ap-calculus-ab

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2005
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2004
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2002
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2001
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1998
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-bc_course-and-exam-description 24 -bc_sample-questions 18
2005 free-response

6 maths questions

Q1 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.
Q2 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
The tide removes sand from Sandy Point Beach at a rate modeled by the function $R$, given by $$R ( t ) = 2 + 5 \sin \left( \frac { 4 \pi t } { 25 } \right)$$ A pumping station adds sand to the beach at a rate modeled by the function $S$, given by $$S ( t ) = \frac { 15 t } { 1 + 3 t }$$ Both $R ( t )$ and $S ( t )$ have units of cubic yards per hour and $t$ is measured in hours for $0 \leq t \leq 6$. At time $t = 0$, the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.
(b) Write an expression for $Y ( t )$, the total number of cubic yards of sand on the beach at time $t$.
(c) Find the rate at which the total amount of sand on the beach is changing at time $t = 4$.
(d) For $0 \leq t \leq 6$, at what time $t$ is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
Q3 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( { } ^ { \circ } \mathrm { C } \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.
\begin{tabular}{ c } Distance
$x ( \mathrm {~cm} )$
& 0 & 1 & 5 & 6 & 8 \hline
Temperature
$T ( x ) \left( { } ^ { \circ } \mathrm { C } \right)$
& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.
Q4 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table below, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.
$x$0$0 < x < 1$1$1 < x < 2$2$2 < x < 3$3$3 < x < 4$
$f ( x )$- 1Negative0Positive2Positive0Negative
$f ^ { \prime } ( x )$4Positive0PositiveDNENegative- 3Negative
$f ^ { \prime \prime } ( x )$- 2Negative0PositiveDNENegative0Positive

(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.
Q5 Travel graphs View
A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) \, dt$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, dt$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?
Q6 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = - \frac { 2 x } { y }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 1 ) = - 1$. Write an equation for the line tangent to the graph of $f$ at $( 1 , - 1 )$ and use it to approximate $f ( 1.1 )$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 1 ) = - 1$.