Exercise 1 - Part A

Here are two curves $\mathcal { C } _ { 1 }$ and $\mathcal { C } _ { 2 }$ which give for two people $P _ { 1 }$ and $P _ { 2 }$ of different body compositions the concentration $C$ of alcohol in the blood (blood alcohol level) as a function of time $t$ after ingestion of the same quantity of alcohol. The instant $t = 0$ corresponds to the moment when the two individuals ingest the alcohol. $C$ is expressed in grams per litre and $t$ in hours.

1. The function $C$ is defined on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $C ^ { \prime }$ its derivative function. At an instant $t$ positive or zero, the rate of appearance of alcohol in the blood is given by $C ^ { \prime } ( t )$. At what instant is this rate maximal? It is often said that a person of weak body composition experiences the effects of alcohol more quickly.
2. On the previous graph, identify the curve corresponding to the person with the largest body composition. Justify the choice made.
3. A person on an empty stomach ingests alcohol. It is admitted that the concentration $C$ of alcohol in their blood can be modelled by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = A t \mathrm { e } ^ { - t }$$ where $A$ is a positive constant that depends on the body composition and the quantity of alcohol ingested. a. We denote $f ^ { \prime }$ the derivative function of the function $f$. Determine $f ^ { \prime } ( 0 )$. b. Is the following statement true? ``For equal quantities of alcohol ingested, the larger $A$ is, the more corpulent the person is.''

Part B - A particular case

Paul, a 19-year-old student of average body composition and a young driver, drinks two glasses of rum. The concentration $C$ of alcohol in his blood is modelled as a function of time $t$, expressed in hours, by the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( t ) = 2 t \mathrm { e } ^ { - t } .$$

1. Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
2. At what instant is the concentration of alcohol in Paul's blood maximal? What is its value then? Round to $10 ^ { - 2 }$ near.
3. Recall the limit of $\frac { \mathrm { e } ^ { t } } { t }$ as $t$ tends to $+ \infty$ and deduce from it that of $f ( t )$ at $+ \infty$. Interpret the result in the context of the exercise.
4. Paul wants to know after how much time he can take his car. We recall that the legislation allows a maximum concentration of alcohol in the blood of $0,2 \mathrm {~g} . \mathrm { L } ^ { - 1 }$ for a young driver. a. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 0,2$. b. What minimum duration must Paul wait before he can take the wheel in full compliance with the law? Give the result rounded to the nearest minute.
5. The minimum concentration of alcohol detectable in the blood is estimated at $5 \times 10 ^ { - 3 }$ g.L${}^{ - 1 }$. a. Justify that there exists an instant $T$ from which the concentration of alcohol in the blood is no longer detectable. b. The following algorithm is given where $f$ is the function defined by $f ( t ) = 2 t \mathrm { e } ^ { - t }$.
   

   Initialization:	$t$ takes the value 3,5
   	$p$ takes the value 0,25
   	$C$ takes the value 0,21
   Processing:	While $C > 5 \times 10 ^ { - 3 }$ do:
   	$\quad \mid \quad t$ takes the value $t + p$
   	$\quad C$ takes the value $f ( t )$
   Output:	End While
   Display $t$

   
   Copy and complete the following table of values by executing this algorithm. Round the values to $10 ^ { - 2 }$ near.
   

   	Initialization	Step 1	Step 2
   $p$	0,25
   $t$	3,5
   $C$	0,21

   
   What does the value displayed by this algorithm represent?

Part A

In the orthonormal coordinate system above, the representative curves of a function $f$ and its derivative function, denoted $f ^ { \prime }$, are drawn, both defined on $] 3 ; + \infty [$.

1. Associate each curve with the function it represents. Justify.
2. Determine graphically the possible solution(s) of the equation $f ( x ) = 3$.
3. Indicate, by graphical reading, the convexity of the function $f$.

Part B

1. Justify that the quantity $\ln \left( x ^ { 2 } - x - 6 \right)$ is well defined for values $x$ in the interval ]3; $+ \infty$ [, which we will call $I$ in the following.
2. We admit that the function $f$ from Part A is defined by $f ( x ) = \ln \left( x ^ { 2 } - x - 6 \right)$ on $I$. Calculate the limits of the function $f$ at the two endpoints of the interval $I$. Deduce an equation of an asymptote to the representative curve of the function $f$ on $I$.
3. a. Calculate $f ^ { \prime } ( x )$ for all $x$ belonging to $I$. b. Study the direction of variation of the function $f$ on $I$.

Draw the variation table of the function $f$ showing the limits at the endpoints of $I$.
4. a. Justify that the equation $f ( x ) = 3$ admits a unique solution $\alpha$ on the interval ]5; 6[. b. Determine, using a calculator, an approximation of $\alpha$ to within $10 ^ { - 2 }$.
5. a. Justify that $f ^ { \prime \prime } ( x ) = \frac { - 2 x ^ { 2 } + 2 x - 13 } { \left( x ^ { 2 } - x - 6 \right) ^ { 2 } }$. b. Study the convexity of the function $f$ on $I$.

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as shown in the graphs.