ap-calculus-bc

Papers (38)
2025
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2024
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2023
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2019
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2015
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2014
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2011
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2010
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2009
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2008
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2007
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2006
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2005
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2004
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2003
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2002
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1999
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1998
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2010 free-response_formB

5 maths questions

Q2 Variable acceleration (vectors) View
2. The velocity vector of a particle moving in the $x y$-plane has components given by
$$\frac { d x } { d t } = 14 \cos \left( t ^ { 2 } \right) \sin \left( e ^ { t } \right) \text { and } \frac { d y } { d t } = 1 + 2 \sin \left( t ^ { 2 } \right) , \text { for } 0 \leq t \leq 1.5 .$$
At time $t = 0$, the position of the particle is $( - 2,3 )$.
(a) For $0 < t < 1.5$, find all values of $t$ at which the line tangent to the path of the particle is vertical.
(b) Write an equation for the line tangent to the path of the particle at $t = 1$.
(c) Find the speed of the particle at $t = 1$.
(d) Find the acceleration vector of the particle at $t = 1$.
$t$024681012
$P ( t )$0465357606263

[Figure]
Q3 Numerical integration Multi-Part Applied Integration with Context (Trapezoidal/Numerical Estimation) View
3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time $t = 0$. During the time interval $0 \leq t \leq 12$ hours, water is pumped into the pool at the rate $P ( t )$ cubic feet per hour. The table above gives values of $P ( t )$ for selected values of $t$. During the same time interval, water is leaking from the pool at the rate $R ( t )$ cubic feet per hour, where $R ( t ) = 25 e ^ { - 0.05 t }$. (Note: The volume $V$ of a cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval $0 \leq t \leq 12$ hours. Show the computations that lead to your answer.
(b) Calculate the total amount of water that leaked out of the pool during the time interval $0 \leq t \leq 12$ hours.
(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time $t = 12$ hours. Round your answer to the nearest cubic foot.
(d) Find the rate at which the volume of water in the pool is increasing at time $t = 8$ hours. How fast is the water level in the pool rising at $t = 8$ hours? Indicate units of measure in both answers.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
© 2010 The College Board. Visit the College Board on the Web: \href{http://www.collegeboard.com}{www.collegeboard.com}.
No calculator is allowed for these problems. [Figure]
Q4 Applied differentiation Velocity-time or acceleration-time graph interpretation View
4. A squirrel starts at building $A$ at time $t = 0$ and travels along a straight, horizontal wire connected to building $B$. For $0 \leq t \leq 18$, the squirrel's velocity is modeled by the piecewise-linear function defined by the graph above.
(a) At what times in the interval $0 < t < 18$, if any, does the squirrel change direction? Give a reason for your answer.
(b) At what time in the interval $0 \leq t \leq 18$ is the squirrel farthest from building $A$ ? How far from building $A$ is the squirrel at that time?
(c) Find the total distance the squirrel travels during the time interval $0 \leq t \leq 18$.
(d) Write expressions for the squirrel's acceleration $a ( t )$, velocity $v ( t )$, and distance $x ( t )$ from building $A$ that are valid for the time interval $7 < t < 10$.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2010 The College Board. Visit the College Board on the Web: \href{http://www.collegeboard.com}{www.collegeboard.com}.
Q5 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
5. Let $f$ and $g$ be the functions defined by $f ( x ) = \frac { 1 } { x }$ and $g ( x ) = \frac { 4 x } { 1 + 4 x ^ { 2 } }$, for all $x > 0$.
(a) Find the absolute maximum value of $g$ on the open interval $( 0 , \infty )$ if the maximum exists. Find the absolute minimum value of $g$ on the open interval $( 0 , \infty )$ if the minimum exists. Justify your answers.
(b) Find the area of the unbounded region in the first quadrant to the right of the vertical line $x = 1$, below the graph of $f$, and above the graph of $g$.
Q6 Taylor series Verifying a particular solution satisfies a second-order ODE View
6. The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { n } } { n - 1 }$ on its interval of convergence.
(a) Find the interval of convergence for the Maclaurin series of $f$. Justify your answer.
(b) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } - y = \frac { 4 x ^ { 2 } } { 1 + 2 x }$ for $| x | < R$, where $R$ is the radius of convergence from part (a).
WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM