$\int \frac { x } { \sqrt { 1 - 9 x ^ { 2 } } } d x =$
(A) $- \frac { 1 } { 9 } \sqrt { 1 - 9 x ^ { 2 } } + C$
(B) $- \frac { 1 } { 18 } \ln \sqrt { 1 - 9 x ^ { 2 } } + C$
(C) $\frac { 1 } { 3 } \arcsin ( 3 x ) + C$
(D) $\frac { x } { 3 } \arcsin ( 3 x ) + C$

Using the substitution $u = \sqrt { x }$, $\int _ { 1 } ^ { 4 } \frac { e ^ { \sqrt { x } } } { \sqrt { x } } d x$ is equal to which of the following?
(A) $2 \int _ { 1 } ^ { 16 } e ^ { u } d u$
(B) $2 \int _ { 1 } ^ { 4 } e ^ { u } d u$
(C) $2 \int _ { 1 } ^ { 2 } e ^ { u } d u$
(D) $\frac { 1 } { 2 } \int _ { 1 } ^ { 2 } e ^ { u } d u$
(E) $\int _ { 1 } ^ { 4 } e ^ { u } d u$

Let $f$ be a function such that $\int _ { 6 } ^ { 12 } f ( 2 x ) d x = 10$. Which of the following must be true?
(A) $\int _ { 12 } ^ { 24 } f ( t ) d t = 5$
(B) $\int _ { 12 } ^ { 24 } f ( t ) d t = 20$
(C) $\int _ { 6 } ^ { 12 } f ( t ) d t = 5$
(D) $\int _ { 6 } ^ { 12 } f ( t ) d t = 20$
(E) $\int _ { 3 } ^ { 6 } f ( t ) d t = 5$

Let $f$ be a function that is twice differentiable for all real numbers. The table below gives values of $f$ for selected points in the closed interval $2 \leq x \leq 13$.

$x$	2	3	5	8	13
$f(x)$	1	4	$-2$	3	6

(a) Estimate $f'(4)$. Show the work that leads to your answer.
(b) Evaluate $\int_{2}^{13} \left(3 - 5f'(x)\right) dx$. Show the work that leads to your answer.
(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate $\int_{2}^{13} f(x) \, dx$. Show the work that leads to your answer.
(d) Suppose $f'(5) = 3$ and $f''(x) < 0$ for all $x$ in the closed interval $5 \leq x \leq 8$. Use the line tangent to the graph of $f$ at $x = 5$ to show that $f(7) \leq 4$. Use the secant line for the graph of $f$ on $5 \leq x \leq 8$ to show that $f(7) \geq \frac{4}{3}$.

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon $(t = 0)$ and 8 P.M. $(t = 8)$. The number of entries in the box $t$ hours after noon is modeled by a differentiable function $E$ for $0 \leq t \leq 8$. Values of $E(t)$, in hundreds of entries, at various times $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(hours)

& 0 & 2 & 5 & 7 & 8 \hline

$E(t)$

(hundreds of

entries)

& 0 & 4 & 13 & 21 & 23 \hline \end{tabular}
(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time $t = 6$. Show the computations that lead to your answer.
(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$. Using correct units, explain the meaning of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$ in terms of the number of entries.
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function $P$, where $P(t) = t^3 - 30t^2 + 298t - 976$ hundreds of entries per hour for $8 \leq t \leq 12$. According to the model, how many entries had not yet been processed by midnight $(t = 12)$?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.

As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H(t)$ is measured in degrees Celsius. Values of $H(t)$ at selected values of time $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 9 & 10 \hline

$H(t)$

(degrees Celsius)

& 66 & 60 & 52 & 44 & 43 \hline \end{tabular}
(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer.
(b) Using correct units, explain the meaning of $\frac{1}{10}\int_{0}^{10} H(t)\, dt$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac{1}{10}\int_{0}^{10} H(t)\, dt$.
(c) Evaluate $\int_{0}^{10} H'(t)\, dt$. Using correct units, explain the meaning of the expression in the context of this problem.
(d) At time $t = 0$, biscuits with temperature $100^\circ\mathrm{C}$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B'(t) = -13.84e^{-0.173t}$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?

