ^ { - x }$
(D) $e ^ { - x } \cos \left( e ^ { - x } \right)$
(E) $\quad - e ^ { - x } \cos \left( e ^ { - x } \right)$
[Figure] 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) $f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 ) < f ( 1 )$ 18. An equation of the line tangent to the graph of $y = x + \cos x$ at the point $( 0,1 )$ is
(A) $y = 2 x + 1$
(B) $y = x + 1$
(C) $y = x$
(D) $y = x - 1$
(E) $y = 0$ 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 20. What are all values of $k$ for which $\int _ { - 3 } ^ { k } x ^ { 2 } d x = 0$ ?
(A) - 3
(B) 0
(C) 3
(D) -3 and 3
(E) $- 3,0$, and 3 21. If $\frac { d y } { d t } = k y$ and $k$ is a nonzero constant, then $y$ could be
(A) $2 e ^ { k t y }$
(B) $2 e ^ { k t }$
(C) $e ^ { k t } + 3$
(D) $k t y + 5$
(E) $\frac { 1 } { 2 } k y ^ { 2 } + \frac { 1 } { 2 }$ 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] 24. The maximum acceleration attained on the interval $0 \leq t \leq 3$ by the particle whose velocity is given by $v ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 12 t + 4$ is
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40 25. What is the area of the region between the graphs of $y = x ^ { 2 }$ and $y = - x$ from $x = 0$ to $x = 2$ ?
(A) $\frac { 2 } { 3 }$
(B) $\frac { 8 } { 3 }$
(C) 4
(D) $\frac { 14 } { 3 }$
(E) $\frac { 16 } { 3 }$
- The function $f$ is continuous on the closed interval $[ 0,2 ]$ and has values that are given in the table above. The equation $f ( x ) = \frac { 1 } { 2 }$ must have at least two solutions in the interval $[ 0,2 ]$ if $k =$
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) 3 - What is the average value of $y = x ^ { 2 } \sqrt { x ^ { 3 } + 1 }$ on the interval $[ 0,2 ]$ ?
(A) $\frac { 26 } { 9 }$
(B) $\frac { 52 } { 9 }$
(C) $\frac { 26 } { 3 }$
(D) $\frac { 52 } { 3 }$
(E) 24 - If $f ( x ) = \tan ( 2 x )$, then $f ^ { \prime } \left( \frac { \pi } { 6 } \right) =$
(A) $\sqrt { 3 }$
(B) $2 \sqrt { 3 }$
(C) 4
(D) $4 \sqrt { 3 }$
(E) 8
50 Minutes-Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.
[Figure] 76. The graph of a function $f$ is shown above. Which of the following statements about $f$ is false?
(A) $f$ is continuous at $x = a$.
(B) $f$ has a relative maximum at $x = a$.
(C) $x = a$ is in the domain of $f$.
(D) $\lim _ { x \rightarrow a ^ { + } } f ( x )$ is equal to $\lim _ { x \rightarrow a ^ { - } } f ( x )$.
(E) $\lim _ { x \rightarrow a } f ( x )$ exists. 77. Let $f$ be the function given by $f ( x ) = 3 e ^ { 2 x }$ and let $g$ be the function given by $g ( x ) = 6 x ^ { 3 }$. At what value of $x$ do the graphs of $f$ and $g$ have parallel tangent lines?
(A) - 0.701
(B) - 0.567
(C) - 0.391
(D) - 0.302
(E) - 0.258 78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference $C$, what is the rate of change of the area of the circle, in square centimeters per second?
(A) $- ( 0.2 ) \pi C$
(B) $- ( 0.1 ) C$
(C) $- \frac { ( 0.1 ) C } { 2 \pi }$
(D) $\quad ( 0.1 ) ^ { 2 } C$
(E) $\quad ( 0.1 ) ^ { 2 } \pi C$
[Figure] [Figure] [Figure] 79. The graphs of the derivatives of the functions $f , g$, and $h$ are shown above. Which of the functions $f , g$, or $h$ have a relative maximum on the open interval $a < x < b$ ?
