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 $t$.

Let $f$ be a continuous function defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.
(b) Let $P$ be the function defined by $P ( x ) = G ( x ) \cdot f ( x )$. Find $P ^ { \prime } ( 3 )$.
(c) Find $\lim _ { x \rightarrow 2 } \frac { G ( x ) } { x ^ { 2 } - 2 x }$.
(d) Find the average rate of change of $G$ on the interval $[ - 4,2 ]$. Does the Mean Value Theorem guarantee a value $c , - 4 < c < 2$, for which $G ^ { \prime } ( c )$ is equal to this average rate of change? Justify your answer.

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A ( t ) = 450 \sqrt { \sin ( 0.62 t ) }$, where $t$ is the number of hours after 5 A.M. and $A ( t )$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.
(a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. $( t = 1 )$ to 10 A.M. $( t = 5 )$.
(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. $( t = 1 )$ to 10 A.M. $( t = 5 )$.
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ( $t = 1$ ) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever $A ( t ) \geq 400$. The number of vehicles in line at time $t$, for $a \leq t \leq 4$, is given by $N ( t ) = \int _ { a } ^ { t } ( A ( x ) - 400 ) d x$, where $a$ is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval $a \leq t \leq 4$. Justify your answer.

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r ( t )$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r ^ { \prime } ( t )$, the rate of change of the radius, over the time interval $0 \leq t \leq 12$.

\begin{tabular}{ c } $t$

(days)

& 0 & 3 & 7 & 10 & 12 \hline

$r ^ { \prime } ( t )$

(centimeters per day)

& - 6.1 & - 5.0 & - 4.4 & - 3.8 & - 3.5 \hline \end{tabular}
(a) Approximate $r ^ { \prime \prime } ( 8.5 )$ using the average rate of change of $r ^ { \prime }$ over the interval $7 \leq t \leq 10$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t , 0 \leq t \leq 3$, for which $r ^ { \prime } ( t ) = - 6$ ? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int _ { 0 } ^ { 12 } r ^ { \prime } ( t ) d t$.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time $t = 3$ days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function $f$, where $f(t)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. Selected values of $f(t)$ are given in the table.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 60 & 90 & 120 & 135 & 150 \hline

$f ( t )$

(gallons per second)

& 0 & 0.1 & 0.15 & 0.1 & 0.05 & 0 \hline \end{tabular}
(a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\, dt$ in the context of the problem. Use a right Riemann sum with the three subintervals $[60,90]$, $[90, 120]$, and $[120, 135]$ to approximate the value of $\int_{60}^{135} f(t)\, dt$.
(b) Must there exist a value of $c$, for $60 < c < 120$, such that $f'(c) = 0$? Justify your answer.
(c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t) = \left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^{2}\right)$ for $0 \leq t \leq 150$. Using this model, find the average rate of flow of gasoline over the time interval $0 \leq t \leq 150$. Show the setup for your calculations.
(d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem.

The temperature of coffee in a cup at time $t$ minutes is modeled by a decreasing differentiable function $C$, where $C(t)$ is measured in degrees Celsius. For $0 \leq t \leq 12$, selected values of $C(t)$ are given in the table shown.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 3 & 7 & 12 \hline

$C(t)$

(degrees Celsius)

& 100 & 85 & 69 & 55 \hline \end{tabular}
(a) Approximate $C'(5)$ using the average rate of change of $C$ over the interval $3 \leq t \leq 7$. Show the work that leads to your answer and include units of measure.
(b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of $\int_{0}^{12} C(t)\, dt$. Interpret the meaning of $\frac{1}{12} \int_{0}^{12} C(t)\, dt$ in the context of the problem.
(c) For $12 \leq t \leq 20$, the rate of change of the temperature of the coffee is modeled by $C'(t) = \frac{-24.55 e^{0.01t}}{t}$, where $C'(t)$ is measured in degrees Celsius per minute. Find the temperature of the coffee at time $t = 20$. Show the setup for your calculations.
(d) For the model defined in part (c), it can be shown that $C''(t) = \frac{0.2455 e^{0.01t}(100 - t)}{t^2}$. For $12 < t < 20$, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x = -6$, and the $x$- and $y$-axes. Region $R$ has area 12.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.

