15. If $F ( x ) = \int _ { 0 } ^ { x } \sqrt { t ^ { 3 } + 1 } d t$, then $F ^ { \prime } ( 2 ) =$
(A) - 3
(B) - 2
(C) 2
(D) 3
(E) 18

Let $f$ be a differentiable function, defined for all real numbers $x$, with the following properties. (i) $f ^ { \prime } ( x ) = a x ^ { 2 } + b x$ (ii) $f ^ { \prime } ( 1 ) = 6$ and $f ^ { \prime \prime } ( 1 ) = 18$ (iii) $\int _ { 1 } ^ { 2 } f ( x ) d x = 18$
Find $f ( x )$. Show your work.

$\quad \int \sec x \tan x \, d x =$
(A) $\sec x + C$
(B) $\tan x + C$
(C) $\frac { \sec ^ { 2 } x } { 2 } + C$
(D) $\frac { \tan ^ { 2 } x } { 2 } + C$
(E) $\frac { \sec ^ { 2 } x \tan ^ { 2 } x } { 2 } + C$

The figure above shows the graph of $f$. If $f ( x ) = \int _ { 2 } ^ { x } g ( t ) d t$, which of the following could be the graph of $y = g ( x )$ ?
(A) [graph A]
(B) [graph B]
(C) [graph C]
(D) [graph D]
(E) [graph E]

Let $g$ be a continuously differentiable function with $g ( 1 ) = 6$ and $g ^ { \prime } ( 1 ) = 3$. What is $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } g ( t ) d t } { g ( x ) - 6 }$ ?
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) The limit does not exist.

We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 } { x } ( 1 + \ln x )$$

1. In the three situations below, we have drawn, in an orthonormal coordinate system, the representative curve $\mathscr { C } _ { f }$ of the function $f$ and a curve $\mathscr { C } _ { F }$. In only one situation, the curve $\mathscr { C } _ { F }$ is the representative curve of a primitive $F$ of the function $f$. Which one? Justify the answer.

Consider the function $f$ defined on the interval $[0; +\infty[$ by $f(x) = k\mathrm{e}^{-kx}$ where $k$ is a strictly positive real number. We call $\mathcal{C}_f$ its graph in the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider point A on the curve $\mathcal{C}_f$ with x-coordinate 0 and point B on the curve $\mathcal{C}_f$ with x-coordinate 1. Point C has coordinates $(1; 0)$.

1. Determine an antiderivative of function $f$ on the interval $[0; +\infty[$.
2. Express, as a function of $k$, the area of triangle OCB and that of the region $\mathcal{D}$ bounded by the y-axis, the curve $\mathcal{C}_f$ and the segment $[OB]$.
3. Show that there exists a unique value of the strictly positive real number $k$ such that the area of region $\mathcal{D}$ is twice that of triangle OCB.

Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2-3}$$ One of the antiderivatives $F$ of the function $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 2x\mathrm{e}^{x^2-3}$ b. $F(x) = \left(2x^2+1\right)\mathrm{e}^{x^2-3}$ c. $F(x) = \frac{1}{2}x\mathrm{e}^{x^2-3}$ d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2-3}$

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.

1. We consider the function $f$ defined on $\mathbb{R}$ by: $f(x) = (x + 1)e^x$.
   An antiderivative $F$ of $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 1 + xe^x$ b. $F(x) = (1 + x)e^x$ c. $F(x) = (2 + x)e^x$ d. $F(x) = \left(\frac{x^2}{2} + x\right)e^x$.
   
2. We consider the lines $(d_1)$ and $(d_2)$ whose parametric representations are respectively: $$\left(d_1\right) \left\{\begin{array}{l} x = 2 + r \\ y = 1 + r \\ z = -r \end{array} \quad (r \in \mathbb{R}) \quad \text{and} \quad (d_2) \left\{\begin{array}{rl} x & = 1 - s \\ y & = -1 + s \\ z & = 2 - s \end{array} \quad (s \in \mathbb{R})\right.\right.$$ The lines $(d_1)$ and $(d_2)$ are: a. secant. b. strictly parallel. c. coincident. d. non-coplanar.
   
3. We consider the plane $(P)$ whose Cartesian equation is: $$2x - y + z - 1 = 0$$ We consider the line $(\Delta)$ whose parametric representation is: $$\left\{\begin{array}{l} x = 2 + u \\ y = 4 + u \quad (u \in \mathbb{R}) \\ z = 1 - u \end{array}\right.$$ The line $(\Delta)$ is: a. secant and non-orthogonal to the plane $(P)$. b. included in the plane $(P)$. c. strictly parallel to the plane $(P)$. d. orthogonal to the plane $(P)$.
   
4. We consider the plane $(P_1)$ whose Cartesian equation is $x - 2y + z + 1 = 0$, as well as the plane $(P_2)$ whose Cartesian equation is $2x + y + z - 6 = 0$. The planes $(P_1)$ and $(P_2)$ are: a. secant and perpendicular. b. coincident. c. secant and non-perpendicular. d. strictly parallel.
   
