ap-calculus-ab

Papers (46)
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
free-response 6
2016
free-response 6
2015
free-response 6
2014
free-response 6
2013
free-response 6
2012
free-response 6 practice-exam 50
2011
free-response 6 free-response_formB 6
2010
free-response 6 free-response_formB 3
2009
free-response 6 free-response_formB 6
2008
free-response 6 free-response_formB 6
2007
free-response 6 free-response_formB 6
2006
free-response 6 free-response_formB 6
2005
free-response 6 free-response_formB 6
2004
free-response 3 free-response_formB 6
2003
free-response 6
2002
free-response 5 free-response_formB 6
2001
free-response 6 free-response_formB 6
2000
free-response 6 free-response_formB 6
1999
free-response 6 free-response_formB 7
1998
free-response 6 free-response_formB 5 full-exam 18
Unknown
-bc_1969-1998_multiple-choice-collection 29 -bc_course-and-exam-description 26 -bc_sample-questions 18 1988_full-exam 3
2000 free-response_formB

6 maths questions

Q1 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $f$ and $g$ be the functions given by $f ( x ) = 1 + \sin ( 2 x )$ and $g ( x ) = e ^ { x / 2 }$. Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$ 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, 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 $v ( t ) = \ln \left( t ^ { 2 } - 3 t + 3 \right)$. 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 $f$ above consists of three line segments.
(a) Let $g$ be the function given by $g ( x ) = \int _ { - 4 } ^ { x } f ( t ) d t$.
For each of $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$, 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 $h ( x ) = \int _ { x } ^ { 3 } f ( t ) d t$. 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 $y ^ { 2 } = 2 + x y$.
(a) Show that $\frac { d y } { d x } = \frac { y } { 2 y - x }$.
(b) Find all points $( x , y )$ on the curve where the line tangent to the curve has slope $\frac { 1 } { 2 }$.
(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 $y ^ { 2 } = 2 + x y$. At time $t = 5$, the value of $y$ is 3 and $\frac { d y } { d t } = 6$. Find the value of $\frac { d x } { d t }$ at time $t = 5$.
Q6 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = \frac { - x y ^ { 2 } } { 2 }$. 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.
(Note: Use the axes provided in the test booklet.)
(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 $f ( - 1 ) = 2$.