An invasive species of plant appears in a fruit grove at time $t = 0$ and begins to spread. The function $C$ defined by $C ( t ) = 7.6 \arctan ( 0.2 t )$ models the number of acres in the fruit grove affected by the species $t$ weeks after the species appears. It can be shown that $C ^ { \prime } ( t ) = \frac { 38 } { 25 + t ^ { 2 } }$.
(Note: Your calculator should be in radian mode.)
A. Find the average number of acres affected by the invasive species from time $t = 0$ to time $t = 4$ weeks. Show the setup for your calculations.
B. Find the time $t$ when the instantaneous rate of change of $C$ equals the average rate of change of $C$ over the time interval $0 \leq t \leq 4$. Show the setup for your calculations.
C. Assume that the invasive species continues to spread according to the given model for all times $t > 0$. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.
D. At time $t = 4$ weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function $A$, defined by $A ( t ) = C ( t ) - \int _ { 4 } ^ { t } 0.1 \cdot \ln ( x ) d x$, models the number of acres affected by the species over the time interval $4 \leq t \leq 36$. At what time $t$, for $4 \leq t \leq 36$, does $A$ attain its maximum value? Justify your answer.

A student starts reading a book at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the student reads is modeled by the differentiable function $R$, where $R ( t )$ is measured in words per minute. Selected values of $R ( t )$ are given in the table shown.

$t$ (minutes)	0	2	8	10
$R ( t )$ (words per minute)	90	100	150	162

A. Approximate $R ^ { \prime } ( 1 )$ using the average rate of change of $R$ over the interval $0 \leq t \leq 2$. Show the work that leads to your answer. Indicate units of measure.
B. Must there be a value $c$, for $0 < c < 10$, such that $R ( c ) = 155$ ? Justify your answer.
C. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 10 } R ( t ) d t$. Show the work that leads to your answer.
D. A teacher also starts reading at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function $W$ defined by $W ( t ) = - \frac { 3 } { 10 } t ^ { 2 } + 8 t + 100$, where $W ( t )$ is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.
Let $g$ be the function defined by $g ( x ) = \int _ { 6 } ^ { x } f ( t ) d t$.
A. Find $g ^ { \prime } ( 8 )$. Give a reason for your answer.
B. Find all values of $x$ in the open interval $- 6 < x < 12$ at which the graph of $g$ has a point of inflection. Give a reason for your answer.
C. Find $g ( 12 )$ and $g ( 0 )$. Label your answers.
D. Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $- 6 \leq x \leq 12$. Justify your answer.

Consider the function $f$ defined on $[0; +\infty[$ by $$f(x) = \frac{\ln(x + 3)}{x + 3}$$

1. Show that $f$ is differentiable on $[0; +\infty[$. Study the sign of its derivative function $f'$, its possible limit at $+\infty$, and draw up the table of its variations.
2. We define the sequence $(u_n)_{n \geqslant 0}$ by its general term $u_n = \int_n^{n+1} f(x)\,\mathrm{d}x$. a. Justify that, if $n \leqslant x \leqslant n+1$, then $f(n+1) \leqslant f(x) \leqslant f(n)$. b. Show, without attempting to calculate $u_n$, that, for every natural integer $n$, $$f(n+1) \leqslant u_n \leqslant f(n).$$ c. Deduce that the sequence $(u_n)$ is convergent and determine its limit.
3. Let $F$ be the function defined on $[0; +\infty[$ by $$F(x) = [\ln(x+3)]^2.$$ a. Justify the differentiability on $[0; +\infty[$ of the function $F$ and determine, for every positive real $x$, the number $F'(x)$. b. We set, for every natural integer $n$, $I_n = \int_0^n f(x)\,\mathrm{d}x$. Calculate $I_n$.
4. We set, for every natural integer $n$, $S_n = u_0 + u_1 + \cdots + u_{n-1}$. Calculate $S_n$. Is the sequence $(S_n)$ convergent?

4. We define the number $I = \int _ { 0 } ^ { 1 } f _ { 1 } ( x ) \mathrm { d } x$.
Show that $I = \ln \left( \frac { 1 + \mathrm { e } } { 2 } \right)$. Give a graphical interpretation of $I$.

