$f ( x ) = \frac { 2 x } { \sqrt { x ^ { 2 } + x + 1 } }$
a) $x ^ { 2 } + x + 1 > 0$
$x ^ { 2 } + x + ( 1 / 2 ) ^ { 2 } + 1 > 0$
$( x + 1 / 2 ) ^ { 2 } + 1 - 1 / 4 > 0$
$( x + 1 / 2 ) ^ { 2 } + 3 / 4 > 0$ for oll real $x$
Domain: All real $x$
b) see graph
c) $\lim _ { x \rightarrow \infty } \frac { 2 x } { \sqrt { x ^ { 2 } + x + 1 } } = \lim _ { x \rightarrow \infty } \frac { 2 x } { \sqrt { \frac { x } { x ^ { 2 } } + x + 1 } } = \lim _ { x \rightarrow \infty } \frac { 2 } { \sqrt { 1 + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } } } = \frac { 2 } { \sqrt { 1 + 0 + 0 } } = 2$
$\lim _ { x \rightarrow - \infty } \frac { 2 x } { \sqrt { x } ^ { 2 } + x + 1 } = \lim _ { x \rightarrow - \infty } \frac { 2 } { \sqrt { 1 + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } } } = - \frac { 2 } { \sqrt { 1 + 0 + 0 } } = - 2$
$y = 2 , y = - 2$
d) $f ^ { \prime } ( x ) = \frac { x + 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 3 / 2 } }$
$f ^ { \prime } ( x ) = 0$
$x + 2 = 0$
$x = - 2$
$f ( - 2 ) = - \frac { 4 } { \sqrt { 3 } } = - \frac { 4 \sqrt { 3 } } { 3 }$
Range: $\{ 4 : - 4 \sqrt { 3 } / 3 \leq 4 < 2 \}$

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by: $$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$ where $a, b$ and $c$ are fixed real numbers.
The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.

1. Give the values of $u(1)$ and $u(4)$.
2. Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
3. Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

The English physiologist Archibald Vivian Hill proposed, in his studies, that the velocity $v$ of contraction of a muscle when subjected to a weight $p$ is given by the equation $( p + a )( v + b ) = K$, with $a$, $b$, and $K$ constants.
A physiotherapist, with the intention of maximizing the beneficial effect of the exercises he would recommend to one of his patients, wanted to study this equation and classified it as follows:

Type of curve

Oblique half-line

Horizontal half-line

Branch of parabola

Arc of circle

Branch of hyperbola

The physiotherapist analyzed the dependence between $v$ and $p$ in Hill's equation and classified it according to its geometric representation in the Cartesian plane, using the coordinate pair ($p$; $v$). Assume that $K > 0$.
The graph of the equation that the physiotherapist used to maximize the effect of the exercises is of the type
(A) oblique half-line.
(B) horizontal half-line.
(C) branch of parabola.
(D) arc of circle.
(E) branch of hyperbola.

Let $$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$ where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones. a) How many vertical asymptotes does the graph of $f$ have?
Options: $n$ less than $n$ more than $n$ impossible to decide
Answer: $\_\_\_\_$ b) What can you deduce about the value of $n$?
Options: $n < 4$ $n = 4$ $n > 4$ impossible to decide
Answer: $\_\_\_\_$ c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?
Options: $x = 0$ $x = 1$ $x = n - 1$ $x = n$
Answer: $\_\_\_\_$ d) How many inflection points does the graph of $f$ have in the region $x < 0$?
Options: none 1 more than 1 impossible to decide
(Hint: Sketching is better than calculating.)
Answer: $\_\_\_\_$

Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions $$f(x) = \frac{x^{3}}{x^{2}-x} \qquad g(x) = \frac{x^{2}-x}{x^{3}} \qquad h(x) = \frac{x^{3}-x}{x^{3}+x}$$ (i) Does $f$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does $g$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(iii) Does $h$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

