ap-calculus-bc

Papers (38)
2025
free-response 6
2024
free-response 6
2023
free-response 6
2022
free-response 5
2021
free-response 6
2019
free-response 6
2018
free-response 6
2017
free-response 6
2016
free-response 6
2015
free-response 6
2014
free-response 5
2013
free-response 6
2012
free-response 6 practice-exam 49
2011
free-response 5 free-response_formB 6
2010
free-response 6 free-response_formB 5
2009
free-response 6 free-response_formB 6
2008
free-response 5 free-response_formB 6
2007
free-response 6 free-response_formB 6
2006
free-response 6 free-response_formB 6
2005
free-response 6 free-response_formB 4
2004
free-response 6 free-response_formB 6
2003
free-response 3 free-response_formB 5
2002
free-response 5 free-response_formB 6
2001
free-response 6
2000
free-response 5
1999
free-response 6
1998
free-response 5
2011 free-response_formB

6 maths questions

Q1 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function $S$, where $S ( t )$ is measured in millimeters and $t$ is measured in days for $0 \leq t \leq 60$. The rate at which the height of the water is rising in the can is given by $S ^ { \prime } ( t ) = 2 \sin ( 0.03 t ) + 1.5$.
(a) According to the model, what is the height of the water in the can at the end of the 60-day period?
(b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure.
(c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time $t = 7$ ? Indicate units of measure.
(d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function $M$, where $M ( t ) = \frac { 1 } { 400 } \left( 3 t ^ { 3 } - 30 t ^ { 2 } + 330 t \right)$. The height $M ( t )$ is measured in millimeters, and $t$ is measured in days for $0 \leq t \leq 60$. Let $D ( t ) = M ^ { \prime } ( t ) - S ^ { \prime } ( t )$. Apply the Intermediate Value Theorem to the function $D$ on the interval $0 \leq t \leq 60$ to justify that there exists a time $t , 0 < t < 60$, at which the heights of water in the two cans are changing at the same rate.
Q2 Polar coordinates View
The polar curve $r$ is given by $r ( \theta ) = 3 \theta + \sin \theta$, where $0 \leq \theta \leq 2 \pi$.
(a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of $r$.
(b) For $\frac { \pi } { 2 } \leq \theta \leq \pi$, there is one point $P$ on the polar curve $r$ with $x$-coordinate - 3 . Find the angle $\theta$ that corresponds to point $P$. Find the $y$-coordinate of point $P$. Show the work that leads to your answers.
(c) A particle is traveling along the polar curve $r$ so that its position at time $t$ is $( x ( t ) , y ( t ) )$ and such that $\frac { d \theta } { d t } = 2$. Find $\frac { d y } { d t }$ at the instant that $\theta = \frac { 2 \pi } { 3 }$, and interpret the meaning of your answer in the context of the problem.
Q3 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
The functions $f$ and $g$ are given by $f ( x ) = \sqrt { x }$ and $g ( x ) = 6 - x$. Let $R$ be the region bounded by the $x$-axis and the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) The region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 2$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose base lies in $R$ and whose height is $2 y$. Write, but do not evaluate, an integral expression that gives the volume of the solid.
(c) There is a point $P$ on the graph of $f$ at which the line tangent to the graph of $f$ is perpendicular to the graph of $g$. Find the coordinates of point $P$.
Q4 Areas by integration Accumulation Function Analysis View
The graph of the differentiable function $y = f ( x )$ with domain $0 \leq x \leq 10$ is shown in the figure above. The area of the region enclosed between the graph of $f$ and the $x$-axis for $0 \leq x \leq 5$ is 10 , and the area of the region enclosed between the graph of $f$ and the $x$-axis for $5 \leq x \leq 10$ is 27 . The arc length for the portion of the graph of $f$ between $x = 0$ and $x = 5$ is 11, and the arc length for the portion of the graph of $f$ between $x = 5$ and $x = 10$ is 18 . The function $f$ has exactly two critical points that are located at $x = 3$ and $x = 8$.
(a) Find the average value of $f$ on the interval $0 \leq x \leq 5$.
(b) Evaluate $\int _ { 0 } ^ { 10 } ( 3 f ( x ) + 2 ) d x$. Show the computations that lead to your answer.
(c) Let $g ( x ) = \int _ { 5 } ^ { x } f ( t ) d t$. On what intervals, if any, is the graph of $g$ both concave up and decreasing? Explain your reasoning.
(d) The function $h$ is defined by $h ( x ) = 2 f \left( \frac { x } { 2 } \right)$. The derivative of $h$ is $h ^ { \prime } ( x ) = f ^ { \prime } \left( \frac { x } { 2 } \right)$. Find the arc length of the graph of $y = h ( x )$ from $x = 0$ to $x = 20$.
Q5 Variable acceleration (1D) Multi-part particle motion analysis (table-based velocity) View
Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function $B$ models Ben's position on the track, measured in meters from the western end of the track, at time $t$, measured in seconds from the start of the ride. The table above gives values for $B ( t )$ and Ben's velocity, $v ( t )$, measured in meters per second, at selected times $t$.
(a) Use the data in the table to approximate Ben's acceleration at time $t = 5$ seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of $\int _ { 0 } ^ { 60 } | v ( t ) | d t$ in the context of this problem. Approximate $\int _ { 0 } ^ { 60 } | v ( t ) | d t$ using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For $40 \leq t \leq 60$, must there be a time $t$ when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time $t$, the distance $L ( t )$ between Ben and the light satisfies $( L ( t ) ) ^ { 2 } = 12 ^ { 2 } + ( B ( t ) ) ^ { 2 }$. At what rate is the distance between Ben and the light changing at time $t = 40$ ?
Q6 Taylor series Construct series for a composite or related function View
Let $f ( x ) = \ln \left( 1 + x ^ { 3 } \right)$.
(a) The Maclaurin series for $\ln ( 1 + x )$ is $x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } + \cdots + ( - 1 ) ^ { n + 1 } \cdot \frac { x ^ { n } } { n } + \cdots$. Use the series to write the first four nonzero terms and the general term of the Maclaurin series for $f$.
(b) The radius of convergence of the Maclaurin series for $f$ is 1 . Determine the interval of convergence. Show the work that leads to your answer.
(c) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime } \left( t ^ { 2 } \right)$. If $g ( x ) = \int _ { 0 } ^ { x } f ^ { \prime } \left( t ^ { 2 } \right) d t$, use the first two nonzero terms of the Maclaurin series for $g$ to approximate $g ( 1 )$.
(d) The Maclaurin series for $g$, evaluated at $x = 1$, is a convergent alternating series with individual terms that decrease in absolute value to 0 . Show that your approximation in part (c) must differ from $g ( 1 )$ by less than $\frac { 1 } { 5 }$.