(A) $f$ only
(B) $g$ only
(C) $h$ only
(D) $f$ and $g$ only
(E) $f , g$, and $h$ 80. The first derivative of the function $f$ is given by $f ^ { \prime } ( x ) = \frac { \cos ^ { 2 } x } { x } - \frac { 1 } { 5 }$. How many critical values does $f$ have on the open interval $( 0,10 )$ ?
(A) One
(B) Three
(C) Four
(D) Five
(E) Seven 81. Let $f$ be the function given by $f ( x ) = | x |$. Which of the following statements about $f$ are true? I. $f$ is continuous at $x = 0$. II. $\quad f$ is differentiable at $x = 0$. III. $f$ has an absolute minimum at $x = 0$.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only 82. If $f$ is a continuous function and if $F ^ { \prime } ( x ) = f ( x )$ for all real numbers $x$, then $\int _ { 1 } ^ { 3 } f ( 2 x ) d x =$
(A) $2 F ( 3 ) - 2 F ( 1 )$
(B) $\frac { 1 } { 2 } F ( 3 ) - \frac { 1 } { 2 } F ( 1 )$
(C) $2 F ( 6 ) - 2 F ( 2 )$
(D) $F ( 6 ) - F ( 2 )$
(E) $\frac { 1 } { 2 } F ( 6 ) - \frac { 1 } { 2 } F ( 2 )$ 83. If $a \neq 0$, then $\lim _ { x \rightarrow a } \frac { x ^ { 2 } - a ^ { 2 } } { x ^ { 4 } - a ^ { 4 } }$ is
(A) $\frac { 1 } { a ^ { 2 } }$
(B) $\frac { 1 } { 2 a ^ { 2 } }$
(C) $\frac { 1 } { 6 a ^ { 2 } }$
(D) 0
(E) nonexistent 84. Population $y$ grows according to the equation $\frac { d y } { d t } = k y$, where $k$ is a constant and $t$ is measured in years. If the population doubles every 10 years, then the value of $k$ is
(A) 0.069
(B) 0.200
(C) 0.301
(D) 3.322
(E) 5.000
- The function $f$ is continuous on the closed interval $[ 2,8 ]$ and has values that are given in the table above. Using the subintervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of $\int _ { 2 } ^ { 8 } f ( x ) d x ?$
(A) 110
(B) 130
(C) 160
(D) 190
(E) 210 [Figure] - The base of a solid is a region in the first quadrant bounded by the $x$-axis, the $y$-axis, and the line $x + 2 y = 8$, as shown in the figure above. If cross sections of the solid perpendicular to the $x$-axis are semicircles, what is the volume of the solid?
(A) 12.566
(B) 14.661
(C) 16.755
(D) 67.021
(E) 134.041 - Which of the following is an equation of the line tangent to the graph of $f ( x ) = x ^ { 4 } + 2 x ^ { 2 }$ at the point where $f ^ { \prime } ( x ) = 1$ ?
(A) $y = 8 x - 5$
(B) $y = x + 7$
(C) $y = x + 0.763$
(D) $y = x - 0.122$
(E) $y = x - 2.146$ - Let $F ( x )$ be an antiderivative of $\frac { ( \ln x ) ^ { 3 } } { x }$. If $F ( 1 ) = 0$, then $F ( 9 ) =$
(A) 0.048
(B) 0.144
(C) 5.827
(D) 23.308
(E) $1,640.250$ - If $g$ is a differentiable function such that $g ( x ) < 0$ for all real numbers $x$ and if $f ^ { \prime } ( x ) = \left( x ^ { 2 } - 4 \right) g ( x )$, which of the following is true?
(A) $f$ has a relative maximum at $x = - 2$ and a relative minimum at $x = 2$.
