ap-calculus-bc

Papers (28)
2025
free-response 6
2024
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2023
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2022
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2021
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2019
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2018
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2017
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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
free-response 6 practice-exam 51
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
free-response 6
2004
free-response_formB 6
2003
free-response_formB 6
2002
free-response 6 free-response_formB 6
2001
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1999
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1998
free-response 6
1999 free-response

6 maths questions

Q1 Parametric differentiation View
A particle moves in the $xy$-plane so that its position at any time $t$, $0 \leq t \leq \pi$, is given by $x(t) = \frac{t^2}{2} - \ln(1+t)$ and $y(t) = 3\sin t$.
(a) Sketch the path of the particle in the $xy$-plane below. Indicate the direction of motion along the path.
(b) At what time $t$, $0 \leq t \leq \pi$, does $x(t)$ attain its minimum value? What is the position $(x(t), y(t))$ of the particle at this time?
(c) At what time $t$, $0 < t < \pi$, is the particle on the $y$-axis? Find the speed and the acceleration vector of the particle at this time.
Q2 Volumes of Revolution Multi-Part Area-and-Volume Free Response View
The shaded region, $R$, is bounded by the graph of $y = x^2$ and the line $y = 4$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated by revolving $R$ about the $x$-axis.
(c) There exists a number $k$, $k > 4$, such that when $R$ is revolved about the line $y = k$, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.
Q3 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.
\begin{tabular}{ c } $t$
(hours)
&
$R(t)$
(gallons per hour)
\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int_{0}^{24} R(t)\, dt$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t$, $0 < t < 24$, such that $R'(t) = 0$? Justify your answer.
(c) The rate of water flow $R(t)$ can be approximated by $Q(t) = \frac{1}{79}\left(768 + 23t - t^2\right)$. Use $Q(t)$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.
Q4 Taylor series Lagrange error bound application View
The function $f$ has derivatives of all orders for all real numbers $x$. Assume $f(2) = -3$, $f'(2) = 5$, $f''(2) = 3$, and $f'''(2) = -8$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 2$ and use it to approximate $f(1.5)$.
(b) The fourth derivative of $f$ satisfies the inequality $\left|f^{(4)}(x)\right| \leq 3$ for all $x$ in the closed interval $[1.5, 2]$. Use the Lagrange error bound on the approximation to $f(1.5)$ found in part (a) to explain why $f(1.5) \neq -5$.
(c) Write the fourth-degree Taylor polynomial, $P(x)$, for $g(x) = f\left(x^2 + 2\right)$ about $x = 0$. Use $P$ to explain why $g$ must have a relative minimum at $x = 0$.
Q5 Indefinite & Definite Integrals Accumulation Function Analysis View
The graph of the function $f$, consisting of three line segments, is given above. Let $g(x) = \int_{1}^{x} f(t)\, dt$.
(a) Compute $g(4)$ and $g(-2)$.
(b) Find the instantaneous rate of change of $g$, with respect to $x$, at $x = 1$.
(c) Find the absolute minimum value of $g$ on the closed interval $[-2, 4]$. Justify your answer.
(d) The second derivative of $g$ is not defined at $x = 1$ and $x = 2$. How many of these values are $x$-coordinates of points of inflection of the graph of $g$? Justify your answer.
Q6 Tangents, normals and gradients Tangent line used for linear approximation or Euler's method View
Let $f$ be the function whose graph goes through the point $(3, 6)$ and whose derivative is given by $f'(x) = \frac{1 + e^x}{x^2}$.
(a) Write an equation of the line tangent to the graph of $f$ at $x = 3$ and use it to approximate $f(3.1)$.
(b) Use Euler's method, starting at $x = 3$ with a step size of 0.05, to approximate $f(3.1)$. Use $f''$ to explain why this approximation is less than $f(3.1)$.
(c) Use $\int_{3}^{3.1} f'(x)\, dx$ to evaluate $f(3.1)$.