ap-calculus-bc

Papers (28)
2025
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2024
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2023
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2019
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2018
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2016
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2015
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2014
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2013
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2012
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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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2001
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1999
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1998
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2001 free-response

6 maths questions

Q1 Parametric differentiation View
An object moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ with $$\frac{dx}{dt} = \cos\left(t^3\right) \text{ and } \frac{dy}{dt} = 3\sin\left(t^2\right)$$ for $0 \leq t \leq 3$. At time $t = 2$, the object is at position $(4,5)$.
(a) Write an equation for the line tangent to the curve at $(4,5)$.
(b) Find the speed of the object at time $t = 2$.
(c) Find the total distance traveled by the object over the time interval $0 \leq t \leq 1$.
(d) Find the position of the object at time $t = 3$.
Q2 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.
\begin{tabular}{ c } $t$
(days)
&
$W(t)$
$\left({}^{\circ}\mathrm{C}\right)$
\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W'(12)$. Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.
(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P'(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.
(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.
Q3 Variable acceleration (1D) Multi-part particle motion analysis (graph-based velocity) View
A car is traveling on a straight road with velocity $55\,\mathrm{ft/sec}$ at time $t = 0$. For $0 \leq t \leq 18$ seconds, the car's acceleration $a(t)$, in $\mathrm{ft/sec}^2$, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at $t = 2$ seconds? Why or why not?
(b) At what time in the interval $0 \leq t \leq 18$, other than $t = 0$, is the velocity of the car $55\,\mathrm{ft/sec}$? Why?
(c) On the time interval $0 \leq t \leq 18$, what is the car's absolute maximum velocity, in $\mathrm{ft/sec}$, and at what time does it occur? Justify your answer.
(d) At what times in the interval $0 \leq t \leq 18$, if any, is the car's velocity equal to zero? Justify your answer.
Q4 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h'(x) = \dfrac{x^2 - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Let $f$ be the function satisfying $f'(x) = -3x f(x)$, for all real numbers $x$, with $f(1) = 4$ and $\lim_{x \rightarrow \infty} f(x) = 0$.
(a) Evaluate $\displaystyle\int_{1}^{\infty} -3x f(x)\, dx$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 1$ with a step size of $0.5$, to approximate $f(2)$.
(c) Write an expression for $y = f(x)$ by solving the differential equation $\dfrac{dy}{dx} = -3xy$ with the initial condition $f(1) = 4$.
Q6 Sequences and Series Power Series Expansion and Radius of Convergence View
A function $f$ is defined by $$f(x) = \frac{1}{3} + \frac{2}{3^2}x + \frac{3}{3^3}x^2 + \cdots + \frac{n+1}{3^{n+1}}x^n + \cdots$$ for all $x$ in the interval of convergence of the given power series.
(a) Find the interval of convergence for this power series. Show the work that leads to your answer.
(b) Find $\displaystyle\lim_{x \rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}$.
(c) Write the first three nonzero terms and the general term for an infinite series that represents $\displaystyle\int_0^1 f(x)\, dx$.
(d) Find the sum of the series determined in part (c).