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
2002 free-response_formB

6 maths questions

Q1 Parametric differentiation View
A particle moves in the $xy$-plane so that its position at any time $t$, for $-\pi \leq t \leq \pi$, is given by $x(t) = \sin(3t)$ and $y(t) = 2t$.
(a) Sketch the path of the particle in the $xy$-plane provided. Indicate the direction of motion along the path.
(b) Find the range of $x(t)$ and the range of $y(t)$.
(c) Find the smallest positive value of $t$ for which the $x$-coordinate of the particle is a local maximum. What is the speed of the particle at this time?
(d) Is the distance traveled by the particle from $t = -\pi$ to $t = \pi$ greater than $5\pi$? Justify your answer.
Q2 Differential equations Tangent Line Approximation from a DE View
The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P'(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?
Q3 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the region in the first quadrant bounded by the $y$-axis and the graphs of $y = 4x - x^3 + 1$ and $y = \frac{3}{4}x$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) Write an expression involving one or more integrals that gives the perimeter of $R$. Do not evaluate.
Q4 Indefinite & Definite Integrals Accumulation Function Analysis View
The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g'(6)$, and $g''(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3 - x}{y}$.
(a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer.
(b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.
Q6 Taylor series Construct series for a composite or related function View
The Maclaurin series for $\ln\left(\dfrac{1}{1-x}\right)$ is $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ with interval of convergence $-1 \leq x < 1$.
(a) Find the Maclaurin series for $\ln\left(\dfrac{1}{1+3x}\right)$ and determine the interval of convergence.
(b) Find the value of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$.
(c) Give a value of $p$ such that $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n^p}$ converges, but $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2p}}$ diverges. Give reasons why your value of $p$ is correct.
(d) Give a value of $p$ such that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverges, but $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2p}}$ converges. Give reasons why your value of $p$ is correct.