Using the substitution $u = x ^ { 2 } - 3 , \int _ { - 1 } ^ { 4 } x \left( x ^ { 2 } - 3 \right) ^ { 5 } d x$ is equal to which of the following?
(A) $2 \int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(B) $\int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(C) $\frac { 1 } { 2 } \int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(D) $\int _ { - 1 } ^ { 4 } u ^ { 5 } d u$
(E) $\frac { 1 } { 2 } \int _ { - 1 } ^ { 4 } u ^ { 5 } d u$

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.

We define the sequence $\left(u_{n}\right)$ as follows: for every natural integer $n$, $u_{n} = \int_{0}^{1} \frac{x^{n}}{1+x} \mathrm{~d}x$.

1. Calculate $u_{0} = \int_{0}^{1} \frac{1}{1+x} \mathrm{~d}x$.
2. a) Prove that, for every natural integer $n$, $u_{n+1} + u_{n} = \frac{1}{n+1}$. b) Deduce the exact value of $u_{1}$.
3. a) Copy and complete the algorithm below so that it displays as output the term of rank $n$ of the sequence $(u_{n})$ where $n$ is a natural integer entered as input by the user.
   

   Variables :	$i$ and $n$ are natural integers, $u$ is a real number
   Input :	Enter $n$
   Initialization :	Assign to $u$ the value ...
   Processing :	\begin{tabular}{l} For $i$ varying from 1 to... | Assign to $u$ the value . . .
   End For

   \hline & \hline Output : & Display $u$ \hline \end{tabular}
   b) Using this algorithm, the following table of values was obtained:
   

   $n$	0	1	2	3	4	5	10	50	100
   $u_{n}$	0,6931	0,3069	0,1931	0,1402	0,1098	0,0902	0,0475	0,0099	0,0050

   
   What conjectures concerning the behavior of the sequence $(u_{n})$ can be made?
4. a) Prove that the sequence $(u_{n})$ is decreasing. b) Prove that the sequence $(u_{n})$ is convergent.
5. We call $\ell$ the limit of the sequence $(u_{n})$. Prove that $\ell = 0$.

We consider the functions $f(x) = x\mathrm{e}^{1-x^{2}}$ and $g(x) = \mathrm{e}^{1-x}$.

1. Find a primitive $F$ of the function $f$ on $\mathbb{R}$.
2. Deduce the value of $\int_{0}^{1} \left(\mathrm{e}^{1-x} - x\mathrm{e}^{1-x^{2}}\right) \mathrm{d}x$.
3. Give a graphical interpretation of this result.

Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by:
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { 1 - x } }$$

Part A

1. Study the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.
2. Prove that for all real $x$ in the interval $[ 0 ; 1 ] , f ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + \mathrm { e } }$ (recall that $\mathrm { e } = \mathrm { e } ^ { 1 }$ ).
3. Show then that $\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \ln ( 2 ) + 1 - \ln ( 1 + \mathrm { e } )$.

Part B

Let $n$ be a natural number. Consider the functions $f _ { n }$ defined on $[ 0 ; 1 ]$ by:
$$f _ { n } ( x ) = \frac { 1 } { 1 + n \mathrm { e } ^ { 1 - x } }$$
We denote $\mathscr { C } _ { n }$ the representative curve of the function $f _ { n }$ in the plane with an orthonormal coordinate system. Consider the sequence with general term
$$u _ { n } = \int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$

1. The representative curves of the functions $f _ { n }$ for $n$ varying from 1 to 5 are drawn in the appendix. Complete the graph by drawing the curve $\mathscr { C } _ { 0 }$ representative of the function $f _ { 0 }$.
2. Let $n$ be a natural number, interpret graphically $u _ { n }$ and specify the value of $u _ { 0 }$.
3. What conjecture can be made regarding the direction of variation of the sequence $\left( u _ { n } \right)$ ?

