(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.