ap-calculus-bc

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6 maths questions

Q1 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
Fish enter a lake at a rate modeled by the function $E$ given by $E ( t ) = 20 + 15 \sin \left( \frac { \pi t } { 6 } \right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. Both $E ( t )$ and $L ( t )$ are measured in fish per hour, and $t$ is measured in hours since midnight $( t = 0 )$.
(a) How many fish enter the lake over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. ($t = 5$)? Explain your reasoning.
Q2 Polar coordinates View
Let $S$ be the region bounded by the graph of the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$, as shown in the figure above.
(a) Find the area of $S$.
(b) What is the average distance from the origin to a point on the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$?
(c) There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.
(d) For $k > 0$, let $A ( k )$ be the area of the portion of region $S$ that is also inside the circle $r = k \cos \theta$. Find $\lim _ { k \rightarrow \infty } A ( k )$.
Q3 Indefinite & Definite Integrals Accumulation Function Analysis View
The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 5$. The figure above shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $( 5, 3 )$. It is known that the point $( 3, 3 - \sqrt { 5 } )$ is on the graph of $f$.
(a) If $\int _ { - 6 } ^ { 5 } f ( x ) \, dx = 7$, find the value of $\int _ { - 6 } ^ { - 2 } f ( x ) \, dx$. Show the work that leads to your answer.
(b) Evaluate $\int _ { 3 } ^ { 5 } \left( 2 f ^ { \prime } ( x ) + 4 \right) dx$.
(c) The function $g$ is given by $g ( x ) = \int _ { - 2 } ^ { x } f ( t ) \, dt$. Find the absolute maximum value of $g$ on the interval $- 2 \leq x \leq 5$. Justify your answer.
(d) Find $\lim _ { x \rightarrow 1 } \frac { 10 ^ { x } - 3 f ^ { \prime } ( x ) } { f ( x ) - \arctan x }$.
Q4 Differential equations Applied Modeling with Differential Equations View
A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\frac { d h } { d t } = - \frac { 1 } { 10 } \sqrt { h }$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
(c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.
Q5 Integration with Partial Fractions View
Consider the family of functions $f ( x ) = \frac { 1 } { x ^ { 2 } - 2 x + k }$, where $k$ is a constant.
(a) Find the value of $k$, for $k > 0$, such that the slope of the line tangent to the graph of $f$ at $x = 0$ equals 6.
(b) For $k = - 8$, find the value of $\int _ { 0 } ^ { 1 } f ( x ) \, dx$.
(c) For $k = 1$, find the value of $\int _ { 0 } ^ { 2 } f ( x ) \, dx$ or show that it diverges.
Q6 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View
A function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown above, along with the line tangent to the graph of $f$ at $x = 0$. Selected derivatives of $f$ at $x = 0$ are given in the table below.
$n$$f ^ { ( n ) } ( 0 )$
23
3$-\frac { 23 } { 2 }$
454

(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$.
(b) Write the first three nonzero terms of the Maclaurin series for $e ^ { x }$. Write the second-degree Taylor polynomial for $e ^ { x } f ( x )$ about $x = 0$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. Use the Taylor polynomial found in part (a) to find an approximation for $h ( 1 )$.
(d) It is known that the Maclaurin series for $h$ converges to $h ( x )$ for all real numbers $x$. It is also known that the individual terms of the series for $h ( 1 )$ alternate in sign and decrease in absolute value to 0. Use the alternating series error bound to show that the approximation found in part (c) differs from $h ( 1 )$ by at most 0.45.