Prove this conjecture.
4. Does the sequence ( $u _ { n }$ ) have a limit?

We consider the sequence $\left( I _ { n } \right)$ defined by $I _ { 0 } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x } \mathrm {~d} x$ and for every non-zero natural number $n$
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { n } } { 1 - x } \mathrm {~d} x$$

1. Show that $I _ { 0 } = \ln ( 2 )$.
2. a. Calculate $I _ { 0 } - I _ { 1 }$. b. Deduce $I _ { 1 }$.
3. a. Show that, for every natural number $n , I _ { n } - I _ { n + 1 } = \frac { \left( \frac { 1 } { 2 } \right) ^ { n + 1 } } { n + 1 }$. b. Propose an algorithm to determine, for a given natural number $n$, the value of $I _ { n }$.
4. Let $n$ be a non-zero natural number.

It is admitted that if $x$ belongs to the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$ then $0 \leqslant \frac { x ^ { n } } { 1 - x } \leqslant \frac { 1 } { 2 ^ { n - 1 } }$. a. Show that for every non-zero natural number $n$, $0 \leqslant I _ { n } \leqslant \frac { 1 } { 2 ^ { n } }$. b. Deduce the limit of the sequence ( $I _ { n }$ ) as $n$ tends to $+ \infty$.
5. For every non-zero natural number $n$, we set
$$S _ { n } = \frac { 1 } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 } } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 3 } } { 3 } + \ldots + \frac { \left( \frac { 1 } { 2 } \right) ^ { n } } { n }$$
a. Show that for every non-zero natural number $n$, $S _ { n } = I _ { 0 } - I _ { n }$. b. Determine the limit of $S _ { n }$ as $n$ tends to $+ \infty$.

Define a function $f$ as follows: $f ( 0 ) = 0$ and, for any $x > 0$, $$f ( x ) = \lim _ { L \rightarrow \infty } \int _ { \frac { 1 } { x } } ^ { L } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$$ (or, in simpler notation, the improper integral $\int _ { \frac { 1 } { x } } ^ { \infty } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$).
(i) Show that the definition makes sense for any $x > 0$ by justifying why the limit in the definition exists, i.e., why the improper integral converges.
(ii) Find $f ^ { \prime } \left( \frac { 1 } { \pi } \right)$ if it exists. Clearly indicate the basic result(s) you are using.
(iii) Using the hint or otherwise, find $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( h ) - f ( 0 ) } { h }$, i.e., the right hand derivative of $f$ at $x = 0$. We can take the limit only from the right hand side because $f ( x )$ is undefined for negative values of $x$. Hint: Break $f ( h )$ into two terms by using a standard technique with an appropriate choice. Then separately analyze the resulting two terms in the derivative.

The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$.
$$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$
Find the value of $a ^ { 4 }$. [3 points]

In the figure, let $a$ be the area of region $A$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the $y$-axis, and let $b$ be the area of region $B$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the line $x = 2$. What is the value of $b - a$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $e - 1$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) $e$

What is the value of $\int _ { 1 } ^ { e } \ln \frac { x } { e } \, d x$? [3 points]
(1) $\frac { 1 } { e } - 1$
(2) $2 - e$
(3) $\frac { 1 } { e } - 2$
(4) $1 - e$
(5) $\frac { 1 } { 2 } - e$

When the function $f ( x )$ is $$f ( x ) = \int _ { 0 } ^ { x } \frac { 1 } { 1 + e ^ { - t } } d t$$ what is the value of the real number $a$ that satisfies $( f \circ f ) ( a ) = \ln 5$? [4 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$