(B) $f$ has a relative minimum at $x = - 2$ and a relative maximum at $x = 2$.
(C) $f$ has relative minima at $x = - 2$ and at $x = 2$.
(D) $f$ has relative maxima at $x = - 2$ and at $x = 2$.
(E) It cannot be determined if $f$ has any relative extrema. - If the base $b$ of a triangle is increasing at a rate of 3 inches per minute while its height $h$ is decreasing at a rate of 3 inches per minute, which of the following must be true about the area $A$ of the triangle?
(A) $A$ is always increasing.
(B) $A$ is always decreasing.
(C) $A$ is decreasing only when $b < h$.
(D) $A$ is decreasing only when $b > h$.
(E) $A$ remains constant. - Let $f$ be a function that is differentiable on the open interval $( 1,10 )$. If $f ( 2 ) = - 5 , f ( 5 ) = 5$, and $f ( 9 ) = - 5$, which of the following must be true? I. $f$ has at least 2 zeros. II. The graph of $f$ has at least one horizontal tangent. III. For some $c , 2 < c < 5 , f ( c ) = 3$.
(A) None
(B) I only
(C) I and II only
(D) I and III only
(E) I, II, and III - If $0 \leq k < \frac { \pi } { 2 }$ and the area under the curve $y = \cos x$ from $x = k$ to $x = \frac { \pi } { 2 }$ is 0.1 , then $k =$
(A) 1.471
(B) 1.414
(C) 1.277
(D) 1.120
(E) 0.436
$\mathbf { 5 5 }$ Minutes-No Calculator
Note: 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.
- What are all values of $x$ for which the function $f$ defined by $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 9 x + 7$ is increasing?
(A) $- 3 < x < 1$
(B) $- 1 < x < 1$
(C) $x < - 3$ or $x > 1$
(D) $x < - 1$ or $x > 3$
(E) All real numbers - In the $x y$-plane, the graph of the parametric equations $x = 5 t + 2$ and $y = 3 t$, for $- 3 \leq t \leq 3$, is a line segment with slope
(A) $\frac { 3 } { 5 }$
(B) $\frac { 5 } { 3 }$
(C) 3
(D) 5
(E) 13 - The slope of the line tangent to the curve $y ^ { 2 } + ( x y + 1 ) ^ { 3 } = 0$ at $( 2 , - 1 )$ is
(A) $- \frac { 3 } { 2 }$
(B) $- \frac { 3 } { 4 }$
(C) 0
(D) $\frac { 3 } { 4 }$
(E) $\frac { 3 } { 2 }$ - $\int \frac { 1 } { x ^ { 2 } - 6 x + 8 } d x =$
(A) $\quad \frac { 1 } { 2 } \ln \left| \frac { x - 4 } { x - 2 } \right| + C$
(B) $\quad \frac { 1 } { 2 } \ln \left| \frac { x - 2 } { x - 4 } \right| + C$
(C) $\frac { 1 } { 2 } \ln | ( x - 2 ) ( x - 4 ) | + C$
(D) $\frac { 1 } { 2 } \ln | ( x - 4 ) ( x + 2 ) | + C$
(E) $\quad \ln | ( x - 2 ) ( x - 4 ) | + C$ - If $f$ and $g$ are twice differentiable and if $h ( x ) = f ( g ( x ) )$, then $h ^ { \prime \prime } ( x ) =$
(A) $f ^ { \prime \prime } ( g ( x ) ) \left[ g ^ { \prime } ( x ) \right] ^ { 2 } + f ^ { \prime } ( g ( x ) ) g ^ { \prime \prime } ( x )$
(B) $f ^ { \prime \prime } ( g ( x ) ) g ^ { \prime } ( x ) + f ^ { \prime } ( g ( x ) ) g ^ { \prime \prime } ( x )$
(C) $f ^ { \prime \prime } ( g ( x ) ) \left[ g ^ { \prime } ( x ) \right] ^ { 2 }$
(D) $f ^ { \prime \prime } ( g ( x ) ) g ^ { \prime \prime } ( x )$
(E) $f ^ { \prime \prime } ( g ( x ) )$ [Figure] - The graph of $y = h ( x )$ is shown above. Which of the following could be the graph of $y = h ^ { \prime } ( x )$ ?
