ap-calculus-bc

Papers (38)
2025
free-response 6
2024
free-response 6
2023
free-response 6
2022
free-response 5
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 5
2013
free-response 6
2012
free-response 6 practice-exam 49
2011
free-response 5 free-response_formB 6
2010
free-response 6 free-response_formB 5
2009
free-response 6 free-response_formB 6
2008
free-response 5 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 4
2004
free-response 6 free-response_formB 6
2003
free-response 3 free-response_formB 5
2002
free-response 5 free-response_formB 6
2001
free-response 6
2000
free-response 5
1999
free-response 6
1998
free-response 5
2007 free-response_formB

6 maths questions

Q1 Areas Between Curves 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 (vectors) View
An object moving along a curve in the $x y$-plane is at position $( x ( t ) , y ( t ) )$ at time $t$ with $$\frac { d x } { d t } = \arctan \left( \frac { t } { 1 + t } \right) \text { and } \frac { d y } { d t } = \ln \left( t ^ { 2 } + 1 \right)$$ for $t \geq 0$. At time $t = 0$, the object is at position $( - 3 , - 4 )$. (Note: $\tan ^ { - 1 } x = \arctan x$ ) (a) Find the speed of the object at time $t = 4$. (b) Find the total distance traveled by the object over the time interval $0 \leq t \leq 4$. (c) Find $x ( 4 )$. (d) For $t > 0$, there is a point on the curve where the line tangent to the curve has slope 2 . At what time $t$ is the object at this point? Find the acceleration vector at this point.
Q3 Applied differentiation Table-Based Estimation with Rate of Change Interpretation 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 Second order differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = 3 x + 2 y + 1$. (a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. (b) Find the values of the constants $m , b$, and $r$ for which $y = m x + b + e ^ { r x }$ is a solution to the differential equation. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = - 2$. Use Euler's method, starting at $x = 0$ with a step size of $\frac { 1 } { 2 }$, to approximate $f ( 1 )$. Show the work that leads to your answer. (d) Let $y = g ( x )$ be another solution to the differential equation with the initial condition $g ( 0 ) = k$, where $k$ is a constant. Euler's method, starting at $x = 0$ with a step size of 1 , gives the approximation $g ( 1 ) \approx 0$. Find the value of $k$.
Q6 Taylor series Derive series via differentiation or integration of a known series View
Let $f$ be the function given by $f ( x ) = 6 e ^ { - x / 3 }$ for all $x$. (a) Find the first four nonzero terms and the general term for the Taylor series for $f$ about $x = 0$. (b) Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Find the first four nonzero terms and the general term for the Taylor series for $g$ about $x = 0$. (c) The function $h$ satisfies $h ( x ) = k f ^ { \prime } ( a x )$ for all $x$, where $a$ and $k$ are constants. The Taylor series for $h$ about $x = 0$ is given by $$h ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots .$$ Find the values of $a$ and $k$.