ap-calculus-ab

Papers (31)
2025
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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
practice-exam 50
2011
free-response 6 free-response_formB 6
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
free-response 6
Unknown
-bc_course-and-exam-description 24 -bc_sample-questions 18
1999 free-response

6 maths questions

Q1 Variable acceleration (1D) Multi-part particle motion analysis (formula-based velocity) View
A particle moves along the $y$-axis with velocity given by $v ( t ) = t \sin \left( t ^ { 2 } \right)$ for $t \geq 0$.
(a) In which direction (up or down) is the particle moving at time $t = 1.5$ ? Why?
(b) Find the acceleration of the particle at time $t = 1.5$. Is the velocity of the particle increasing at $t = 1.5$ ? Why or why not?
(c) Given that $y ( t )$ is the position of the particle at time $t$ and that $y ( 0 ) = 3$, find $y ( 2 )$.
(d) Find the total distance traveled by the particle from $t = 0$ to $t = 2$.
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 $y = 4$, as shown in the figure above.
(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 $\int _ { 0 } ^ { 24 } R ( t ) d t$. 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 $R ^ { \prime } ( t ) = 0$ ? Justify your answer.
(c) The rate of water flow $R ( t )$ can be approximated by $Q ( t ) = \frac { 1 } { 79 } \left( 768 + 23 t - t ^ { 2 } \right)$. Use $Q ( t )$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.
Q4 Tangents, normals and gradients Find tangent line equation at a given point View
Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$.
(a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$.
(b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer.
(c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$.
(d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.
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 Connected Rates of Change Tangent Line Geometry with Connected Rates View
In the figure above, line $\ell$ is tangent to the graph of $y = \frac { 1 } { x ^ { 2 } }$ at point $P$, with coordinates $\left( w , \frac { 1 } { w ^ { 2 } } \right)$, where $w > 0$. Point $Q$ has coordinates $( w , 0 )$. Line $\ell$ crosses the $x$-axis at point $R$, with coordinates $( k , 0 )$.
(a) Find the value of $k$ when $w = 3$.
(b) For all $w > 0$, find $k$ in terms of $w$.
(c) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of $k$ with respect to time?
(d) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of the area of $\triangle PQR$ with respect to time? Determine whether the area is increasing or decreasing at this instant.