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
(E) [Figure] - $\int _ { 1 } ^ { e } \left( \frac { x ^ { 2 } - 1 } { x } \right) d x =$
(A) $e - \frac { 1 } { e }$
(B) $e ^ { 2 } - e$
(C) $\frac { e ^ { 2 } } { 2 } - e + \frac { 1 } { 2 }$
(D) $e ^ { 2 } - 2$
(E) $\frac { e ^ { 2 } } { 2 } - \frac { 3 } { 2 }$ - If $\frac { d y } { d x } = \sin x \cos ^ { 2 } x$ and if $y = 0$ when $x = \frac { \pi } { 2 }$, what is the value of $y$ when $x = 0$ ?
(A) - 1
(B) $- \frac { 1 } { 3 }$
(C) 0
(D) $\frac { 1 } { 3 }$
(E) 1 [Figure] - The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?
(A) 500
(B) 600
(C) 2,400
(D) 3,000
(E) 4,800 - A particle moves on a plane curve so that at any time $t > 0$ its $x$-coordinate is $t ^ { 3 } - t$ and its $y$-coordinate is $( 2 t - 1 ) ^ { 3 }$. The acceleration vector of the particle at $t = 1$ is
(A) $( 0,1 )$
(B) $( 2,3 )$
(C) $( 2,6 )$
(D) $( 6,12 )$
(E) $( 6,24 )$ - If $f$ is a linear function and $0 < a < b$, then $\int _ { a } ^ { b } f ^ { \prime \prime } ( x ) d x =$
(A) 0
(B) 1
(C) $\frac { a b } { 2 }$
(D) $b - a$
(E) $\frac { b ^ { 2 } - a ^ { 2 } } { 2 }$ - If $f ( x ) = \left\{ \begin{aligned} \ln x & \text { for } 0 < x \leq 2 \\ x ^ { 2 } \ln 2 & \text { for } 2 < x \leq 4 , \end{aligned} \right.$ then $\lim _ { x \rightarrow 2 } f ( x )$ is
(A) $\quad \ln 2$
(B) $\quad \ln 8$
(C) $\quad \ln 16$
(D) 4
(E) nonexistent [Figure] - 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 - What is the approximation of the value of $\sin 1$ obtained by using the fifth-degree Taylor polynomial about $x = 0$ for $\sin x$ ?
(A) $1 - \frac { 1 } { 2 } + \frac { 1 } { 24 }$
(B) $1 - \frac { 1 } { 2 } + \frac { 1 } { 4 }$
(C) $1 - \frac { 1 } { 3 } + \frac { 1 } { 5 }$
(D) $1 - \frac { 1 } { 4 } + \frac { 1 } { 8 }$
(E) $\quad 1 - \frac { 1 } { 6 } + \frac { 1 } { 120 }$ - $\int x \cos x d x =$
(A) $\quad x \sin x - \cos x + C$
(B) $x \sin x + \cos x + C$
(C) $- x \sin x + \cos x + C$
(D) $x \sin x + C$
(E) $\frac { 1 } { 2 } x ^ { 2 } \sin x + C$ - If $f$ is the function defined by $f ( x ) = 3 x ^ { 5 } - 5 x ^ { 4 }$, what are all the $x$-coordinates of points of inflection for the graph of $f$ ?
