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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2019
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2018
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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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2007 free-response

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the region in the first and second quadrants bounded above by the graph of $y = \frac{20}{1 + x^2}$ and below by the horizontal line $y = 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated about the $x$-axis.
(c) The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are semicircles. Find the volume of this solid.
Q2 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval $0 \leq t \leq 7$, where $t$ is measured in hours. In this model, rates are given as follows:
(i) The rate at which water enters the tank is $f(t) = 100t^2 \sin(\sqrt{t})$ gallons per hour for $0 \leq t \leq 7$.
(ii) The rate at which water leaves the tank is $$g(t) = \left\{ \begin{array}{r} 250 \text{ for } 0 \leq t < 3 \\ 2000 \text{ for } 3 < t \leq 7 \end{array} \right. \text{ gallons per hour.}$$
The graphs of $f$ and $g$, which intersect at $t = 1.617$ and $t = 5.076$, are shown in the figure above. At time $t = 0$, the amount of water in the tank is 5000 gallons.
(a) How many gallons of water enter the tank during the time interval $0 \leq t \leq 7$? Round your answer to the nearest gallon.
(b) For $0 \leq t \leq 7$, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.
(c) For $0 \leq t \leq 7$, at what time $t$ is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.
Q3 Polar coordinates View
The graphs of the polar curves $r = 2$ and $r = 3 + 2\cos\theta$ are shown in the figure above. The curves intersect when $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$.
(a) Let $R$ be the region that is inside the graph of $r = 2$ and also inside the graph of $r = 3 + 2\cos\theta$, as shaded in the figure above. Find the area of $R$.
(b) A particle moving with nonzero velocity along the polar curve given by $r = 3 + 2\cos\theta$ has position $(x(t), y(t))$ at time $t$, with $\theta = 0$ when $t = 0$. This particle moves along the curve so that $\frac{dr}{dt} = \frac{dr}{d\theta}$. Find the value of $\frac{dr}{dt}$ at $\theta = \frac{\pi}{3}$ and interpret your answer in terms of the motion of the particle.
(c) For the particle described in part (b), $\frac{dy}{dt} = \frac{dy}{d\theta}$. Find the value of $\frac{dy}{dt}$ at $\theta = \frac{\pi}{3}$ and interpret your answer in terms of the motion of the particle.
Q4 Integration by Parts Integration by Parts within Function Analysis View
Let $f$ be the function defined for $x > 0$, with $f(e) = 2$ and $f'$, the first derivative of $f$, given by $f'(x) = x^2 \ln x$.
(a) Write an equation for the line tangent to the graph of $f$ at the point $(e, 2)$.
(b) Is the graph of $f$ concave up or concave down on the interval $1 < x < 3$? Give a reason for your answer.
(c) Use antidifferentiation to find $f(x)$.
Q5 Connected Rates of Change Table-Based Estimation with Rate of Change Interpretation View
The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r'(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$. The radius of the balloon is 30 feet when $t = 5$.
$t$ (minutes)02571112
$r'(t)$ (feet per minute)5.74.02.01.20.60.5

(Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^3$.)
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r'(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r'(t)\, dt$ in terms of the radius of the balloon.
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r'(t)\, dt$? Give a reason for your answer.
Q6 Taylor series Construct series for a composite or related function View
Let $f$ be the function given by $f(x) = e^{-x^2}$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use your answer to part (a) to find $\lim_{x \to 0} \frac{1 - x^2 - f(x)}{x^4}$.
(c) Write the first four nonzero terms of the Taylor series for $\int_{0}^{x} e^{-t^2}\, dt$ about $x = 0$. Use the first two terms of your answer to estimate $\int_{0}^{1/2} e^{-t^2}\, dt$.
(d) Explain why the estimate found in part (c) differs from the actual value of $\int_{0}^{1/2} e^{-t^2}\, dt$ by less than $\frac{1}{200}$.