ap-calculus-bc

Papers (28)
2025
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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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1998
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2010 free-response

6 maths questions

Q1 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
There is no snow on Janet's driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by $f(t) = 7te^{\cos t}$ cubic feet per hour, where $t$ is measured in hours since midnight. Janet starts removing snow at 6 A.M. $(t = 6)$. The rate $g(t)$, in cubic feet per hour, at which Janet removes snow from the driveway at time $t$ hours after midnight is modeled by $$g(t) = \begin{cases} 0 & \text{for } 0 \leq t < 6 \\ 125 & \text{for } 6 \leq t < 7 \\ 108 & \text{for } 7 \leq t \leq 9. \end{cases}$$
(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.?
(b) Find the rate of change of the volume of snow on the driveway at 8 A.M.
(c) Let $h(t)$ represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time $t$ hours after midnight. Express $h$ as a piecewise-defined function with domain $0 \leq t \leq 9$.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?
Q2 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon $(t = 0)$ and 8 P.M. $(t = 8)$. The number of entries in the box $t$ hours after noon is modeled by a differentiable function $E$ for $0 \leq t \leq 8$. Values of $E(t)$, in hundreds of entries, at various times $t$ are shown in the table below.
\begin{tabular}{ c } $t$
(hours)
& 0 & 2 & 5 & 7 & 8 \hline
$E(t)$
(hundreds of
entries)
& 0 & 4 & 13 & 21 & 23 \hline \end{tabular}
(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time $t = 6$. Show the computations that lead to your answer.
(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$. Using correct units, explain the meaning of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$ in terms of the number of entries.
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function $P$, where $P(t) = t^3 - 30t^2 + 298t - 976$ hundreds of entries per hour for $8 \leq t \leq 12$. According to the model, how many entries had not yet been processed by midnight $(t = 12)$?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.
Q3 Variable acceleration (vectors) View
A particle is moving along a curve so that its position at time $t$ is $(x(t), y(t))$, where $x(t) = t^2 - 4t + 8$ and $y(t)$ is not explicitly given. Both $x$ and $y$ are measured in meters, and $t$ is measured in seconds. It is known that $\frac{dy}{dt} = te^{t-3} - 1$.
(a) Find the speed of the particle at time $t = 3$ seconds.
(b) Find the total distance traveled by the particle for $0 \leq t \leq 4$ seconds.
(c) Find the time $t$, $0 \leq t \leq 4$, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer.
(d) There is a point with $x$-coordinate 5 through which the particle passes twice. Find each of the following.
(i) The two values of $t$ when that occurs
(ii) The slopes of the lines tangent to the particle's path at that point
(iii) The $y$-coordinate of that point, given $y(2) = 3 + \frac{1}{e}$
Q4 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 $y = 2\sqrt{x}$, the horizontal line $y = 6$, and the $y$-axis.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 7$.
(c) Region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 6$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose height is 3 times the length of its base in region $R$. Write, but do not evaluate, an integral expression that gives the volume of the solid.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac{dy}{dx} = 1 - y$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(1) = 0$. For this particular solution, $f(x) < 1$ for all values of $x$.
(a) 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.
(b) Find $\lim_{x \to 1} \frac{f(x)}{x^3 - 1}$. Show the work that leads to your answer.
(c) Find the particular solution $y = f(x)$ to the differential equation $\frac{dy}{dx} = 1 - y$ with the initial condition $f(1) = 0$.
Q6 Taylor series Use series to analyze function properties (extrema, monotonicity, concavity) View
The function $f$, defined by $$f(x) = \begin{cases} \frac{\cos x - 1}{x^2} & \text{for } x \neq 0 \\ -\frac{1}{2} & \text{for } x = 0 \end{cases}$$ has derivatives of all orders. Let $g$ be the function defined by $g(x) = 1 + \int_{0}^{x} f(t)\,dt$.
(a) Write the first three nonzero terms and the general term of the Taylor series for $\cos x$ about $x = 0$. Use this series to write the first three nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use the Taylor series for $f$ about $x = 0$ found in part (a) to determine whether $f$ has a relative maximum, relative minimum, or neither at $x = 0$. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for $g$ about $x = 0$.
(d) The Taylor series for $g$ about $x = 0$, evaluated at $x = 1$, is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for $g$ about $x = 0$ to estimate the value of $g(1)$. Explain why this estimate differs from the actual value of $g(1)$ by less than $\frac{1}{6!}$.