(A) - 1
(B) 0
(C) 1
(D) 0 and 1
(E) -1, 0, and 1 [Figure] - 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) $f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 ) < f ( 1 )$
1998 AP Calculus BC: Section I, Part A
- Which of the following series converge? I. $\quad \sum _ { n = 1 } ^ { \infty } \frac { n } { n + 2 }$ II. $\quad \sum _ { n = 1 } ^ { \infty } \frac { \cos ( n \pi ) } { n }$ III. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$
(A) None
(B) II only
(C) III only
(D) I and II only
(E) I and III only - The area of the region inside the polar curve $r = 4 \sin \theta$ and outside the polar curve $r = 2$ is given by
(A) $\frac { 1 } { 2 } \int _ { 0 } ^ { \pi } ( 4 \sin \theta - 2 ) ^ { 2 } d \theta$
(B) $\frac { 1 } { 2 } \int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } ( 4 \sin \theta - 2 ) ^ { 2 } d \theta$
(C) $\frac { 1 } { 2 } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } ( 4 \sin \theta - 2 ) ^ { 2 } d \theta$
(D) $\frac { 1 } { 2 } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } \left( 16 \sin ^ { 2 } \theta - 4 \right) d \theta$
(E) $\frac { 1 } { 2 } \int _ { 0 } ^ { \pi } \left( 16 \sin ^ { 2 } \theta - 4 \right) d \theta$ - When $x = 8$, the rate at which $\sqrt [ 3 ] { x }$ is increasing is $\frac { 1 } { k }$ times the rate at which $x$ is increasing. What is the value of $k$ ?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 12 - The length of the path described by the parametric equations $x = \frac { 1 } { 3 } t ^ { 3 }$ and $y = \frac { 1 } { 2 } t ^ { 2 }$, where $0 \leq t \leq 1$, is given by
(A) $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 2 } + 1 } d t$
(B) $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 2 } + t } d t$
(C) $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 4 } + t ^ { 2 } } d t$
(D) $\frac { 1 } { 2 } \int _ { 0 } ^ { 1 } \sqrt { 4 + t ^ { 4 } } d t$
(E) $\frac { 1 } { 6 } \int _ { 0 } ^ { 1 } t ^ { 2 } \sqrt { 4 t ^ { 2 } + 9 } d t$ - If $\lim _ { b \rightarrow \infty } \int _ { 1 } ^ { b } \frac { d x } { x ^ { p } }$ is finite, then which of the following must be true?
(A) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } }$ converges
(B) $\quad \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } }$ diverges
(C) $\quad \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p - 2 } }$ converges
(D) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p - 1 } }$ converges
(E) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p + 1 } }$ diverges - Let $f$ be a function defined and continuous on the closed interval $[ a , b ]$. If $f$ has a relative maximum at $c$ and $a < c < b$, which of the following statements must be true? I. $f ^ { \prime } ( c )$ exists. II. If $f ^ { \prime } ( c )$ exists, then $f ^ { \prime } ( c ) = 0$. III. If $f ^ { \prime \prime } ( c )$ exists, then $f ^ { \prime \prime } ( c ) \leq 0$.
(A) II only
(B) III only
(C) I and II only
(D) I and III only
(E) II and III only [Figure] - Shown above is a slope field for which of the following differential equations?
(A) $\frac { d y } { d x } = 1 + x$
(B) $\frac { d y } { d x } = x ^ { 2 }$
(C) $\frac { d y } { d x } = x + y$
(D) $\frac { d y } { d x } = \frac { x } { y }$
(E) $\frac { d y } { d x } = \ln y$ - $\int _ { 0 } ^ { \infty } x ^ { 2 } e ^ { - x ^ { 3 } } d x$ is
(A) $- \frac { 1 } { 3 }$
(B) 0
(C) $\frac { 1 } { 3 }$
(D) 1
(E) divergent - The population $P ( t )$ of a species satisfies the logistic differential equation $\frac { d P } { d t } = P \left( 2 - \frac { P } { 5000 } \right)$, where the initial population $P ( 0 ) = 3,000$ and $t$ is the time in years. What is $\lim _ { t \rightarrow \infty } P ( t )$ ?
