ap-calculus-bc

Papers (38)
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 49
2011
free-response 5 free-response_formB 6
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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2000
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1999
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1998
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2004 free-response

6 maths questions

Q1 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function $F$ defined by
$$F ( t ) = 82 + 4 \sin \left( \frac { t } { 2 } \right) \text { for } 0 \leq t \leq 30$$
where $F ( t )$ is measured in cars per minute and $t$ is measured in minutes.
(a) To the nearest whole number, how many cars pass through the intersection over the 30 -minute period?
(b) Is the traffic flow increasing or decreasing at $t = 7$ ? Give a reason for your answer.
(c) What is the average value of the traffic flow over the time interval $10 \leq t \leq 15$ ? Indicate units of measure.
(d) What is the average rate of change of the traffic flow over the time interval $10 \leq t \leq 15$ ? Indicate units of measure.
Q2 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $f$ and $g$ be the functions given by $f ( x ) = 2 x ( 1 - x )$ and $g ( x ) = 3 ( x - 1 ) \sqrt { x }$ for $0 \leq x \leq 1$. The graphs of $f$ and $g$ are shown in the figure above.
(a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$.
(b) Find the volume of the solid generated when the shaded region enclosed by the graphs of $f$ and $g$ is revolved about the horizontal line $y = 2$.
(c) Let $h$ be the function given by $h ( x ) = k x ( 1 - x )$ for $0 \leq x \leq 1$. For each $k > 0$, the region (not shown) enclosed by the graphs of $h$ and $g$ is the base of a solid with square cross sections perpendicular to the $x$-axis. There is a value of $k$ for which the volume of this solid is equal to 15 . Write, but do not solve, an equation involving an integral expression that could be used to find the value of $k$.
Q3 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View
An object moving along a curve in the $x y$-plane has position $( x ( t ) , y ( t ) )$ at time $t \geq 0$ with $\frac { d x } { d t } = 3 + \cos \left( t ^ { 2 } \right)$. The derivative $\frac { d y } { d t }$ is not explicitly given. At time $t = 2$, the object is at position $( 1,8 )$.
(a) Find the $x$-coordinate of the position of the object at time $t = 4$.
(b) At time $t = 2$, the value of $\frac { d y } { d t }$ is - 7 . Write an equation for the line tangent to the curve at the point $( x ( 2 ) , y ( 2 ) )$.
(c) Find the speed of the object at time $t = 2$.
(d) For $t \geq 3$, the line tangent to the curve at $( x ( t ) , y ( t ) )$ has a slope of $2 t + 1$. Find the acceleration vector of the object at time $t = 4$.
Q4 Implicit equations and differentiation Second derivative via implicit differentiation View
Consider the curve given by $x ^ { 2 } + 4 y ^ { 2 } = 7 + 3 x y$.
(a) Show that $\frac { d y } { d x } = \frac { 3 y - 2 x } { 8 y - 3 x }$.
(b) Show that there is a point $P$ with $x$-coordinate 3 at which the line tangent to the curve at $P$ is horizontal. Find the $y$-coordinate of $P$.
(c) Find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $P$ found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point $P$ ? Justify your answer.
Q5 First order differential equations (integrating factor) Applied Modeling with Differential Equations View
A population is modeled by a function $P$ that satisfies the logistic differential equation
$$\frac { d P } { d t } = \frac { P } { 5 } \left( 1 - \frac { P } { 12 } \right) .$$
(a) If $P ( 0 ) = 3$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
If $P ( 0 ) = 20$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
(b) If $P ( 0 ) = 3$, for what value of $P$ is the population growing the fastest?
(c) A different population is modeled by a function $Y$ that satisfies the separable differential equation
$$\frac { d Y } { d t } = \frac { Y } { 5 } \left( 1 - \frac { t } { 12 } \right)$$
Find $Y ( t )$ if $Y ( 0 ) = 3$.
(d) For the function $Y$ found in part (c), what is $\lim _ { t \rightarrow \infty } Y ( t )$ ?
Q6 Taylor series Lagrange error bound application View
Let $f$ be the function given by $f ( x ) = \sin \left( 5 x + \frac { \pi } { 4 } \right)$, and let $P ( x )$ be the third-degree Taylor polynomial for $f$ about $x = 0$.
(a) Find $P ( x )$.
(b) Find the coefficient of $x ^ { 22 }$ in the Taylor series for $f$ about $x = 0$.
(c) Use the Lagrange error bound to show that $\left| f \left( \frac { 1 } { 10 } \right) - P \left( \frac { 1 } { 10 } \right) \right| < \frac { 1 } { 100 }$.
(d) Let $G$ be the function given by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Write the third-degree Taylor polynomial for $G$ about $x = 0$.