5. We consider the points $E(1; 2; 1)$, $F(2; 4; 3)$ and $G(-2; 2; 5)$.
   We can affirm that the measure $\alpha$ of the angle $\widehat{FEG}$ satisfies: a. $\alpha = 90°$ b. $\alpha > 90°$ c. $\alpha = 0°$ d. $\alpha \approx 71°$

Let $a$ be a strictly positive real number. We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = a\ln(x)$$ We denote $\mathscr{C}$ its representative curve in an orthonormal coordinate system. Let $x_0$ be a real number strictly greater than 1.

1. Determine the abscissa of the point of intersection of the curve $\mathscr{C}$ and the x-axis.
2. Verify that the function $F$ defined by $F(x) = a[x\ln(x) - x]$ is a primitive of the function $f$ on the interval $]0; +\infty[$.
3. Deduce the area of the blue region as a function of $a$ and $x_0$.

We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point $M$ with abscissa $x_0$. We call $A$ the point of intersection of the tangent line $T$ with the y-axis and $B$ the orthogonal projection of $M$ onto the y-axis.

1. Prove that the length AB is equal to a constant (that is, to a number that does not depend on $x_0$) which we will determine. The candidate will take care to make their approach explicit.

We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$ and we call $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.

1. We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$. a. Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$. b. For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that: $$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
2. a. Determine the limit of the function $f$ at 0. b. Interpret this result graphically.
3. a. Determine the limit of the function $f$ at $+ \infty$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$. Draw the variation table of the function $f$ showing the limits at the boundaries of the domain of definition. c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
4. We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$. a. Calculate $I$. b. Interpret the result graphically.
5. We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that: $$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } } .$$ a. By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.

Evaluate $\int_{0}^{\infty} \left(1 + x^{2}\right)^{-(m+1)} dx$, where $m$ is a natural number.

Let $f ( u ) = \tan ^ { - 1 } ( u )$, a function whose domain is the set of all real numbers and whose range is $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Let $g ( v ) = \int _ { 0 } ^ { v } f ( t ) \, d t$.
(a) $f ( 1 ) = \frac { \pi } { 4 }$.
(b) $f ( 1 ) + f ( 2 ) + f ( 3 ) = \pi$.
(c) $g$ is an increasing function on the entire real line.
(d) $g$ is an odd function, i.e., $g ( - x ) = - g ( x )$ for all real $x$.

For the function $F ( x ) = \int _ { 0 } ^ { x } \left( t ^ { 3 } - 1 \right) d t$, what is the value of $F ^ { \prime } ( 2 )$? [3 points]
(1) 11
(2) 9
(3) 7
(4) 5
(5) 3

The derivative $f ^ { \prime } ( x )$ of a polynomial function $f ( x )$ is $f ^ { \prime } ( x ) = 6 x ^ { 2 } + 4$. If the graph of $y = f ( x )$ passes through the point $( 0,6 )$, find the value of $f ( 1 )$. [4 points]

What is the value of $\int _ { 0 } ^ { e } \frac { 5 } { x + e } d x$? [3 points]
(1) $\ln 2$
(2) $2 \ln 2$
(3) $3 \ln 2$
(4) $4 \ln 2$
(5) $5 \ln 2$

For a function $f ( x )$ with $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x$ and $f ( 0 ) = 2$, find the value of $f ( 1 )$. [3 points]

A function $f ( x )$ that is continuous on the set of all real numbers satisfies the following condition. When $n - 1 \leq x < n$, $| f ( x ) | = | 6 ( x - n + 1 ) ( x - n ) |$. (Here, $n$ is a natural number.)
For the function $$g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t - \int _ { x } ^ { 4 } f ( t ) d t$$ defined on the open interval $(0, 4)$, when $g ( x )$ has a minimum value of 0 at $x = 2$, what is the value of $\int _ { \frac { 1 } { 2 } } ^ { 4 } f ( x ) d x$? [4 points]
(1) $- \frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
(5) $\frac { 5 } { 2 }$

For a function $f ( x )$, if $f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x$ and $f ( 0 ) = 3$, what is the value of $f ( 2 )$? [3 points]

A polynomial function $f(x)$ satisfies $$\int_{0}^{x} f(t)\, dt = 3x^{3} + 2x$$ for all real numbers $x$. What is the value of $f(1)$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

For a polynomial function $f(x)$, $f'(x) = 9x^{2} + 4x$ and $f(1) = 6$. What is the value of $f(2)$? [3 points]

We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Let $t \in ]0,\pi[$. Show that the function $$H_t : x \mapsto \frac{1}{1 - 2x\cos(t) + x^2}$$ is expandable as a power series on $]-1,1[$.

We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Using the expansion of $H_t$ as a power series, deduce that $$\forall n \in \mathbb{N},\, \forall t \in ]0,\pi[, \quad V_n(\cos t) = \frac{\sin((n+1)t)}{\sin t}$$

Let $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$. Using the results of question III.C.6), deduce the value of $\int_{-\infty}^{+\infty} f(x) \mathrm{d}x$.

For $n \in \mathbb{N}$, we set $$I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx.$$ Determine by induction $I_{n}$ for all $n \in \mathbb{N}$.