Part A
We consider the function $f$ defined on the interval $[0; +\infty[$ by $$f(x) = \sqrt{x+1}.$$ We admit that this function is differentiable on this same interval.

1. Prove that the function $f$ is increasing on the interval $[0; +\infty[$.
2. Prove that for every real number $x$ belonging to the interval $[0; +\infty[$: $$f(x) - x = \frac{-x^2 + x + 1}{\sqrt{x+1} + x}.$$
3. Deduce from this that on the interval $[0; +\infty[$ the equation $f(x) = x$ admits as unique solution: $$\ell = \frac{1+\sqrt{5}}{2}.$$

Part B
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, by $u_{n+1} = f(u_n)$ where $f$ is the function studied in part A. We admit that the sequence with general term $u_n$ is well defined for every natural number $n$.

1. Prove by induction that for every natural number $n$, we have $$1 \leqslant u_{n+1} \leqslant u_n.$$
2. Deduce from this that the sequence $(u_n)$ converges.
3. Prove that the sequence $(u_n)$ converges to $\ell = \frac{1+\sqrt{5}}{2}$.
4. We consider the Python script below: \begin{verbatim} from math import * def seuil(n): u=5 i=0 while abs(u-l)>=10**(-n): u=sqrt(u+1) i=i+1 return(i) \end{verbatim} We recall that the command $\mathbf{abs}(\mathbf{x})$ returns the absolute value of $x$.
   1. [a.] Give the value returned by \texttt{seuil(2)}.
   2. [b.] The value returned by \texttt{seuil(4)} is 9. Interpret this value in the context of the exercise.

We consider $n$ a non-zero natural integer. We consider the function $f_n$ defined on the interval $[0; 1]$ by:
$$f_n(x) = x^n e^{1-x}$$
We admit that the function $f_n$ is differentiable on $[0; 1]$ and we denote $f_n'$ its derivative function.

Part A

In this part we study the case where $n = 1$. We thus study the function $f_1$ defined on $[0; 1]$ by:
$$f_1(x) = x e^{1-x}$$

1. Show that $f_1'(x)$ is strictly positive for all real $x$ in $[0; 1[$.
2. Deduce the table of variations of the function $f_1$ on the interval $[0; 1]$.
3. Deduce that the equation $f_1(x) = 0.1$ admits a unique solution in the interval $[0; 1]$.

Part B

We consider the sequence $(u_n)$ defined for all non-zero natural integers $n$ by
$$u_n = \int_0^1 f_n(x) \, dx \quad \text{that is} \quad u_n = \int_0^1 x^n e^{1-x} \, dx$$
We admit that $u_1 = e - 2$.

1. a. Justify that for all $x \in [0; 1]$ and for all non-zero natural integers $n$, $$0 \leq x^{n+1} \leq x^n$$ b. Deduce that for all non-zero natural integers $n$, $$0 \leq u_{n+1} \leq u_n.$$ c. Show that the sequence $(u_n)$ is convergent.
2. a. Using integration by parts, prove that for all non-zero natural integers $n$ we have: $$u_{n+1} = (n+1)u_n - 1$$ b. Consider the Python script below defining the function suite(): \begin{verbatim} from math import exp def suite(): u = ... for n in range (1, ...): u = ... return \end{verbatim} Copy and complete the Python script above so that the function suite() returns the value of $\int_0^1 x^8 e^{1-x} \, dx$.
3. a. Prove that for all non-zero natural integers $n$ we have: $$u_n \leq \frac{e}{n+1}$$ b. Deduce the limit of the sequence $(u_n)$.

A derivada da função $f(x) = x^3 - 3x^2 + 2x - 1$ é
(A) $f'(x) = 3x^2 - 6x + 2$ (B) $f'(x) = 3x^2 - 3x + 2$ (C) $f'(x) = x^2 - 6x + 2$ (D) $f'(x) = 3x^2 + 6x + 2$ (E) $f'(x) = 3x^2 - 6x - 2$

O limite $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$ é igual a
(A) 0 (B) 1 (C) 2 (D) 4 (E) indefinido

QUESTION 164
The derivative of $f(x) = x^3 - 4x^2 + 5x - 2$ is
(A) $f'(x) = 3x^2 - 8x + 5$
(B) $f'(x) = 3x^2 - 4x + 5$
(C) $f'(x) = 3x^2 + 8x + 5$
(D) $f'(x) = x^2 - 8x + 5$
(E) $f'(x) = 3x^2 - 8x - 5$

The derivative of $f(x) = x^3 - 3x^2 + 2x$ at $x = 1$ is:
(A) $-2$
(B) $-1$
(C) $0$
(D) $1$
(E) $2$

A function $f$ is defined by $f ( x ) = e ^ { x }$ if $x < 1$ and $f ( x ) = \log _ { e } ( x ) + a x ^ { 2 } + b x$ if $x \geq 1$. Here $a$ and $b$ are unknown real numbers. Can $f$ be differentiable at $x = 1$ ?
(A) $f$ is not differentiable at $x = 1$ for any $a$ and $b$.
(B) There exist unique numbers $a$ and $b$ for which $f$ is differentiable at $x = 1$.
(C) $f$ is differentiable at $x = 1$ whenever $a + b = e$.
(D) $f$ is differentiable at $x = 1$ regardless of the values of $a$ and $b$.

