ap-calculus-ab

Papers (31)
2025
free-response 6
2024
free-response 6
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
practice-exam 50
2011
free-response 6 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
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
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1999
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1998
free-response 6
Unknown
-bc_course-and-exam-description 24 -bc_sample-questions 18
2003 free-response

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the shaded region bounded by the graphs of $y = \sqrt{x}$ and $y = e^{-3x}$ and the vertical line $x = 1$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the horizontal line $y = 1$.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a rectangle whose height is 5 times the length of its base in region $R$. Find the volume of this solid.
Q2 Variable acceleration (1D) Multi-part particle motion analysis (formula-based velocity) View
A particle moves along the $x$-axis so that its velocity at time $t$ is given by $$v(t) = -(t+1)\sin\left(\frac{t^2}{2}\right)$$ At time $t = 0$, the particle is at position $x = 1$.
(a) Find the acceleration of the particle at time $t = 2$. Is the speed of the particle increasing at $t = 2$? Why or why not?
(b) Find all times $t$ in the open interval $0 < t < 3$ when the particle changes direction. Justify your answer.
(c) Find the total distance traveled by the particle from time $t = 0$ until time $t = 3$.
(d) During the time interval $0 \leq t \leq 3$, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.
Q3 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function $R$ of time $t$. The graph of $R$ and a table of selected values of $R(t)$, for the time interval $0 \leq t \leq 90$ minutes, are shown above.
(a) Use data from the table to find an approximation for $R'(45)$. Show the computations that lead to your answer. Indicate units of measure.
(b) The rate of fuel consumption is increasing fastest at time $t = 45$ minutes. What is the value of $R''(45)$? Explain your reasoning.
(c) Approximate the value of $\int_{0}^{90} R(t)\,dt$ using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of $\int_{0}^{90} R(t)\,dt$? Explain your reasoning.
(d) For $0 < b \leq 90$ minutes, explain the meaning of $\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Explain the meaning of $\frac{1}{b}\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Indicate units of measure in both answers.
Q4 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$.
(d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.
Q5 Differential equations Applied Modeling with Differential Equations View
A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t$, measured in seconds. The volume $V$ of coffee in the pot is changing at the rate of $-5\pi\sqrt{h}$ cubic inches per second. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Show that $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$.
(b) Given that $h = 17$ at time $t = 0$, solve the differential equation $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$ for $h$ as a function of $t$.
(c) At what time $t$ is the coffeepot empty?
Q6 Indefinite & Definite Integrals Average Value of a Function View
Let $f$ be the function defined by $$f(x) = \begin{cases} \sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\ 5 - x & \text{for } 3 < x \leq 5. \end{cases}$$ (a) Is $f$ continuous at $x = 3$? Explain why or why not.
(b) Find the average value of $f(x)$ on the closed interval $0 \leq x \leq 5$.
(c) Suppose the function $g$ is defined by $$g(x) = \begin{cases} k\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\ mx + 2 & \text{for } 3 < x \leq 5, \end{cases}$$ where $k$ and $m$ are constants. If $g$ is differentiable at $x = 3$, what are the values of $k$ and $m$?