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
2007 free-response_formB

6 maths questions

Q1 Volumes of Revolution Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal line $y = 2$, and let $S$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal lines $y = 1$ and $y = 2$, as shown above. (a) Find the area of $R$. (b) Find the area of $S$. (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 1$.
Q2 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 $t \geq 0$ is given by $v ( t ) = \sin \left( t ^ { 2 } \right)$. The graph of $v$ is shown above for $0 \leq t \leq \sqrt { 5 \pi }$. The position of the particle at time $t$ is $x ( t )$ and its position at time $t = 0$ is $x ( 0 ) = 5$. (a) Find the acceleration of the particle at time $t = 3$. (b) Find the total distance traveled by the particle from time $t = 0$ to $t = 3$. (c) Find the position of the particle at time $t = 3$. (d) For $0 \leq t \leq \sqrt { 5 \pi }$, find the time $t$ at which the particle is farthest to the right. Explain your answer.
Q3 Applied differentiation Applied modeling with differentiation View
The wind chill is the temperature, in degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ), a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity $v$, in miles per hour (mph). If the air temperature is $32 ^ { \circ } \mathrm { F }$, then the wind chill is given by $W ( v ) = 55.6 - 22.1 v ^ { 0.16 }$ and is valid for $5 \leq v \leq 60$. (a) Find $W ^ { \prime } ( 20 )$. Using correct units, explain the meaning of $W ^ { \prime } ( 20 )$ in terms of the wind chill. (b) Find the average rate of change of $W$ over the interval $5 \leq v \leq 60$. Find the value of $v$ at which the instantaneous rate of change of $W$ is equal to the average rate of change of $W$ over the interval $5 \leq v \leq 60$. (c) Over the time interval $0 \leq t \leq 4$ hours, the air temperature is a constant $32 ^ { \circ } \mathrm { F }$. At time $t = 0$, the wind velocity is $v = 20 \mathrm { mph }$. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at $t = 3$ hours? Indicate units of measure.
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 $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 2 } x + y - 1$. (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.) (b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Describe the region in the $x y$-plane in which all solution curves to the differential equation are concave up. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 0$ ? Justify your answer. (d) Find the values of the constants $m$ and $b$, for which $y = m x + b$ is a solution to the differential equation.
Q6 Chain Rule Deduction or Consequence from Prior Results View
Let $f$ be a twice-differentiable function such that $f ( 2 ) = 5$ and $f ( 5 ) = 2$. Let $g$ be the function given by $g ( x ) = f ( f ( x ) )$. (a) Explain why there must be a value $c$ for $2 < c < 5$ such that $f ^ { \prime } ( c ) = - 1$. (b) Show that $g ^ { \prime } ( 2 ) = g ^ { \prime } ( 5 )$. Use this result to explain why there must be a value $k$ for $2 < k < 5$ such that $g ^ { \prime \prime } ( k ) = 0$. (c) Show that if $f ^ { \prime \prime } ( x ) = 0$ for all $x$, then the graph of $g$ does not have a point of inflection. (d) Let $h ( x ) = f ( x ) - x$. Explain why there must be a value $r$ for $2 < r < 5$ such that $h ( r ) = 0$.