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
2014 free-response

5 maths questions

Q1 Exponential Functions 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 $x = - 1$.
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 Taylor 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 $| x - 1 | < R$.