ap-calculus-bc

Papers (28)
2025
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2024
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2023
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2022
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2021
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2019
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2018
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2017
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2016
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2015
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2014
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2013
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2012
free-response 6 practice-exam 51
2011
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2010
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2009
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2008
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2007
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2006
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2005
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2004
free-response_formB 6
2003
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2002
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2001
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1999
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1998
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2008 free-response

6 maths questions

Q1 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the region bounded by the graphs of $y = \sin ( \pi x )$ and $y = x ^ { 3 } - 4 x$, as shown in the figure above.
(a) Find the area of $R$.
(b) The horizontal line $y = - 2$ splits the region $R$ into two parts. Write, but do not evaluate, an integral expression for the area of the part of $R$ that is below this horizontal line.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.
(d) The region $R$ models the surface of a small pond. At all points in $R$ at a distance $x$ from the $y$-axis, the depth of the water is given by $h ( x ) = 3 - x$. Find the volume of water in the pond.
Q2 Indefinite & Definite Integrals Multi-Part Applied Integration with Context (Trapezoidal/Numerical Estimation) View
Concert tickets went on sale at noon $( t = 0 )$ and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time $t$ is modeled by a twice-differentiable function $L$ for $0 \leq t \leq 9$. Values of $L ( t )$ at various times $t$ are shown in the table below.
$t$ (hours)0134789
$L ( t )$ (people)120156176126150800

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. $( t = 5.5 )$. Show the computations that lead to your answer. Indicate units of measure.
(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.
(c) For $0 \leq t \leq 9$, what is the fewest number of times at which $L ^ { \prime } ( t )$ must equal 0 ? Give a reason for your answer.
(d) The rate at which tickets were sold for $0 \leq t \leq 9$ is modeled by $r ( t ) = 550 t e ^ { - t / 2 }$ tickets per hour. Based on the model, how many tickets were sold by 3 P.M. ( $t = 3$ ), to the nearest whole number?
Q3 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View
Let $h$ be a function having derivatives of all orders for $x > 0$. Selected values of $h$ and its first four derivatives are indicated in the table below. The function $h$ and these four derivatives are increasing on the interval $1 \leq x \leq 3$.
$x$$h ( x )$$h ^ { \prime } ( x )$$h ^ { \prime \prime } ( x )$$h ^ { \prime \prime \prime } ( x )$$h ^ { ( 4 ) } ( x )$
11130429918
280128$\frac { 488 } { 3 }$$\frac { 448 } { 3 }$$\frac { 584 } { 9 }$
3317$\frac { 753 } { 2 }$$\frac { 1383 } { 4 }$$\frac { 3483 } { 16 }$$\frac { 1125 } { 16 }$

(a) Write the first-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$. Is this approximation greater than or less than $h ( 1.9 )$ ? Explain your reasoning.
(b) Write the third-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$.
(c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for $h$ about $x = 2$ approximates $h ( 1.9 )$ with error less than $3 \times 10 ^ { - 4 }$.
Q4 Variable acceleration (1D) Multi-part particle motion analysis (graph-based velocity) View
A particle moves along the $x$-axis so that its velocity at time $t$, for $0 \leq t \leq 6$, is given by a differentiable function $v$ whose graph is shown above. The velocity is 0 at $t = 0 , t = 3$, and $t = 5$, and the graph has horizontal tangents at $t = 1$ and $t = 4$. The areas of the regions bounded by the $t$-axis and the graph of $v$ on the intervals $[ 0,3 ] , [ 3,5 ]$, and $[ 5,6 ]$ are 8, 3, and 2, respectively. At time $t = 0$, the particle is at $x = - 2$.
(a) For $0 \leq t \leq 6$, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.
(b) For how many values of $t$, where $0 \leq t \leq 6$, is the particle at $x = - 8$ ? Explain your reasoning.
(c) On the interval $2 < t < 3$, is the speed of the particle increasing or decreasing? Give a reason for your answer.
(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.
Q5 Stationary points and optimisation Find critical points and classify extrema of a given function View
The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = ( x - 3 ) e ^ { x }$ for $x > 0$, and $f ( 1 ) = 7$.
(a) The function $f$ has a critical point at $x = 3$. At this point, does $f$ have a relative minimum, a relative maximum, or neither? Justify your answer.
(b) On what intervals, if any, is the graph of $f$ both decreasing and concave up? Explain your reasoning.
(c) Find the value of $f ( 3 )$.
Q6 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the logistic differential equation $\frac { d y } { d t } = \frac { y } { 8 } ( 6 - y )$. Let $y = f ( t )$ be the particular solution to the differential equation with $f ( 0 ) = 8$.
(a) A slope field for this differential equation is given below. Sketch possible solution curves through the points $( 3,2 )$ and $( 0,8 )$.
(b) Use Euler's method, starting at $t = 0$ with two steps of equal size, to approximate $f ( 1 )$.
(c) Write the second-degree Taylor polynomial for $f$ about $t = 0$, and use it to approximate $f ( 1 )$.
(d) What is the range of $f$ for $t \geq 0$ ?