(A) 2,500
(B) 3,000
(C) 4,200
(D) 5,000
(E) 10,000 - If $\sum _ { n = 0 } ^ { \infty } a _ { n } x ^ { n }$ is a Taylor series that converges to $f ( x )$ for all real $x$, then $f ^ { \prime } ( 1 ) =$
(A) 0
(B) $\quad a _ { 1 }$
(C) $\sum _ { n = 0 } ^ { \infty } a _ { n }$
(D) $\sum _ { n = 1 } ^ { \infty } n a _ { n }$
(E) $\sum _ { n = 1 } ^ { \infty } n a _ { n } { } ^ { n - 1 }$ - $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } e ^ { t ^ { 2 } } d t } { x ^ { 2 } - 1 }$ is
(A) 0
(B) 1
(C) $\frac { e } { 2 }$
(D) $e$
(E) nonexistent
50 Minutes-Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number. 76. For what integer $k , k > 1$, will both $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k n } } { n }$ and $\sum _ { n = 1 } ^ { \infty } \left( \frac { k } { 4 } \right) ^ { n }$ converge?
(A) 6
(B) 5
(C) 4
(D) 3
(E) 2 77. If $f$ is a vector-valued function defined by $f ( t ) = \left( e ^ { - t } , \cos t \right)$, then $f ^ { \prime \prime } ( t ) =$
(A) $- e ^ { - t } + \sin t$
(B) $e ^ { - t } - \cos t$
(C) $\left( - e ^ { - t } , - \sin t \right)$
(D) $\left( e ^ { - t } , \cos t \right)$
(E) $\left( e ^ { - t } , - \cos t \right)$ 78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference $C$, what is the rate of change of the area of the circle, in square centimeters per second?
(A) $- ( 0.2 ) \pi C$
(B) $- ( 0.1 ) C$
(C) $- \frac { ( 0.1 ) C } { 2 \pi }$
(D) $( 0.1 ) ^ { 2 } C$
(E) $( 0.1 ) ^ { 2 } \pi C$ 79. Let $f$ be the function given by $f ( x ) = \frac { ( x - 1 ) \left( x ^ { 2 } - 4 \right) } { x ^ { 2 } - a }$. For what positive values of $a$ is $f$ continuous for all real numbers $x$ ?
(A) None
(B) 1 only
(C) 2 only
(D) 4 only
(E) 1 and 4 only 80. Let $R$ be the region enclosed by the graph of $y = 1 + \ln \left( \cos ^ { 4 } x \right)$, the $x$-axis, and the lines $x = - \frac { 2 } { 3 }$ and $x = \frac { 2 } { 3 }$. The closest integer approximation of the area of $R$ is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4 81. If $\frac { d y } { d x } = \sqrt { 1 - y ^ { 2 } }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } =$
(A) $- 2 y$
(B) $- y$
(C) $\frac { - y } { \sqrt { 1 - y ^ { 2 } } }$
(D) $y$
(E) $\frac { 1 } { 2 }$ 82. If $f ( x ) = g ( x ) + 7$ for $3 \leq x \leq 5$, then $\int _ { 3 } ^ { 5 } [ f ( x ) + g ( x ) ] d x =$
(A) $\quad 2 \int _ { 3 } ^ { 5 } g ( x ) d x + 7$
(B) $2 \int _ { 3 } ^ { 5 } g ( x ) d x + 14$
(C) $2 \int _ { 3 } ^ { 5 } g ( x ) d x + 28$
(D) $\int _ { 3 } ^ { 5 } g ( x ) d x + 7$
(E) $\int _ { 3 } ^ { 5 } g ( x ) d x + 14$ 83. The Taylor series for $\ln x$, centered at $x = 1$, is $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { ( x - 1 ) ^ { n } } { n }$. Let $f$ be the function given by the sum of the first three nonzero terms of this series. The maximum value of $| \ln x - f ( x ) |$ for $0.3 \leq x \leq 1.7$ is
(A) 0.030
(B) 0.039
(C) 0.145
(D) 0.153
(E) 0.529 84. What are all values of $x$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { ( x + 2 ) ^ { n } } { \sqrt { n } }$ converges?
