ap-calculus-bc

Papers (28)
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 51
2011
free-response 6
2010
free-response 6
2009
free-response 6
2008
free-response 6
2007
free-response 6
2006
free-response 6
2005
free-response 6
2004
free-response_formB 6
2003
free-response_formB 6
2002
free-response 6 free-response_formB 6
2001
free-response 6
1999
free-response 6
1998
free-response 6
2006 free-response

6 maths questions

Q1 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the shaded region bounded by the graph of $y = \ln x$ and the line $y = x - 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = -3$.
(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $R$ is rotated about the $y$-axis.
Q2 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
At an intersection in Thomasville, Oregon, cars turn left at the rate $L(t) = 60\sqrt{t}\sin^{2}\left(\frac{t}{3}\right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L(t) \geq 150$ cars per hour. Find all values of $t$ for which $L(t) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.
Q3 Variable acceleration (vectors) View
An object moving along a curve in the $xy$-plane is at position $(x(t), y(t))$ at time $t$, where $$\frac{dx}{dt} = \sin^{-1}\left(1 - 2e^{-t}\right) \text{ and } \frac{dy}{dt} = \frac{4t}{1 + t^{3}}$$ for $t \geq 0$. At time $t = 2$, the object is at the point $(6, -3)$. (Note: $\sin^{-1} x = \arcsin x$)
(a) Find the acceleration vector and the speed of the object at time $t = 2$.
(b) The curve has a vertical tangent line at one point. At what time $t$ is the object at this point?
(c) Let $m(t)$ denote the slope of the line tangent to the curve at the point $(x(t), y(t))$. Write an expression for $m(t)$ in terms of $t$ and use it to evaluate $\lim_{t \rightarrow \infty} m(t)$.
(d) The graph of the curve has a horizontal asymptote $y = c$. Write, but do not evaluate, an expression involving an improper integral that represents this value $c$.
Q4 Constant acceleration (SUVAT) Velocity-time or acceleration-time graph interpretation View
Rocket $A$ has positive velocity $v(t)$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table below.
$t$ (seconds)01020304050607080
$v(t)$ (feet per second)51422293540444749

(a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{10}^{70} v(t)\, dt$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int_{10}^{70} v(t)\, dt$.
(c) Rocket $B$ is launched upward with an acceleration of $a(t) = \frac{3}{\sqrt{t+1}}$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac{dy}{dx} = 5x^{2} - \frac{6}{y-2}$ for $y \neq 2$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(-1) = -4$.
(a) Evaluate $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ at $(-1, -4)$.
(b) Is it possible for the $x$-axis to be tangent to the graph of $f$ at some point? Explain why or why not.
(c) Find the second-degree Taylor polynomial for $f$ about $x = -1$.
(d) Use Euler's method, starting at $x = -1$ with two steps of equal size, to approximate $f(0)$. Show the work that leads to your answer.
Q6 Taylor series Use series to analyze function properties (extrema, monotonicity, concavity) View
The function $f$ is defined by the power series $$f(x) = -\frac{x}{2} + \frac{2x^{2}}{3} - \frac{3x^{3}}{4} + \cdots + \frac{(-1)^{n} n x^{n}}{n+1} + \cdots$$ for all real numbers $x$ for which the series converges. The function $g$ is defined by the power series $$g(x) = 1 - \frac{x}{2!} + \frac{x^{2}}{4!} - \frac{x^{3}}{6!} + \cdots + \frac{(-1)^{n} x^{n}}{(2n)!} + \cdots$$ for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The graph of $y = f(x) - g(x)$ passes through the point $(0, -1)$. Find $y'(0)$ and $y''(0)$. Determine whether $y$ has a relative minimum, a relative maximum, or neither at $x = 0$. Give a reason for your answer.