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
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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
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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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2005 free-response

6 maths questions

Q1 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.
Q2 Polar coordinates View
The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r = \theta + \sin ( 2 \theta )$ for $0 \leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by $\frac { d r } { d \theta } = 1 + 2 \cos ( 2 \theta )$.
(a) Find the area bounded by the curve and the $x$-axis.
(b) Find the angle $\theta$ that corresponds to the point on the curve with $x$-coordinate $-2$.
(c) For $\frac { \pi } { 3 } < \theta < \frac { 2 \pi } { 3 } , \frac { d r } { d \theta }$ is negative. What does this fact say about $r$ ? What does this fact say about the curve?
(d) Find the value of $\theta$ in the interval $0 \leq \theta \leq \frac { \pi } { 2 }$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.
Q3 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( {}^{\circ} \mathrm{C} \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.
\begin{tabular}{ c } Distance
$x ( \mathrm{~cm} )$
& 0 & 1 & 5 & 6 & 8 \hline
Temperature
$T ( x ) \left( {}^{\circ} \mathrm{C} \right)$
& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.
Q4 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point $( 0, 1 )$.
(b) The solution curve that passes through the point $( 0, 1 )$ has a local minimum at $x = \ln \left( \frac { 3 } { 2 } \right)$. What is the $y$-coordinate of this local minimum?
(c) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f ( - 0.4 )$. Show the work that leads to your answer.
(d) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine whether the approximation found in part (c) is less than or greater than $f ( - 0.4 )$. Explain your reasoning.
Q5 Variable acceleration (1D) Multi-part particle motion analysis (graph-based velocity) View
A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) \, d t$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, d t$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?
Q6 Taylor series Determine radius or interval of convergence View
Let $f$ be a function with derivatives of all orders and for which $f ( 2 ) = 7$. When $n$ is odd, the $n$th derivative of $f$ at $x = 2$ is 0. When $n$ is even and $n \geq 2$, the $n$th derivative of $f$ at $x = 2$ is given by $f ^ { ( n ) } ( 2 ) = \frac { ( n - 1 ) ! } { 3 ^ { n } }$.
(a) Write the sixth-degree Taylor polynomial for $f$ about $x = 2$.
(b) In the Taylor series for $f$ about $x = 2$, what is the coefficient of $( x - 2 ) ^ { 2 n }$ for $n \geq 1$ ?
(c) Find the interval of convergence of the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.