Throughout this question every mentioned function is required to be a differentiable function from $\mathbb { R }$ to $\mathbb { R }$. The symbol $\circ$ denotes composition of functions.
(a) Suppose $f \circ f = f$. Then for each $x$, one must have $f ^ { \prime } ( x ) = $ \_\_\_\_ or $f ^ { \prime } ( f ( x ) ) = $ \_\_\_\_. Complete the sentence and justify.
(b) For a non-constant $f$ satisfying $f \circ f = f$, it is known and you may assume that the range of $f$ must have one of the following forms: $\mathbb { R } , ( - \infty , b ] , [ a , \infty )$ or $[ a , b ]$. Show that in fact the range must be all of $\mathbb { R }$ and deduce that there is a unique such function $f$. (Possible hints: For each $y$ in the range of $f$, what can you say about $f ( y )$? If the range has a maximum element $b$ what can you say about the derivative of $f$?)
(c) Suppose that $g \circ g \circ g = g$ and that $g \circ g$ is a non-constant function. Show that $g$ must be onto, $g$ must be strictly increasing or strictly decreasing and that there is a unique such increasing $g$.

Two real numbers $a$ and $b$ satisfy $\lim _ { x \rightarrow 2 } \frac { \sqrt { x ^ { 2 } + a } - b } { x - 2 } = \frac { 2 } { 5 }$. Find the value of $a + b$. [3 points]

For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]

The value of $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x + 3 } - 2 }$ is? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

When the function $f ( x )$ is $$f ( x ) = \begin{cases} 1 - x & ( x < 0 ) \\ x ^ { 2 } - 1 & ( 0 \leqq x < 1 ) \\ \frac { 2 } { 3 } \left( x ^ { 3 } - 1 \right) & ( x \geqq 1 ) \end{cases}$$ which of the following statements in are correct? [3 points]
Remarks ㄱ. $f ( x )$ is differentiable at $x = 1$. ㄴ. $| f ( x ) |$ is differentiable at $x = 0$. ㄷ. The minimum natural number $k$ such that $x ^ { k } f ( x )$ is differentiable at $x = 0$ is 2.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

$\lim _ { x \rightarrow a } \frac { 2 ^ { x } - 1 } { 3 \sin ( x - a ) } = b \ln 2$ is satisfied by two constants $a , b$. What is the value of $a + b$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$

A polynomial function $f ( x )$ and two natural numbers $m , n$ satisfy $$\begin{array} { l l } \lim _ { x \rightarrow \infty } \frac { f ( x ) } { x ^ { m } } = 1 , & \lim _ { x \rightarrow \infty } \frac { f ^ { \prime } ( x ) } { x ^ { m - 1 } } = a \\ \lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { n } } = b , & \lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x ^ { n - 1 } } = 9 \end{array}$$ Which of the following are correct? Select all that apply from . (where $a , b$ are real numbers.) [4 points]
ㄱ. $m \geqq n$ ㄴ. $a b \geqq 9$ ㄷ. If $f ( x )$ is a cubic function, then $a m = b n$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a polynomial function $f ( x )$, if $\lim _ { x \rightarrow 2 } \frac { f ( x + 1 ) - 8 } { x ^ { 2 } - 4 } = 5$, find the value of $f ( 3 ) + f ^ { \prime } ( 3 )$. [3 points]

(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$

For two constants $a , b$, if $\lim _ { x \rightarrow 3 } \frac { \sqrt { x + a } - b } { x - 3 } = \frac { 1 } { 4 }$, what is the value of $a + b$? [2 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

For the function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + x - 2 \right)$, find the value of $f ^ { \prime } ( 2 )$. [3 points]

On the coordinate plane, for a natural number $n$, let $\mathrm { A } _ { n }$ be the point where the two lines $y = \frac { 1 } { n } x$ and $x = n$ meet, and let $\mathrm { B } _ { n }$ be the point where the line $x = n$ and the $x$-axis meet. Let $\mathrm { C } _ { n }$ be the center of the circle inscribed in triangle $\mathrm { A } _ { n } \mathrm { OB } _ { n }$, and let $S _ { n }$ be the area of triangle $\mathrm { A } _ { n } \mathrm { OC } _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } } { n }$? [4 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

For the function $f ( x ) = 2 x \cos x$ with domain $\{ x \mid 0 \leq x \leq \pi \}$, which of the following are correct? Choose all that apply from . [4 points]
Remarks ㄱ. If $f ^ { \prime } ( a ) = 0$, then $\tan a = \frac { 1 } { a }$. ㄴ. There exists $a$ in the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 3 } \right)$ where the function $f ( x )$ has a local maximum value at $x = a$. ㄷ. On the interval $\left[ 0 , \frac { \pi } { 2 } \right]$, the number of distinct real roots of the equation $f ( x ) = 1$ is 2.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { \sin 3 x }$? [2 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$

For the function $f ( x ) = x ^ { 3 } + 7 x + 3$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For a function $f ( x )$ differentiable on the set of all real numbers, let the function $g ( x )$ be defined as $$g ( x ) = \frac { f ( x ) } { e ^ { x - 2 } }$$ If $\lim _ { x \rightarrow 2 } \frac { f ( x ) - 3 } { x - 2 } = 5$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f ( x ) = 2 x ^ { 3 } + x + 1$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 8$, find the value of $f ^ { \prime } ( 2 )$. [3 points]