For a positive number $t$, the function $f ( x )$ defined on the interval $[ 1 , \infty )$ is $$f ( x ) = \begin{cases} \ln x & ( 1 \leq x < e ) \\ - t + \ln x & ( x \geq e ) \end{cases}$$ Among linear functions $g ( x )$ satisfying the following condition, let $h ( t )$ be the minimum value of the slope of the line $y = g ( x )$.
For all real numbers $x \geq 1$, $( x - e ) \{ g ( x ) - f ( x ) \} \geq 0$.
For a differentiable function $h ( t )$, a positive number $a$ satisfies $h ( a ) = \frac { 1 } { e + 2 }$. What is the value of $h ^ { \prime } \left( \frac { 1 } { 2 e } \right) \times h ^ { \prime } ( a )$? [4 points]
(1) $\frac { 1 } { ( e + 1 ) ^ { 2 } }$
(2) $\frac { 1 } { e ( e + 1 ) }$
(3) $\frac { 1 } { e ^ { 2 } }$
(4) $\frac { 1 } { ( e - 1 ) ( e + 1 ) }$
(5) $\frac { 1 } { e ( e - 1 ) }$

What is the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \sqrt { \frac { 3 n } { 3 n + k } }$? [3 points]
(1) $4 \sqrt { 3 } - 6$
(2) $\sqrt { 3 } - 1$
(3) $5 \sqrt { 3 } - 8$
(4) $2 \sqrt { 3 } - 3$
(5) $3 \sqrt { 3 } - 5$

Two functions $f(x)$ and $g(x)$ are defined and differentiable on the set of all positive real numbers. $g(x)$ is the inverse function of $f(x)$, and $g'(x)$ is continuous on the set of all positive real numbers. For all positive numbers $a$, $$\int_1^a \frac{1}{g'(f(x))f(x)}\,dx = 2\ln a + \ln(a+1) - \ln 2$$ and $f(1) = 8$. Find the value of $f(2)$. [3 points]
(1) 36
(2) 40
(3) 44
(4) 48
(5) 52

What is the value of $\int_{0}^{10} \frac{x+2}{x+1}\, dx$? [3 points]
(1) $10 + \ln 5$
(2) $10 + \ln 7$
(3) $10 + 2\ln 3$
(4) $10 + \ln 11$
(5) $10 + \ln 13$

Let $n \in \mathbb { N }$ and let $f$ be a real function of class $C ^ { \infty }$ on $[ n , + \infty [$. We assume that $f$ and all its derivatives have constant sign on $[ n , + \infty [$ and tend to 0 as $+ \infty$.
By applying, for $k \geqslant n$, the result of III.B.1 to $f _ { k } ( t ) = f ( k + t )$, show $$\sum _ { k = n } ^ { + \infty } f ^ { \prime } ( k ) = - f ( n ) + \frac { 1 } { 2 } f ^ { \prime } ( n ) - \sum _ { j = 1 } ^ { p } a _ { 2 j } f ^ { ( 2 j ) } ( n ) + \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$ where we have set $A _ { j } ^ { * } ( t ) = A _ { j } ( t - [ t ] )$ for every $t \in \mathbb { R }$.
Show that $$\left| \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t \right| \leqslant \left| \frac { a _ { 2 p } } { 2 } \right| \left| f ^ { ( 2 p + 1 ) } ( n ) \right|$$

Show that, in the expression of $R _ { n } ( \alpha )$ from II.B.2, the term $O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$ can be written in the form of an integral.

Let $g$ be a piecewise continuous increasing function on $[ 0,1 ]$.
By noting $\int _ { 0 } ^ { 1 } = \int _ { 0 } ^ { 1 / 2 } + \int _ { 1 / 2 } ^ { 1 }$, show that

• if $n \equiv 1 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \geqslant 0$;
• if $n \equiv 3 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \leqslant 0$.

Using the notation from II.B.2, where $\widetilde { S } _ { n , 2 p - 2 } ( \alpha ) = \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { k ^ { \alpha } } - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right)$, show that for every natural integer $p \geqslant 1$ $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p + 2 } ( \alpha )$$ and that $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p - 2 } ( \alpha )$$ Deduce that the error $\left| S ( \alpha ) - \widetilde { S } _ { n , 2 p } ( \alpha ) \right|$ is bounded by $\left| a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) \right|$.