Part B

In this part, we choose $k = - 1$ and we wish to sketch the curve $\mathscr { C } _ { - 1 }$ representing the function $f _ { - 1 }$. For all real $x$, we call $P$ the point on $\mathscr { C } _ { 1 }$ with abscissa $x$ and $M$ the point on $\mathscr { C } _ { - 1 }$ with abscissa $x$. We denote by $K$ the midpoint of segment [ $M P$ ].

1. Show that, for all real $x , f _ { 1 } ( x ) + f _ { - 1 } ( x ) = 1$.
2. Deduce that point $K$ belongs to the line with equation $y = \frac { 1 } { 2 }$.
3. Sketch the curve $\mathscr { C } _ { - 1 }$ on the APPENDIX, to be returned with your answer sheet.
4. Deduce the area, in square units, of the region bounded by the curves $\mathscr { C } _ { 1 } , \mathscr { C } _ { - 1 }$, the $y$-axis and the line with equation $x = 1$.

Part C

In this part, we do not privilege any particular value of the parameter $k$. For each of the following statements, say whether it is true or false and justify your answer.

1. Whatever the value of the real number $k$, the graph of the function $f _ { k }$ is strictly between the lines with equations $y = 0$ and $y = 1$.
2. Whatever the value of the real $k$, the function $f _ { k }$ is strictly increasing.
3. For all real $u _ { n }$ & 4502 & 13378 & 39878 & 119122 & 356342 & 1066978 & 3196838 & 9582322 & 28730582 \hline \end{tabular}
   b. What conjecture can be made concerning the monotonicity of the sequence $\left( u _ { n } \right)$ ?

Let $a$ be a real number between 0 and 1. We denote by $f _ { a }$ the function defined on $\mathbf { R }$ by:
$$f _ { a } ( x ) = a \mathrm { e } ^ { a x } + a .$$
We denote by $I ( a )$ the integral of the function $f _ { a }$ between 0 and 1:
$$I ( a ) = \int _ { 0 } ^ { 1 } f _ { a } ( x ) \mathrm { d } x$$

1. In this question, we set $a = 0$. Determine $I ( 0 )$.
2. In this question, we set $a = 1$. We therefore study the function $f _ { 1 }$ defined on $\mathbf { R }$ by: $$f _ { 1 } ( x ) = \mathrm { e } ^ { x } + 1$$ a. Without detailed study, sketch the graph of the function $f _ { 1 }$ on your paper in an orthogonal coordinate system and show the number $I ( 1 )$. b. Calculate the exact value of $I ( 1 )$, then round to the nearest tenth.
3. Does there exist a value of $a$ for which $I ( a )$ equals 2? If so, give an interval of width $10 ^ { - 2 }$ containing this value.

The two parts of this exercise are independent.

Part A

Let the function $f$ defined on the set of real numbers by
$$f ( x ) = 2 \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C }$ its representative curve in an orthonormal coordinate system. We admit that, for all $x$ belonging to $[ 0 ; \ln ( 2 ) ] , f ( x )$ is positive. Indicate whether the following proposition is true or false by justifying your answer.
Proposition A: The area of the region bounded by the lines with equations $x = 0$ and $x = \ln ( 2 )$, the $x$-axis and the curve $\mathscr { C }$ is equal to 1 unit of area.

Part B

Let $n$ be a strictly positive integer. Let the function $f _ { n }$ defined on the set of real numbers by
$$f _ { n } ( x ) = 2 n \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C } _ { n }$ its representative curve in an orthonormal coordinate system. We admit that $f _ { n }$ is differentiable and that $\mathscr { C } _ { n }$ admits a horizontal tangent at a unique point $S _ { n }$. Indicate whether the following proposition is true or false by justifying your answer.
Proposition B: For all strictly positive integer $n$, the ordinate of the point $S _ { n }$ is $n ^ { 2 }$.

