ap-calculus-ab

Papers (31)
2025
free-response 6
2024
free-response 6
2023
free-response 6
2022
free-response 6
2021
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2019
free-response 6
2018
free-response 6
2017
free-response 6
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
free-response 6
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
free-response 6
1998
free-response 6
Unknown
-bc_course-and-exam-description 24 -bc_sample-questions 18
2002 free-response_formB

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the region bounded by the $y$-axis and the graphs of $y = \frac{x^3}{1+x^2}$ and $y = 4 - 2x$, 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 $x$-axis.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.
Q2 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P^{\prime}(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?
Q3 Variable acceleration (1D) Multi-part particle motion analysis (formula-based velocity) View
A particle moves along the $x$-axis so that its velocity $v$ at any time $t$, for $0 \leq t \leq 16$, is given by $v(t) = e^{2\sin t} - 1$. At time $t = 0$, the particle is at the origin.
(a) On the axes provided, sketch the graph of $v(t)$ for $0 \leq t \leq 16$.
(b) During what intervals of time is the particle moving to the left? Give a reason for your answer.
(c) Find the total distance traveled by the particle from $t = 0$ to $t = 4$.
(d) Is there any time $t$, $0 < t \leq 16$, at which the particle returns to the origin? Justify your answer.
Q4 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g^{\prime}(6)$, and $g^{\prime\prime}(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3-x}{y}$.
(a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer.
(b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.
Q6 Connected Rates of Change Geometric Related Rates with Distance or Angle View
Ship $A$ is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour ($\mathrm{km/hr}$). Ship $B$ is traveling due north away from Lighthouse Rock at a speed of $10\mathrm{~km/hr}$. Let $x$ be the distance between Ship $A$ and Lighthouse Rock at time $t$, and let $y$ be the distance between Ship $B$ and Lighthouse Rock at time $t$, as shown in the figure above.
(a) Find the distance, in kilometers, between Ship $A$ and Ship $B$ when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(b) Find the rate of change, in $\mathrm{km/hr}$, of the distance between the two ships when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(c) Let $\theta$ be the angle shown in the figure. Find the rate of change of $\theta$, in radians per hour, when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.