ap-calculus-bc

Papers (28)
2025
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2024
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2023
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2019
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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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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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2014 free-response

6 maths questions

Q1 Applied differentiation Applied modeling with differentiation View
Grass clippings are placed in a bin, where they decompose. For $0 \leq t \leq 30$, the amount of grass clippings remaining in the bin is modeled by $A ( t ) = 6.687 ( 0.931 ) ^ { t }$, where $A ( t )$ is measured in pounds and $t$ is measured in days.
(a) Find the average rate of change of $A ( t )$ over the interval $0 \leq t \leq 30$. Indicate units of measure.
(b) Find the value of $A ^ { \prime } ( 15 )$. Using correct units, interpret the meaning of the value in the context of the problem.
(c) Find the time $t$ for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval $0 \leq t \leq 30$.
(d) For $t > 30$, $L ( t )$, the linear approximation to $A$ at $t = 30$, is a better model for the amount of grass clippings remaining in the bin. Use $L ( t )$ to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.
Q2 Polar coordinates View
The graphs of the polar curves $r = 3$ and $r = 3 - 2 \sin ( 2 \theta )$ are shown in the figure above for $0 \leq \theta \leq \pi$.
(a) Let $R$ be the shaded region that is inside the graph of $r = 3$ and inside the graph of $r = 3 - 2 \sin ( 2 \theta )$. Find the area of $R$.
(b) For the curve $r = 3 - 2 \sin ( 2 \theta )$, find the value of $\frac { d x } { d \theta }$ at $\theta = \frac { \pi } { 6 }$.
(c) The distance between the two curves changes for $0 < \theta < \frac { \pi } { 2 }$. Find the rate at which the distance between the two curves is changing with respect to $\theta$ when $\theta = \frac { \pi } { 3 }$.
(d) A particle is moving along the curve $r = 3 - 2 \sin ( 2 \theta )$ so that $\frac { d \theta } { d t } = 3$ for all times $t \geq 0$. Find the value of $\frac { d r } { d t }$ at $\theta = \frac { \pi } { 6 }$.
Q3 Indefinite & Definite Integrals Accumulation Function Analysis View
The function $f$ is defined on the closed interval $[ - 5, 4 ]$. The graph of $f$ consists of three line segments and is shown in the figure above. Let $g$ be the function defined by $g ( x ) = \int _ { - 3 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( 3 )$.
(b) On what open intervals contained in $- 5 < x < 4$ is the graph of $g$ both increasing and concave down? Give a reason for your answer.
(c) The function $h$ is defined by $h ( x ) = \frac { g ( x ) } { 5 x }$. Find $h ^ { \prime } ( 3 )$.
(d) The function $p$ is defined by $p ( x ) = f \left( x ^ { 2 } - x \right)$. Find the slope of the line tangent to the graph of $p$ at the point where $x = - 1$.
Q4 Variable acceleration (vectors) View
Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.
\begin{tabular}{ c } $t$
(minutes)
& 0 & 2 & 5 & 8 & 12 \hline
$v _ { A } ( t )$
(meters/minute)
& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.
Q5 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the shaded region bounded by the graph of $y = x e ^ { x ^ { 2 } }$, the line $y = - 2 x$, and the vertical line $x = 1$, as shown in the figure above.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = - 2$.
(c) Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of $R$.
Q6 Sequences and Series Power Series Expansion and Radius of Convergence View
The Taylor series for a function $f$ about $x = 1$ is given by $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n } ( x - 1 ) ^ { n }$ and converges to $f ( x )$ for $| x - 1 | < R$, where $R$ is the radius of convergence of the Taylor series.
(a) Find the value of $R$.
(b) Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 1$.
(c) The Taylor series for $f ^ { \prime }$ about $x = 1$, found in part (b), is a geometric series. Find the function $f ^ { \prime }$ to which the series converges for $| x - 1 | < R$. Use this function to determine $f$ for $| x - 1 | < R$.