The velocity of a particle moving along a straight line is given by $v ( t ) = 1.3 t \ln ( 0.2 t + 0.4 )$ for time $t \geq 0$. What is the acceleration of the particle at time $t = 1.2$?
(A) - 0.580
(B) - 0.548
(C) - 0.093
(D) 0.660

If $f ( x ) = 7 x - 3 + \ln x$, then $f ^ { \prime } ( 1 ) =$
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

For every natural number $n$, the function $f_{n}$ is defined for all real $x$ in the interval $[0; 1]$ by:
$$f_{n}(x) = x + \mathrm{e}^{n(x-1)}$$
Let $\mathscr{C}_{n}$ denote the graph of the function $f_{n}$ in an orthogonal coordinate system.

Part A: generalities on the functions $\boldsymbol{f}_{\boldsymbol{n}}$

1. Prove that, for every natural number $n$, the function $f_{n}$ is increasing and positive on the interval $[0; 1]$.
2. Show that the curves $\mathscr{C}_{n}$ all have a common point A, and specify its coordinates.
3. Using the graphical representations, can one conjecture the behavior of the slopes of the tangent lines at A to the curves $\mathscr{C}_{n}$ for large values of $n$? Prove this conjecture.

Part B: evolution of $\boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})$ when $x$ is fixed

Let $x$ be a fixed real number in the interval $[0; 1]$. For every natural number $n$, we set $u_{n} = f_{n}(x)$.

1. In this question, assume that $x = 1$. Study the possible limit of the sequence $(u_{n})$.
2. In this question, assume that $0 \leq x < 1$. Study the possible limit of the sequence $(u_{n})$.

Part C: area under the curves $\mathscr{C}_{\boldsymbol{n}}$

For every natural number $n$, let $A_{n}$ denote the area, expressed in square units, of the region located between the $x$-axis, the curve $\mathscr{C}_{n}$ and the lines with equations $x = 0$ and $x = 1$ respectively. Based on the graphical representations, conjecture the limit of the sequence $(A_{n})$ as the integer $n$ tends to $+\infty$, then prove this conjecture.

For the function $f ( x ) = x \ln x + 13 x$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = 5 e ^ { 3 x - 3 }$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = \cos x + 4 e ^ { 2 x }$, find the value of $f ^ { \prime } ( 0 )$. [3 points]

For the function $f ( x ) = 4 \sin 7 x$, find the value of $f ^ { \prime } ( 2 \pi )$. [3 points]

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

For a real number $t$, let $f(t)$ denote the slope of the line passing through the origin and tangent to the curve $y = \frac{1}{e^x} + e^t$. For the constant $a$ satisfying $f(a) = -e\sqrt{e}$, find the value of $f'(a)$. [3 points]
(1) $-\frac{1}{3}e\sqrt{e}$
(2) $-\frac{1}{2}e\sqrt{e}$
(3) $-\frac{2}{3}e\sqrt{e}$
(4) $-\frac{5}{6}e\sqrt{e}$
(5) $-e\sqrt{e}$

Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Then which one of the following statements is correct?
(A) $f$ is not continuous at $x = 0$.
(B) $f$ is continuous but not differentiable at $x = 0$.
(C) $f$ is differentiable at $x = 0$ and $f'(0) = -\frac{1}{2}$.
(D) $f$ is differentiable at $x = 0$ and $f'(0) = \frac{1}{2}$.

For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

If $y = \left( \frac { x } { x + 1 } \right) ^ { x } + x ^ { \left( \frac { x } { x + 1 } \right) }$, find $\frac { d y } { d x }$ at $x = 1$.
(1) $\frac { 1 } { 2 } + \ln 2$
(2) $1 + \frac { 1 } { 2 } \ln 2$
(3) $1 - \frac { 1 } { 2 } \ln 2$
(4) $\frac { 1 } { 2 } - \ln 2$

If $y(\alpha) = \sqrt { 2 \left( \frac { \tan \alpha + \cot \alpha } { 1 + \tan ^ { 2 } \alpha } \right) + \frac { 1 } { \sin ^ { 2 } \alpha } }$, $\alpha \in \left( \frac { 3 \pi } { 4 } , \pi \right)$, then $\frac { d y } { d \alpha }$ at $\alpha = \frac { 5 \pi } { 6 }$ is
(1) 4
(2) $\frac { 4 } { 3 }$
(3) $-4$
(4) $- \frac { 1 } { 4 }$

The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x = \frac{1}{2}$ is:
(1) $\frac{2\sqrt{3}}{5}$
(2) $\frac{\sqrt{3}}{12}$
(3) $\frac{2\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{10}$

If $y ( x ) = \cot ^ { - 1 } \left( \frac { \sqrt { 1 + \sin x } + \sqrt { 1 - \sin x } } { \sqrt { 1 + \sin x } - \sqrt { 1 - \sin x } } \right) , x \in \left( \frac { \pi } { 2 } , \pi \right)$, then $\frac { d y } { d x }$ at $x = \frac { 5 \pi } { 6 }$ is: (1) 0 (2) - 1 (3) $\frac { - 1 } { 2 }$ (4) $\frac { 1 } { 2 }$

The value of $\log _ { e } 2 \cdot \frac { \mathrm { d } } { \mathrm { d } x } \log _ { \cos x } \operatorname { cosec } x$ at $x = \frac { \pi } { 4 }$ is
(1) $- 2 \sqrt { 2 }$
(2) $2 \sqrt { 2 }$
(3) $-4$
(4) $4$

If $a = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 2 n } { n ^ { 2 } + k ^ { 2 } }$ and $f ( x ) = \sqrt { \frac { 1 - \cos x } { 1 + \cos x } } , x \in ( 0,1 )$, then:
(1) $2 \sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(2) $f \left( \frac { a } { 2 } \right) f ^ { \prime } \left( \frac { a } { 2 } \right) = \sqrt { 2 }$
(3) $\sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(4) $f \left( \frac { a } { 2 } \right) = \sqrt { 2 } f ^ { \prime } \left( \frac { a } { 2 } \right)$

$$f(x) = \ln\left(\sin^{2} x + e^{2x}\right)$$
Given this, what is $f'(0)$?
A) $e$
B) $1$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $2$