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
2006 free-response_formB

6 maths questions

Q1 Variable acceleration (vectors) View
At time $t$, a particle moving in the $x y$-plane is at position $( x ( t ) , y ( t ) )$, where $x ( t )$ and $y ( t )$ are not explicitly given. For $t \geq 0 , \frac { d x } { d t } = 4 t + 1$ and $\frac { d y } { d t } = \sin \left( t ^ { 2 } \right)$. At time $t = 0 , x ( 0 ) = 0$ and $y ( 0 ) = - 4$. (a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$. (b) Find the slope of the line tangent to the path of the particle at time $t = 3$. (c) Find the position of the particle at time $t = 3$. (d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.
Q2 Chain Rule Evaluate derivative at a point or find tangent slope View
As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H ( t )$ is measured in degrees Celsius. Values of $H ( t )$ at selected values of time $t$ are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer. (b) Using correct units, explain the meaning of $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$. (c) Evaluate $\int _ { 0 } ^ { 10 } H ^ { \prime } ( t ) d t$. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time $t = 0$, biscuits with temperature $100 ^ { \circ } \mathrm { C }$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B ^ { \prime } ( t ) = - 13.84 e ^ { - 0.173 t }$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?
Q3 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $f ( x ) = e ^ { 2 x }$. Let $R$ be the region in the first quadrant bounded by the graph of $f$, the coordinate axes, and the vertical line $x = k$, where $k > 0$. The region $R$ is shown in the figure above. (a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of $R$ in terms of $k$. (b) The region $R$ is rotated about the $x$-axis to form a solid. Find the volume, $V$, of the solid in terms of $k$. (c) The volume $V$, found in part (b), changes as $k$ changes. If $\frac { d k } { d t } = \frac { 1 } { 3 }$, determine $\frac { d V } { d t }$ when $k = \frac { 1 } { 2 }$.
Q4 Connected Rates of Change Accumulation Function Analysis View
The continuous function $f$ is defined on the interval $- 4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure above. Let $g ( x ) = 2 x + \int _ { 0 } ^ { x } f ( t ) d t$. (a) Find $g ( - 3 )$. Find $g ^ { \prime } ( x )$ and evaluate $g ^ { \prime } ( - 3 )$. (b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $- 4 \leq x \leq 3$. Justify your answer. (c) Find all values of $x$ on the interval $- 4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer. (d) Find the average rate of change of $f$ on the interval $- 4 \leq x \leq 3$. There is no point $c , - 4 < c < 3$, for which $f ^ { \prime } ( c )$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.
Q5 First order differential equations (integrating factor) Applied Modeling with Differential Equations View
At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function $W$ models the total amount of solid waste stored at the landfill. Planners estimate that $W$ will satisfy the differential equation $\frac { d W } { d t } = \frac { 1 } { 25 } ( W - 300 )$ for the next 20 years. $W$ is measured in tons, and $t$ is measured in years from the start of 2010. (a) Use the line tangent to the graph of $W$ at $t = 0$ to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time $t = \frac { 1 } { 4 }$ ). (b) Find $\frac { d ^ { 2 } W } { d t ^ { 2 } }$ in terms of $W$. Use $\frac { d ^ { 2 } W } { d t ^ { 2 } }$ to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time $t = \frac { 1 } { 4 }$. (c) Find the particular solution $W = W ( t )$ to the differential equation $\frac { d W } { d t } = \frac { 1 } { 25 } ( W - 300 )$ with initial condition $W ( 0 ) = 1400$.
Q6 Taylor series Construct series for a composite or related function View
Let $f ( x ) = \sin \left( x ^ { 2 } \right) + \cos x$. The graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ is shown above. (a) Write the first four nonzero terms of the Taylor series for $\sin x$ about $x = 0$, and write the first four nonzero terms of the Taylor series for $\sin \left( x ^ { 2 } \right)$ about $x = 0$. (b) Write the first four nonzero terms of the Taylor series for $\cos x$ about $x = 0$. Use this series and the series for $\sin \left( x ^ { 2 } \right)$, found in part (a), to write the first four nonzero terms of the Taylor series for $f$ about $x = 0$. (c) Find the value of $f ^ { ( 6 ) } ( 0 )$. (d) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Using information from the graph of $y = \left| f ^ { ( 5 ) } ( x ) \right|$ shown above, show that $\left| P _ { 4 } \left( \frac { 1 } { 4 } \right) - f \left( \frac { 1 } { 4 } \right) \right| < \frac { 1 } { 3000 }$.