ap-calculus-bc

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6 maths questions

Q1 Connected Rates of Change In/Out Rate Accumulation Problems View
On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.
Q2 Polar coordinates View
The graphs of the polar curves $r = 3$ and $r = 4 - 2 \sin \theta$ are shown in the figure. The curves intersect when $\theta = \frac { \pi } { 6 }$ and $\theta = \frac { 5 \pi } { 6 }$.
(a) Let $S$ be the shaded region that is inside the graph of $r = 3$ and also inside the graph of $r = 4 - 2 \sin \theta$. Find the area of $S$.
(b) A particle moves along the polar curve $r = 4 - 2 \sin \theta$ so that at time $t$ seconds, $\theta = t ^ { 2 }$. Find the time $t$ in the interval $1 \leq t \leq 2$ for which the $x$-coordinate of the particle's position is $-1$.
(c) For the particle described in part (b), find the position vector in terms of $t$. Find the velocity vector at time $t = 1.5$.
Q3 Connected Rates of Change Table-Based Estimation with Rate of Change Interpretation View
Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time $t$, $0 \leq t \leq 6$, is given by a differentiable function $C$, where $t$ is measured in minutes. Selected values of $C ( t )$, measured in ounces, are given in the table below.
\begin{tabular}{ c } $t$
(minutes)
& 0 & 1 & 2 & 3 & 4 & 5 & 6 \hline
$C ( t )$
(ounces)
& 0 & 5.3 & 8.8 & 11.2 & 12.8 & 13.8 & 14.5 \hline \end{tabular}
(a) Use the data in the table to approximate $C ^ { \prime } ( 3.5 )$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t$, $2 \leq t \leq 4$, at which $C ^ { \prime } ( t ) = 2$? Justify your answer.
(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$. Using correct units, explain the meaning of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$ in the context of the problem.
(d) The amount of coffee in the cup, in ounces, is modeled by $B ( t ) = 16 - 16 e ^ { - 0.4 t }$. Using this model, find the rate at which the amount of coffee in the cup is changing when $t = 5$.
Q4 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $0 \leq x \leq 8$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$, $x = 3$, and $x = 5$. The areas of the regions between the graph of $f ^ { \prime }$ and the $x$-axis are labeled in the figure. The function $f$ is defined for all real numbers and satisfies $f ( 8 ) = 4$.
(a) Find all values of $x$ on the open interval $0 < x < 8$ for which the function $f$ has a local minimum. Justify your answer.
(b) Determine the absolute minimum value of $f$ on the closed interval $0 \leq x \leq 8$. Justify your answer.
(c) On what open intervals contained in $0 < x < 8$ is the graph of $f$ both concave down and increasing? Explain your reasoning.
(d) The function $g$ is defined by $g ( x ) = ( f ( x ) ) ^ { 3 }$. If $f ( 3 ) = - \frac { 5 } { 2 }$, find the slope of the line tangent to the graph of $g$ at $x = 3$.
Q5 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
Consider the differential equation $\frac { d y } { d x } = y ^ { 2 } ( 2 x + 2 )$. Let $y = f ( x )$ be the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.
(a) Find $\lim _ { x \rightarrow 0 } \frac { f ( x ) + 1 } { \sin x }$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f \left( \frac { 1 } { 2 } \right)$.
(c) Find $y = f ( x )$, the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.
Q6 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View
A function $f$ has derivatives of all orders at $x = 0$. Let $P _ { n } ( x )$ denote the $n$th-degree Taylor polynomial for $f$ about $x = 0$.
(a) It is known that $f ( 0 ) = - 4$ and that $P _ { 1 } \left( \frac { 1 } { 2 } \right) = - 3$. Show that $f ^ { \prime } ( 0 ) = 2$.
(b) It is known that $f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$ and $f ^ { \prime \prime \prime } ( 0 ) = \frac { 1 } { 3 }$. Find $P _ { 3 } ( x )$.
(c) The function $h$ has first derivative given by $h ^ { \prime } ( x ) = f ( 2 x )$. It is known that $h ( 0 ) = 7$. Find the third-degree Taylor polynomial for $h$ about $x = 0$.