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
2006 free-response_formB

6 maths questions

Q1 Chain Rule Chain Rule with Composition of Explicit Functions View
Let $f$ be the function given by $f ( x ) = \sqrt { x ^ { 4 } - 16 x ^ { 2 } }$. (a) Find the domain of $f$. (b) Describe the symmetry, if any, of the graph of $f$. (c) Find $f ^ { \prime } ( x )$. (d) Find the slope of the line normal to the graph of $f$ at $x = 5$.
Q2 Variable acceleration (1D) Multi-part particle motion analysis (formula-based velocity) View
A particle moves along the $x$-axis so that its velocity at any time $t \geqq 0$ is given by $v ( t ) = 1 - \sin ( 2 \pi t )$. (a) Find the acceleration $a ( t )$ of the particle at any time $t$. (b) Find all values of $t , 0 \leqq t \leqq 2$, for which the particle is at rest. (c) Find the position $x ( t )$ of the particle at any time $t$ if $x ( 0 ) = 0$.
Q3 Volumes of Revolution Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $f ( x ) = \cos x$ and $g ( x ) = x ^ { 2 } - 1$. (a) Find the coordinates of any points of intersection of $f$ and $g$. (b) Find the area bounded by $f$ and $g$. (c) Find the volume generated when the region in part (b) is rotated around the $y$-axis.
Q4 Stationary points and optimisation Sketching a Curve from Analytical Properties View
Let $f$ be the function defined by $f ( x ) = 2 x e ^ { - x }$ for all real numbers $x$. (a) Write an equation of the horizontal asymptote for the graph of $f$. (b) Find the $x$-coordinate of each critical point of $f$. For each such $x$, determine whether $f ( x )$ relative maximum, a relative minimum, or neither. (c) For what values of $x$ is the graph of $f$ concave down? (d) Using the results found in parts (a), (b), and (c), sketch the graph of $y = f ( x )$ in the $x y$ provided below. Note: The $x y$-plane is provided in the pink test booklet only.
Q5 Areas by integration Average Value of a Function View
Let $R$ be the region in the first quadrant under the graph of $y = \frac { x } { x ^ { 2 } + 2 }$ for $0 \leqq x \leqq$ (a) Find the area of $R \cdot \left( - x - \alpha , \frac { 1 } { 1 } \right)$ (b) If the line $x = k$ divides $R$ into two regions of equal area, what is the value of $k$ ? (c) What is the average value of $y = \frac { x } { x ^ { 2 } + 2 }$ on the interval $0 \leqq x \leqq \sqrt { 6 }$ ?
Q6 Standard Integrals and Reverse Chain Rule Finding a Function from an Integral Equation View
Let $f$ be a differentiable function, defined for all real numbers $x$, with the following properties. (i) $f ^ { \prime } ( x ) = a x ^ { 2 } + b x$ (ii) $f ^ { \prime } ( 1 ) = 6$ and $f ^ { \prime \prime } ( 1 ) = 18$ (iii) $\int _ { 1 } ^ { 2 } f ( x ) d x = 18$
Find $f ( x )$. Show your work.