ap-calculus-ab

Papers (31)
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
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
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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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Unknown
-bc_course-and-exam-description 24 -bc_sample-questions 18
2005 free-response_formB

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $f$ and $g$ be the functions given by $f(x) = 1 + \sin(2x)$ and $g(x) = e^{x/2}$. Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$.
(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, the cross sections perpendicular to the $x$-axis are semicircles with diameters extending from $y = f(x)$ to $y = g(x)$. Find the volume of this solid.
Q2 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
A water tank at Camp Newton holds 1200 gallons of water at time $t = 0$. During the time interval $0 \leq t \leq 18$ hours, water is pumped into the tank at the rate $$W(t) = 95\sqrt{t}\sin^2\left(\frac{t}{6}\right) \text{ gallons per hour.}$$ During the same time interval, water is removed from the tank at the rate $$R(t) = 275\sin^2\left(\frac{t}{3}\right) \text{ gallons per hour.}$$
(a) Is the amount of water in the tank increasing at time $t = 15$? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time $t = 18$?
(c) At what time $t$, for $0 \leq t \leq 18$, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.
(d) For $t > 18$, no water is pumped into the tank, but water continues to be removed at the rate $R(t)$ until the tank becomes empty. Let $k$ be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.
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 time $t$, for $0 \leq t \leq 5$, is given by $v(t) = \ln\left(t^2 - 3t + 3\right)$. The particle is at position $x = 8$ at time $t = 0$.
(a) Find the acceleration of the particle at time $t = 4$.
(b) Find all times $t$ in the open interval $0 < t < 5$ at which the particle changes direction. During which time intervals, for $0 \leq t \leq 5$, does the particle travel to the left?
(c) Find the position of the particle at time $t = 2$.
(d) Find the average speed of the particle over the interval $0 \leq t \leq 2$.
Q4 Indefinite & Definite Integrals Accumulation Function Analysis View
The graph of the function $f$ consists of three line segments.
(a) Let $g$ be the function given by $g(x) = \int_{-4}^{x} f(t)\, dt$. For each of $g(-1)$, $g'(-1)$, and $g''(-1)$, find the value or state that it does not exist.
(b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $-4 < x < 3$. Explain your reasoning.
(c) Let $h$ be the function given by $h(x) = \int_{x}^{3} f(t)\, dt$. Find all values of $x$ in the closed interval $-4 \leq x \leq 3$ for which $h(x) = 0$.
(d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.
Q5 Implicit equations and differentiation Verify implicit derivative and find tangent line features View
Consider the curve given by $y^2 = 2 + xy$.
(a) Show that $\dfrac{dy}{dx} = \dfrac{y}{2y - x}$.
(b) Find all points $(x, y)$ on the curve where the line tangent to the curve has slope $\dfrac{1}{2}$.
(c) Show that there are no points $(x, y)$ on the curve where the line tangent to the curve is horizontal.
(d) Let $x$ and $y$ be functions of time $t$ that are related by the equation $y^2 = 2 + xy$. At time $t = 5$, the value of $y$ is 3 and $\dfrac{dy}{dt} = 6$. Find the value of $\dfrac{dx}{dt}$ at time $t = 5$.
Q6 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\dfrac{dy}{dx} = \dfrac{-xy^2}{2}$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(-1) = 2$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) Write an equation for the line tangent to the graph of $f$ at $x = -1$.
(c) Find the solution $y = f(x)$ to the given differential equation with the initial condition $f(-1) = 2$.