(1a) [2 marks] Given is the function $f : x \mapsto \frac { x ^ { 2 } + 2 x } { x + 1 }$ with maximal domain $D _ { f }$. State $D _ { f }$ and the zeros of $f$.
(1b) [3 marks] Give a term of a rational function $h$ that has the following properties: The function $h$ is defined on $\mathbb { R }$; its graph has the line with equation $y = 3$ as a horizontal asymptote and intersects the y-axis at the point ( $0 \mid 4$ ).
Given is the function $g : x \mapsto \frac { 4 } { x }$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph of $g$.
[Figure]
Fig. 1
(2a) [2 marks] Calculate the value of the integral $\int _ { 1 } ^ { e } g ( x ) \mathrm { dx }$
(2b) [3 marks] Determine graphically the point $x _ { 0 } \in \mathbb { R } ^ { + }$ for which the following holds: The local rate of change of $g$ at the point $x _ { 0 }$ equals the average rate of change of $g$ on the interval $[ 1 ; 4 ]$.
The graph $G _ { f }$ of the polynomial function $f$ defined on $\mathbb { R }$ has a horizontal tangent only at the point $x = 3$ (see Figure 2). Consider the function $g$ defined on $\mathbb { R }$ with $g ( x ) = f ( f ( x ) )$.
[Figure]
Fig. 2
(3a) [2 marks] Using Figure 2, state the function values $f ( 6 )$ and $g ( 6 )$.
(3b) [3 marks] According to the chain rule, $g ^ { \prime } ( x ) = f ^ { \prime } ( f ( x ) ) \cdot f ^ { \prime } ( x )$. Use this and Figure 2 to determine all points where the graph of $g$ has a horizontal tangent.
Given are the functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = a \cdot e ^ { - x } + 3$ and $a \in \mathbb { R } \backslash \{ 0 \}$. (4a) [1 marks] Show that $f _ { a } ^ { \prime } ( 0 ) = - a$ holds.
)$} Consider the tangent to the graph of $f _ { a }$ at the point $\left( 0 \mid f _ { a } ( 0 ) \right)$. Determine those values of $a$ for which this tangent has a positive slope and also intersects the x-axis at a point whose x-coordinate is greater than $\frac { 1 } { 2 }$.
Given is the function $f : x \mapsto 2 \cdot \sqrt { 10 x - x ^ { 2 } }$ defined on $[ 0 ; 10 ]$. The graph of $f$ is denoted by $G _ { f }$. Subtask Part B a (2 marks) Determine the zeros of $f$. (for verification: 0 and 10)
The graph $G _ { f }$ has a horizontal tangent at exactly one point. Determine the coordinates of this point and justify that it is a maximum point. (for verification: $f ^ { \prime } ( x ) = \frac { 10 - 2 x } { \sqrt { 10 x - x ^ { 2 } } } ;$ y-coordinate of the maximum point: 10 )
The graph $G _ { f }$ is concave down. One of the following terms is a term of the second derivative function $f ^ { \prime \prime }$ of $f$. Determine whether this is Term I or Term II without calculating a term of $f ^ { \prime \prime }$.
I $\quad f ^ { \prime \prime } ( x ) = \frac { 50 } { \left( x ^ { 2 } - 10 x \right) \cdot \sqrt { 10 x - x ^ { 2 } } } \quad$ II $\quad f ^ { \prime \prime } ( x ) = \frac { 50 } { \left( 10 x - x ^ { 2 } \right) \cdot \sqrt { 10 x - x ^ { 2 } } }$
Show that for $0 \leq x \leq 5$ the equation $f ( 5 - x ) = f ( 5 + x )$ is satisfied by appropriately transforming the terms $f ( 5 - x )$ and $f ( 5 + x )$. Use this to justify that the graph $G _ { f }$ is symmetric with respect to the line with equation $x = 5$.
State the maximal domain of the term $f ^ { \prime } ( x ) = \frac { 10 - 2 x } { \sqrt { 10 x - x ^ { 2 } } }$. Determine $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x )$ and interpret the result geometrically.
State $f ( 8 )$ and sketch $G _ { f }$ in a coordinate system taking into account the previous results.
Consider the tangent to $G _ { f }$ at the point $( 2 \mid f ( 2 ) )$. Calculate the angle at which this tangent intersects the x-axis.
Of the vertices of rectangle ABCD, the point $A ( s \mid 0 )$ with $s \in ] 0 ; 5 [$ and the point $B$ lie on the x-axis, the points $C$ and $D$ lie on $G _ { f }$. The rectangle thus has the line with equation $x = 5$ as its axis of symmetry. Show that the diagonals of this rectangle each have length 10.
A water storage tank has the shape of a right cylinder and is filled with water up to a level of 10 m above the tank bottom. If a hole is drilled in the wall of the water storage tank below the water level, water immediately flows out after the hole is completed, hitting the ground at a certain distance from the tank wall. This distance is called the spray distance in the following (see Figure). The dependence of the spray distance on the height of the hole is modeled by the function $f$ considered in the previous subtasks. Here $x$ is the height of the hole above the tank bottom in meters and $f ( x )$ is the spray distance in meters. [Figure]
Subtask Part B i (1 mark) The graph $G _ { f }$ passes through the point $( 3,6 \mid 9,6 )$. State the meaning of this statement in the context of the problem.
Subtask Part B j (5 marks) Calculate the heights at which the hole can be drilled so that the spray distance is 6 m. Also state the height at which the hole must be drilled so that the spray distance is maximal.
Now consider a specific hole in the water storage tank. As water flows out, the volume of water in the tank decreases as a function of time. The function $g : t \mapsto 0,25 t - 25$ with $0 \leq t \leq 100$ describes the temporal development of this volume change. Here $t$ is the time elapsed since the hole was completed in seconds and $g ( t )$ is the instantaneous rate of change of the water volume in the tank in liters per second. Calculate the volume of water in liters that flows out of the container during the first minute after the hole is completed.

