ap-calculus-ab

Papers (31)
2025
free-response 6
2024
free-response 6
2023
free-response 6
2022
free-response 6
2021
free-response 6
2019
free-response 6
2018
free-response 6
2017
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2016
free-response 6
2015
free-response 6
2014
free-response 6
2013
free-response 6
2012
practice-exam 50
2011
free-response 6 free-response_formB 6
2010
free-response 6
2009
free-response 6
2008
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2007
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2006
free-response 6
2005
free-response 6 free-response_formB 6
2004
free-response_formB 6
2003
free-response 6
2002
free-response 6 free-response_formB 6
2001
free-response 6
1999
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1998
free-response 6
Unknown
-bc_course-and-exam-description 24 -bc_sample-questions 18
2004 free-response_formB

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the region enclosed by the graph of $y = \sqrt{x-1}$, the vertical line $x = 10$, and the $x$-axis.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the horizontal line $y = 3$.
(c) Find the volume of the solid generated when $R$ is revolved about the vertical line $x = 10$.
Q2 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
For $0 \leq t \leq 31$, the rate of change of the number of mosquitoes on Tropical Island at time $t$ days is modeled by $R(t) = 5\sqrt{t}\cos\left(\frac{t}{5}\right)$ mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time $t = 0$.
(a) Show that the number of mosquitoes is increasing at time $t = 6$.
(b) At time $t = 6$, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer.
(c) According to the model, how many mosquitoes will be on the island at time $t = 31$? Round your answer to the nearest whole number.
(d) To the nearest whole number, what is the maximum number of mosquitoes for $0 \leq t \leq 31$? Show the analysis that leads to your conclusion.
Q3 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
A test plane flies in a straight line with positive velocity $v(t)$, in miles per minute at time $t$ minutes, where $v$ is a differentiable function of $t$. Selected values of $v(t)$ for $0 \leq t \leq 40$ are shown in the table below.
\begin{tabular}{ c } $t$
(minutes)
& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \hline
$v(t)$
(miles per minute)
& 7.0 & 9.2 & 9.5 & 7.0 & 4.5 & 2.4 & 2.4 & 4.3 & 7.3 \hline \end{tabular}
(a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate $\int_{0}^{40} v(t)\,dt$. Show the computations that lead to your answer. Using correct units, explain the meaning of $\int_{0}^{40} v(t)\,dt$ in terms of the plane's flight.
(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval $0 < t < 40$? Justify your answer.
(c) The function $f$, defined by $f(t) = 6 + \cos\left(\frac{t}{10}\right) + 3\sin\left(\frac{7t}{40}\right)$, is used to model the velocity of the plane, in miles per minute, for $0 \leq t \leq 40$. According to this model, what is the acceleration of the plane at $t = 23$? Indicate units of measure.
(d) According to the model $f$, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval $0 \leq t \leq 40$?
Q4 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
The figure above shows the graph of $f'$, the derivative of the function $f$, on the closed interval $-1 \leq x \leq 5$. The graph of $f'$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f(2) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $-1 \leq x \leq 5$? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $-1 \leq x \leq 5$? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g(x) = x f(x)$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\dfrac{dy}{dx} = x^{4}(y-2)$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the $xy$-plane. Describe all points in the $xy$-plane for which the slopes are negative.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(0) = 0$.
Q6 Areas Between Curves Maximize or Optimize Area View
Let $\ell$ be the line tangent to the graph of $y = x^{n}$ at the point $(1,1)$, where $n > 1$.
(a) Find $\displaystyle\int_{0}^{1} x^{n}\,dx$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\dfrac{1}{2n}$.
(c) Let $S$ be the region bounded by the graph of $y = x^{n}$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.