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
1998 free-response

6 maths questions

Q1 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
Let $R$ be the region in the first quadrant bounded by the graph of $y = 8 - x^{\frac{3}{2}}$, the $x$-axis, and the $y$-axis.
(a) Find the area of the region $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) The vertical line $x = k$ divides the region $R$ into two regions such that when these two regions are revolved about the $x$-axis, they generate solids with equal volumes. Find the value of $k$.
Q2 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.
Q3 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View
Let $f$ be a function that has derivatives of all orders for all real numbers. Assume $f(0) = 5$, $f'(0) = -3$, $f''(0) = 1$, and $f'''(0) = 4$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$ and use it to approximate $f(0.2)$.
(b) Write the fourth-degree Taylor polynomial for $g$, where $g(x) = f\left(x^{2}\right)$, about $x = 0$.
(c) Write the third-degree Taylor polynomial for $h$, where $h(x) = \int_{0}^{x} f(t)\, dt$, about $x = 0$.
(d) Let $h$ be defined as in part (c). Given that $f(1) = 3$, either find the exact value of $h(1)$ or explain why it cannot be determined.
Q4 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation given by $\dfrac{dy}{dx} = \dfrac{xy}{2}$.
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(b) Let $y = f(x)$ be the particular solution to the given differential equation with the initial condition $f(0) = 3$. Use Euler's method starting at $x = 0$, with a step size of 0.1, to approximate $f(0.2)$. Show the work that leads to your answer.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(0) = 3$. Use your solution to find $f(0.2)$.
Q5 Indefinite & Definite Integrals Average Value of a Function View
The temperature outside a house during a 24-hour period is given by $$F(t) = 80 - 10\cos\left(\frac{\pi t}{12}\right), \quad 0 \leq t \leq 24,$$ where $F(t)$ is measured in degrees Fahrenheit and $t$ is measured in hours.
(a) Sketch the graph of $F$ on the grid provided.
(b) Find the average temperature, to the nearest degree Fahrenheit, between $t = 6$ and $t = 14$.
(c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of $t$ was the air conditioner cooling the house?
(d) The cost of cooling the house accumulates at the rate of $\$0.05$ per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24-hour period?
Q6 Connected Rates of Change Parametric or Curve-Based Particle Motion Rates View
A particle moves along the curve defined by the equation $y = x^{3} - 3x$. The $x$-coordinate of the particle, $x(t)$, satisfies the equation $\dfrac{dx}{dt} = \dfrac{1}{\sqrt{2t+1}}$, for $t \geq 0$ with initial condition $x(0) = -4$.
(a) Find $x(t)$ in terms of $t$.
(b) Find $\dfrac{dy}{dt}$ in terms of $t$.
(c) Find the location and speed of the particle at time $t = 4$.