Recall, without justification, the expression of $t^{\prime}$ as a function of $t$, where $t(x) = \tan(x)$.

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Prove that for all $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Using the result $A_k'(X) = A_{k-1}(X-a)$, deduce, for $j$ and $k$ elements of $\llbracket 0, n \rrbracket$, the value of $A_k^{(j)}(ja)$. Distinguish according to whether $j < k$, $j = k$ or $j > k$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show carefully that $\varphi$ is differentiable on $]0,+\infty[$ and compute its derivative on this interval.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$ and $M(Q) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\ln\left|Q(e^{i\theta})\right|d\theta\right)$. Compute the limit of $\varphi'$ at $0^+$ then deduce that: $$M_p(Q)^{1/p} \underset{p \rightarrow 0^+}{\longrightarrow} M(Q).$$

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$ and all $r \in \mathbb{R}$, we have: $$P(z) = \frac{1}{2\pi} \int_0^{2\pi} P\left(z + re^{i\theta}\right) d\theta$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$ and $\varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M )$. Show that $\varphi _ { \alpha }$ is twice differentiable at 0 and that
$$\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right) .$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. With $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right)$, deduce that $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) \geq 0$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. Show that, if $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) > 0$, then there exists $\eta > 0$, such that for all $t \in ] - \eta , \eta [$,
$$\frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \geq \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A ) - \operatorname { Tr } \left( A ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A ) t$$

Show the equality: for all $m \geq 0$ and $\mu \geq 0$, $$\left(\frac{d}{dx}\right)^{\mu} \cdot \left(x^m f\right) = \left(x^m \left(\frac{d}{dx}\right)^{\mu} + \sum_{i=1}^{\min(\mu,m)} \frac{m(m-1)\cdots(m-i+1)\,\mu(\mu-1)\cdots(\mu-i+1)}{i!} x^{m-i} \left(\frac{d}{dx}\right)^{\mu-i}\right) \cdot f.$$

111- If $2^a = \displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{\sqrt{\cos x} - \sqrt{\sin x}}{\cos(x + \frac{\pi}{4})}$, then what is $a$?
(1) $-\dfrac{1}{2}$ (2) $-\dfrac{1}{4}$ (3) $\dfrac{1}{4}$ (4) $\dfrac{1}{2}$

114- What is $\displaystyle\lim_{x \to \pi} \dfrac{\sin(1 + \cos x)}{1 - \cos 2x}$?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) $2$

118- If $f(x) = \dfrac{x^3 - 2}{1 + x^3}$, $g(x) = \sqrt[3]{x-1}$, then $f'(g(x)) \cdot g'(x)$ is equal to which of the following?
(1) $\dfrac{3}{x}$ (2) $\dfrac{3}{x^2}$ (3) $\dfrac{1}{3x}$ (4) $\dfrac{x-3}{x^2}$

111- What is $\displaystyle\lim_{x\to 0}\dfrac{\cos^2 x - \sqrt{\cos x}}{x^2}$?
(1) $-\dfrac{3}{2}$ (2) $-\dfrac{3}{4}$ (3) $-\dfrac{1}{4}$ (4) $\dfrac{3}{2}$
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112. The derivative of the function $y = \cos^2(\tan^{-1}x)$, at $x = 1$, is which of the following?
(1) $-\dfrac{1}{2}$ (2) $-\dfrac{1}{4}$ (3) $\dfrac{1}{4}$ (4) $1$

111- The limit of $\dfrac{\sqrt{\cos 3x} - \sqrt{\cos x}}{x^2}$ as $x \to 0$ is which of the following?
(1) $-2$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{2}$ (4) $2$

112- The derivative of $f(x) = \sin\!\left(\dfrac{\pi}{2} + \tan^{-1}\dfrac{x}{2}\right)$ at the point $x = 2\sqrt{3}$ is which of the following?
(1) $-\dfrac{1}{24}$ (2) $-\dfrac{1}{16}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{1}{4}$

111- If $f(x) = \sqrt{x^2 - |x| + |x|}$, then $\displaystyle\lim_{h \to 0^+} \dfrac{f(1+h)-f(1)}{h}$ is which of the following?
(1) $\dfrac{1}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{3}{2}$ (4) $\dfrac{5}{2}$

115. If $\displaystyle\lim_{x \to 2} \dfrac{f(x) - f(2)}{x - 2} = \dfrac{4}{3}$ and $g(x) = x + \sqrt{x}$, what is $(f \circ g)'(1)$?
(1) $\dfrac{2}{3}$ (2) $\dfrac{3}{2}$ (3) $2$ (4) $3$
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117. For the function with formula $f(x) = (x+2)\sqrt{4x+1}$, the average rate of change of the function on the interval $[0,2]$ is how much greater than the instantaneous rate of change at $x = \dfrac{3}{4}$?
(1) $0.1$ (2) $0.15$ (3) $0.20$ (4) $0.25$

119. If $f$ is a differentiable function, $g(x) = f\!\left(\sqrt{1 + \tan^2 x}\right)$, and $g'\!\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$, what is the value of $f'(2)$?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $1$

120. Suppose $f(x) = \cos^2(2x) + ax^2 + b$, $\displaystyle\lim_{x \to 0^+} \frac{f(x)}{x} = 0$, and $\displaystyle\lim_{x \to 0^-} \frac{f'(x)}{x} = 2$. What is $a + b$?
(1) $8$ (2) $6$ (3) $4$ (4) $-8$
121. The tangent lines to the curve $f(x) = |\sin(2x)| + 1$ at the point $x = 0$ with length 4 are drawn. If $A$ and $B$ are respectively the second and fourth intersection points of the tangent lines with the $x$-axis, what is the length of segment $AB$?
(1) zero (2) $\dfrac{2\sqrt{7}}{3}$ (3) $\dfrac{4\sqrt{7}}{3}$ (4) $2\sqrt{2}$

17 -- If $f(x) = \left(\dfrac{-1+\sin x}{1+\sin x}\right)^2$ and $f(x) = xg(x)+1$, then $\displaystyle\lim_{x \to 0} g(x)$ equals:
(1) $4$ (2) $2$ (3) $-4$ (4) $-2$

$\lim_{x \to 2} \left[\frac{e^{x^{2}} - e^{2x}}{(x-2)e^{2x}}\right]$ equals
(a) 0
(b) 1
(c) 2
(d) 3

The limit $\lim \left[ \left\{ 1 - \cos \left( \sin ^ { 2 } a x \right) \right\} / x \right]$ as $x -> 0$
(a) Equals 1
(b) Equals a
(c) Equals 0
(d) Does not exist