The purpose of the problem is the study of the integrals $I$ and $J$ defined by:
$$I = \int_{0}^{1} \frac{1}{1 + x} \mathrm{~d}x \quad \text{and} \quad J = \int_{0}^{1} \frac{1}{1 + x^{2}} \mathrm{~d}x$$
Part A: exact value of the integral $I$

1. Give a geometric interpretation of the integral $I$.
2. Calculate the exact value of $I$.

Part B: estimation of the value of $J$
Let $g$ be the function defined on the interval $[0; 1]$ by $g(x) = \frac{1}{1 + x^{2}}$. We denote $\mathscr{C}_{g}$ its representative curve in an orthonormal frame of the plane. We therefore have: $J = \int_{0}^{1} g(x) \mathrm{d}x$. The purpose of this part is to evaluate the integral $J$ using the probabilistic method described below. We choose at random a point $\mathrm{M}(x; y)$ by drawing independently its coordinates $x$ and $y$ at random according to the uniform distribution on $[0; 1]$. We admit that the probability $p$ that a point drawn in this manner is located below the curve $\mathscr{C}_{g}$ is equal to the integral $J$. In practice, we initialize a counter $c$ to 0, we fix a natural number $n$ and we repeat $n$ times the following process:

• we choose at random and independently two numbers $x$ and $y$, according to the uniform distribution on $[0; 1]$;
• if $\mathrm{M}(x; y)$ is below the curve $\mathscr{C}_{g}$ we increment the counter $c$ by 1.

We admit that $f = \frac{c}{n}$ is an approximate value of $J$. This is the principle of the so-called Monte-Carlo method.

The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood. a. Let $F$ be the function defined on the interval $[0;11]$ by: $$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$ Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$. b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.

Below is the graphical representation $\mathscr { C } _ { g }$ in an orthogonal coordinate system of a function $g$ defined and continuous on $\mathbb { R }$. The curve $\mathscr { C } _ { g }$ is symmetric with respect to the $y$-axis and lies in the half-plane $y > 0$.
For all $t \in \mathbb { R }$ we define: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$

Part A

The justifications of the answers to the following questions may be based on graphical considerations.

1. Is the function $G$ increasing on $[ 0 ; + \infty [$ ? Justify.
2. Justify graphically the inequality $G ( 1 ) \leqslant 0.9$.
3. Is the function $G$ positive on $\mathbb { R }$ ? Justify.

In the rest of the problem, the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$.

Part B

1. Study of $g$ a. Determine the limits of the function $g$ at the boundaries of its domain. b. Calculate the derivative of $g$ and deduce the table of variations of $g$ on $\mathbb { R }$. c. Specify the maximum of $g$ on $\mathbb { R }$. Deduce that $g ( 1 ) \leqslant 1$.
2. We denote $E$ the set of points $M$ located between the curve $\mathscr { C } _ { g }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$. We call $I$ the area of this set. We recall that: $$I = G ( 1 ) = \int _ { 0 } ^ { 1 } g ( u ) \mathrm { d } u$$ We wish to estimate the area $I$ by the method called ``Monte-Carlo'' described below.
   • We choose a point $M ( x ; y )$ by randomly drawing its coordinates $x$ and $y$ independently according to the uniform distribution on the interval $[ 0 ; 1 ]$. It is admitted that the probability that the point $M$ belongs to the set $E$ is equal to $I$.
   • We repeat $n$ times the experiment of choosing a point $M$ at random. We count the number $c$ of points belonging to the set $E$ among the $n$ points obtained.
   • The frequency $f = \frac { c } { n }$ is an estimate of the value of $I$. a. The figure below illustrates the method presented for $n = 100$. Determine the value of $f$ corresponding to this graph. b. The execution of the algorithm below uses the Monte-Carlo method described previously to determine a value of the number $f$. Copy and complete this algorithm. $f , x$ and $y$ are real numbers, $n , c$ and $i$ are natural integers. ALEA is a function that randomly generates a number between 0 and 1. \begin{verbatim} $c \leftarrow 0$ For $i$ varying from 1 to $n$ do: $x \leftarrow$ ALEA $y \leftarrow$ ALEA If $y \leqslant \ldots$ then $c \leftarrow \ldots$ end If end For $f \leftarrow \ldots$ \end{verbatim} c. An execution of the algorithm for $n = 1000$ gives $f = 0.757$. Deduce a confidence interval, at the 95\% confidence level, for the exact value of $I$.

Part C

We recall that the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$ and that the function $G$ is defined on $\mathbb { R }$ by: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$ We propose to determine an upper bound for $G ( t )$ for $t \geqslant 1$.

