In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
In this part, we denote $t$ the time (in hours) elapsed since the moment the kiln was turned off. The temperature of the kiln (in degrees Celsius) at time $t$ is given by the function $f$ defined, for every positive real number $t$, by: $$f(t) = a\mathrm{e}^{-\frac{t}{5}} + b,$$ where $a$ and $b$ are two real numbers. We admit that $f$ satisfies the following relation: $f'(t) + \frac{1}{5}f(t) = 4$.

1. Determine the values of $a$ and $b$ knowing that initially, the temperature of the kiln is $1000^{\circ}\mathrm{C}$, that is $f(0) = 1000$.
2. For the following, we admit that, for every positive real number $t$: $$f(t) = 980\mathrm{e}^{-\frac{t}{5}} + 20.$$ a. Determine the limit of $f$ as $t$ tends to $+\infty$. b. Study the variations of $f$ on $[0; +\infty[$. Deduce its complete table of variations. c. With this model, after how many minutes can the kiln be opened safely for the ceramics?
3. The average temperature (in degrees Celsius) of the kiln between two times $t_1$ and $t_2$ is given by: $\frac{1}{t_2 - t_1}\int_{t_1}^{t_2} f(t)\,\mathrm{d}t$. a. Using the graphical representation of $f$, give an estimate of the average temperature $\theta$ of the kiln over the first 15 hours of cooling. Explain your approach. b. Calculate the exact value of this average temperature $\theta$ and give its value rounded to the nearest degree Celsius.
4. In this question, we are interested in the temperature drop (in degrees Celsius) of the kiln over the course of one hour, that is between two times $t$ and $(t+1)$. This drop is given by the function $d$ defined, for every positive real number $t$, by: $d(t) = f(t) - f(t+1)$. a. Verify that, for every positive real number $t$: $d(t) = 980\left(1 - \mathrm{e}^{-\frac{1}{5}}\right)\mathrm{e}^{-\frac{t}{5}}$. b. Determine the limit of $d(t)$ as $t$ tends to $+\infty$. What interpretation can be given to this?

For two constants $a , b$ with $1 < a < b$, the $y$-intercept of the line passing through the two points $\left( a , \log _ { 2 } a \right) , \left( b , \log _ { 2 } b \right)$ and the $y$-intercept of the line passing through the two points $\left( a , \log _ { 4 } a \right) , \left( b , \log _ { 4 } b \right)$ are equal.
For the function $f ( x ) = a ^ { b x } + b ^ { a x }$ with $f ( 1 ) = 40$, what is the value of $f ( 2 )$? [4 points]
(1) 760
(2) 800
(3) 840
(4) 880
(5) 920

Given the function $f ( x ) = \log _ { 2 } \left( x ^ { 2 } + a \right)$. If $f ( 3 ) = 1$, then $a = $ \_\_\_\_

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right hand limit
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { ( 1 - x ) ^ { \frac { 1 } { x } } - e ^ { - 1 } } { x ^ { a } }$$
is equal to a nonzero real number, is $\_\_\_\_$

Let the function $f(x) = \begin{cases} \frac{\log_e(1 + 5x) - \log_e(1 + \alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$ be continuous at $x = 0$. Then $\alpha$ is equal to
(1) 10
(2) $-10$
(3) 5
(4) $-5$