(A) $- 3 < x < - 1$
(B) $- 3 \leq x < - 1$
(C) $- 3 \leq x \leq - 1$
(D) $- 1 \leq x < 1$
(E) $- 1 \leq x \leq 1$
- The function $f$ is continuous on the closed interval $[ 2,8 ]$ and has values that are given in the table above. Using the subintervals $[ 2,5 ] , [ 5,7 ]$, and $[ 7,8 ]$, what is the trapezoidal approximation of $\int _ { 2 } ^ { 8 } f ( x ) d x ?$
(A) 110
(B) 130
(C) 160
(D) 190
(E) 210 [Figure] - The base of a solid is a region in the first quadrant bounded by the $x$-axis, the $y$-axis, and the line $x + 2 y = 8$, as shown in the figure above. If cross sections of the solid perpendicular to the $x$-axis are semicircles, what is the volume of the solid?
(A) 12.566
(B) 14.661
(C) 16.755
(D) 67.021
(E) 134.041 - Which of the following is an equation of the line tangent to the graph of $f ( x ) = x ^ { 4 } + 2 x ^ { 2 }$ at the point where $f ^ { \prime } ( x ) = 1$ ?
(A) $y = 8 x - 5$
(B) $y = x + 7$
(C) $y = x + 0.763$
(D) $y = x - 0.122$
(E) $y = x - 2.146$ [Figure] - Let $g ( x ) = \int _ { a } ^ { x } f ( t ) d t$, where $a \leq x \leq b$. The figure above shows the graph of $g$ on $[ a , b ]$. Which of the following could be the graph of $f$ on $[ a , b ]$ ?
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
(E) [Figure] - The graph of the function represented by the Maclaurin series $1 - x + \frac { x ^ { 2 } } { 2 ! } - \frac { x ^ { 3 } } { 3 ! } + \ldots + \frac { ( - 1 ) ^ { n } x ^ { n } } { n ! } + \ldots$ intersects the graph of $y = x ^ { 3 }$ at $x =$
(A) 0.773
(B) 0.865
(C) 0.929
(D) 1.000
(E) 1.857 - A particle starts from rest at the point $( 2,0 )$ and moves along the $x$-axis with a constant positive acceleration for time $t \geq 0$. Which of the following could be the graph of the distance $s ( t )$ of the particle from the origin as a function of time $t$ ?
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
E) [Figure]
| $t ( \mathrm { sec } )$ | 0 | 2 | 4 | 6 |
| $a ( t ) \left( \mathrm { ft } / \mathrm { sec } ^ { 2 } \right)$ | 5 | 2 | 8 | 3 |
- The data for the acceleration $a ( t )$ of a car from 0 to 6 seconds are given in the table above. If the velocity at $t = 0$ is 11 feet per second, the approximate value of the velocity at $t = 6$, computed using a left-hand Riemann sum with three subintervals of equal length, is
(A) $26 \mathrm { ft } / \mathrm { sec }$
(B) $30 \mathrm { ft } / \mathrm { sec }$
(C) $37 \mathrm { ft } / \mathrm { sec }$
(D) $39 \mathrm { ft } / \mathrm { sec }$
(E) $\quad 41 \mathrm { ft } / \mathrm { sec }$ - Let $f$ be the function given by $f ( x ) = x ^ { 2 } - 2 x + 3$. The tangent line to the graph of $f$ at $x = 2$ is used to approximate values of $f ( x )$. Which of the following is the greatest value of $x$ for which the error resulting from this tangent line approximation is less than 0.5 ?
(A) 2.4
(B) 2.5
(C) 2.6
(D) 2.7
(E) 2.8