1. A preliminary result. It is admitted that, for all real $u \geqslant 1$, we have $g ( u ) \leqslant \frac { 1 } { u ^ { 2 } }$. Deduce that, for all real $t \geqslant 1$, we have: $$\int _ { 1 } ^ { t } g ( u ) \mathrm { d } u \leqslant 1 - \frac { 1 } { t }$$
2. Show that, for all real $t \geqslant 1$, $$G ( t ) \leqslant 2 - \frac { 1 } { t }$$ What can we say about the possible limit of $G ( t )$ as $t$ tends to $+ \infty$ ?

Consider a function $h$ defined and continuous on $\mathbb{R}$ whose variation table is given below:

$x$	$-\infty$	1	$+\infty$
Variations of $h$		0
	$-\infty$

We denote $H$ the antiderivative of $h$ defined on $\mathbb{R}$ which vanishes at 0. It satisfies the property: a. $H$ is positive on $]-\infty; 0]$. b. $H$ is increasing on $]-\infty; 1]$. c. $H$ is negative on $]-\infty; 1]$. d. $H$ is increasing on $\mathbb{R}$.

For every natural number $n$, we consider the following integrals:
$$I_n = \int_0^{\pi} \mathrm{e}^{-nx} \sin(x) \mathrm{d}x, \quad J_n = \int_0^{\pi} \mathrm{e}^{-nx} \cos(x) \mathrm{d}x$$

1. Calculate $I_0$.
2. a. Justify that, for every natural number $n$, we have $I_n \geqslant 0$. b. Show that, for every natural number $n$, we have $I_{n+1} - I_n \leqslant 0$. c. Deduce from the two previous questions that the sequence $(I_n)$ converges.
3. a. Show that, for every natural number $n$, we have: $$I_n \leqslant \int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x$$ b. Show that, for every natural number $n \geqslant 1$, we have: $$\int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x = \frac{1 - \mathrm{e}^{-n\pi}}{n}.$$ c. Deduce from the two previous questions the limit of the sequence $(I_n)$.
4. a. By integrating by parts the integral $I_n$ in two different ways, establish the two following relations, for every natural number $n \geqslant 1$: $$I_n = 1 + \mathrm{e}^{-n\pi} - nJ_n \quad \text{and} \quad I_n = \frac{1}{n}J_n$$ b. Deduce that, for every natural number $n \geqslant 1$, we have $$I_n = \frac{1 + \mathrm{e}^{-n\pi}}{n^2 + 1}$$
5. It is desired to obtain the rank $n$ from which the sequence $(I_n)$ becomes less than $0.1$. Copy and complete the fifth line of the Python script below with the appropriate command. \begin{verbatim} from math import * def seuil() : n = 0 I = 2 ... n=n+1 I =(1+exp(-n*pi))/(n*n+1) return n \end{verbatim}

Question 150
Um recipiente cilíndrico tem raio da base igual a 10 cm e altura igual a 20 cm. Ele está completamente cheio de água. Toda a água desse recipiente é transferida para outro recipiente cilíndrico de raio da base igual a 20 cm. A altura da água no segundo recipiente, em cm, é
(A) 2 (B) 4 (C) 5 (D) 8 (E) 10

A integral $\int_0^2 (3x^2 + 2x)\,dx$ é igual a
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16

QUESTION 159
The value of $\int_0^2 (3x^2 + 2x)\, dx$ is
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

The value of $\displaystyle\int_0^2 (3x^2 + 2x)\,dx$ is:
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

Prove that $\int _ { 1 } ^ { b } a ^ { \log _ { b } x } d x > \ln b$ where $a , b > 0 , b \neq 1$.

Calculate the following integrals whenever possible. If a given integral does not exist, state so. Note that $[ x ]$ denotes the integer part of $x$, i.e., the unique integer $n$ such that $n \leq x < n + 1$. a) $\int _ { 1 } ^ { 4 } x ^ { 2 } d x$
Answer: $\_\_\_\_$ b) $\int _ { 1 } ^ { 3 } [ x ] ^ { 2 } d x$
Answer: $\_\_\_\_$ c) $\int _ { 1 } ^ { 2 } \left[ x ^ { 2 } \right] d x$
Answer: $\_\_\_\_$ d) $\int _ { - 1 } ^ { 1 } \frac { 1 } { x ^ { 2 } } d x$
Answer: $\_\_\_\_$