122- The figure opposite shows the graph of $f(x) = \dfrac{x^3 + ax^2}{x^2 + bx + c}$. The value of $(bc - a)$ is which of the following?
[Figure: Graph of a rational function with asymptotes, showing a curve with a local feature near the origin]
(1) $-2$ (2) $-1$ (3) $1$ (4) $2$

116. The extensions of the asymptotes of the graph of the function $f(x) = \sqrt{x^2 + 2x} - \sqrt{x^2 - 2x}$, the first and third bisectors intersect at two points A and B. What is the length of AB?
(1) $2\sqrt{7}$ (2) $4$ [6pt] (3) $2\sqrt{5}$ (4) $4\sqrt{7}$

116. The oblique asymptote of the curve $y = \sqrt[2]{8x^2 + 2x^2}$ intersects the $y$-axis at which point?
(4) $\dfrac{5}{6}$ (3) $\dfrac{2}{3}$ (2) $\dfrac{1}{2}$ (1) $\dfrac{1}{6}$

122- The figure opposite shows part of the graph of $f(x) = \dfrac{x^2 + ax + b}{x + c}$. What is $b$?
[Figure: Graph of a rational function with vertical asymptote and oblique asymptote; marked points at $x = -2$ and $x = 7$ on the x-axis]
(1) $1$
(2) $4$
(3) $6$
(4) $9$

115- The $x$-intercept of the oblique asymptote of $y = x\sqrt{\dfrac{3x-3}{x-1}}$ is which of the following?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{3}{4}$

118. The graph of $f(x) = \dfrac{-2x^2 + 3x}{ax^2 + bx + c}$ has asymptotes $y = -1$, $y = -2$, $x = -2$, and $x = 1$. What is $f(-1)$?
(1) $1.25$ (2) $1.5$ (3) $1.75$ (4) $-1.5$

124-- The locus of the intersection of the asymptotes of the hyperbola $y=\dfrac{ax+3}{(a+1)x+(a-1)}$ is $y=\dfrac{3}{2}x^{2}+x+\dfrac{5}{6}$. The graph of this hyperbola intersects the $x$-axis at which length?
(1) $3$ (2) $-3$ (3) $\dfrac{3}{2}$ (4) $-\dfrac{3}{2}$

Given a real parameter $a$, with $a \neq 0$, consider the function $f_{a}$ defined as follows:
$$f_{a}(x) = \frac{x^{2} - ax}{x^{2} - a}$$
whose graph will be denoted by $\Omega_{a}$.
a) As the parameter $a$ varies, determine the domain of $f_{a}$, study any discontinuities and write the equations of all its asymptotes.
b) Show that, for $a \neq 1$, all graphs $\Omega_{a}$ intersect their horizontal asymptote at the same point and share the same tangent line at the origin.
c) As $a < 1$ varies, identify the intervals of monotonicity of the function $f_{a}$. Study the function $f_{-1}(x)$ and sketch its graph $\Omega_{-1}$.
d) Determine the area of the bounded region between the graph $\Omega_{-1}$, the line tangent to it at the origin and the line $x = \sqrt{3}$.

Given $f ( x ) = \frac { \ln ( x ) } { x }$, where ln denotes the natural logarithm, defined for $\mathrm { x } > 0$, find: a) ( 0.5 points) Calculate, if it exists, a horizontal asymptote of the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$. b) (1 point) Find a point on the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ where the tangent line to the curve is horizontal and analyze whether this point is a relative extremum. c) (1 point) Calculate the area of the bounded region limited by the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ and the lines $\boldsymbol { y } = \mathbf { 0 }$ and $\boldsymbol { x } = \boldsymbol { e }$.

$$f ( x ) = \frac { - k x ^ { 3 } + k ^ { 2 } x } { k ^ { 3 } x ^ { 2 } + x - ( k + 1 ) }$$
The function has a vertical asymptote at $x = 1$. Accordingly, what is the value of $f ( 2 )$?
A) - 5
B) - 4
C) - 3
D) - 2
E) - 1

The function
$$f ( x ) = \frac { a x } { | b x + 2 | }$$
defined on a subset of the set of positive real numbers has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Accordingly, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

Let a be a real number, and $$f ( x ) = \ln ( 2 x + 8 )$$ The vertical asymptote of the function $$g ( x ) = \frac { \sin x } { x ^ { 2 } + a x }$$ is also a vertical asymptote of the function.\ Accordingly, what is a?\ A) 0\ B) 1\ C) 2\ D) 3\ E) 4