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

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $y = e ^ { - x ^ { 2 } } , y = 1 - \cos x$, and the $y$-axis, as shown in the figure above.
(a) Find the area of the region $R$.
(b) Find the volume of the solid generated when the region $R$ 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 Variable acceleration (1D) Two-particle comparison problem View
Two runners, $A$ and $B$, run on a straight racetrack for $0 \leq t \leq 10$ seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner $A$. The velocity, in meters per second, of Runner $B$ is given by the function $v$ defined by $v ( t ) = \frac { 24 t } { 2 t + 3 }$.
(a) Find the velocity of Runner $A$ and the velocity of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(b) Find the acceleration of Runner $A$ and the acceleration of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(c) Find the total distance run by Runner $A$ and the total distance run by Runner $B$ over the time interval $0 \leq t \leq 10$ seconds. Indicate units of measure.
Q3 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, for $- 7 \leq x \leq 7$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = - 3 , x = 2$, and $x = 5$, and a vertical tangent line at $x = 3$.
(a) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative minimum. Justify your answer.
(b) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative maximum. Justify your answer.
(c) Find all values of $x$, for $- 7 < x < 7$, at which $f ^ { \prime \prime } ( x ) < 0$.
(d) At what value of $x$, for $- 7 \leq x \leq 7$, does $f$ attain its absolute maximum? Justify your answer.
Q4 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of $\sqrt { t + 1 }$ gallons per minute, for $0 \leq t \leq 120$ minutes. At time $t = 0$, the tank contains 30 gallons of water.
(a) How many gallons of water leak out of the tank from time $t = 0$ to $t = 3$ minutes?
(b) How many gallons of water are in the tank at time $t = 3$ minutes?
(c) Write an expression for $A ( t )$, the total number of gallons of water in the tank at time $t$.
(d) At what time $t$, for $0 \leq t \leq 120$, is the amount of water in the tank a maximum? Justify your answer.
Q5 Implicit equations and differentiation Verify implicit derivative and find tangent line features View
Consider the curve given by $x y ^ { 2 } - x ^ { 3 } y = 6$.
(a) Show that $\frac { d y } { d x } = \frac { 3 x ^ { 2 } y - y ^ { 2 } } { 2 x y - x ^ { 3 } }$.
(b) Find all points on the curve whose $x$-coordinate is 1 , and write an equation for the tangent line at each of these points.
(c) Find the $x$-coordinate of each point on the curve where the tangent line is vertical.
Q6 Differential equations Solving Separable DEs with Initial Conditions View
Consider the differential equation $\frac { d y } { d x } = \frac { 3 x ^ { 2 } } { e ^ { 2 y } }$.
(a) Find a solution $y = f ( x )$ to the differential equation satisfying $f ( 0 ) = \frac { 1 } { 2 }$.
(b) Find the domain and range of the function $f$ found in part (a).