The function $h : x \mapsto x \cdot \ln \left( x ^ { 2 } \right)$ is given with maximum domain $D _ { h }$. (1a) [2 marks] State $D _ { h }$ and show that for the term of the derivative function $h ^ { \prime }$ of $h$ the following holds: $h ^ { \prime } ( x ) = \ln \left( x ^ { 2 } \right) + 2$.
(1b) [3 marks] Determine the coordinates of the local maximum point of the graph of $h$ located in the second quadrant.
Figure 1 shows the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ of a polynomial function $f$ defined on $\mathbb { R }$. Only at the points $\left( - 4 \mid f ^ { \prime } ( - 4 ) \right)$ and $\left( 5 \mid f ^ { \prime } ( 5 ) \right)$ does the graph $G _ { f ^ { \prime } }$ have horizontal tangents. [Figure]
(2a) [2 marks] Justify that $f$ has exactly one inflection point.
(2b) [2 marks] There are tangents to the graph of $f$ that are parallel to the angle bisector of the first and third quadrants. Using the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ in Figure 1, determine approximate values for the $x$-coordinates of those points where the graph of $f$ has such a tangent.
The functions $f : x \mapsto x ^ { 2 } + 4$ and $g _ { m } : x \mapsto m \cdot x$ with $m \in \mathbb { R }$ are given, both defined on $\mathbb { R }$. The graph of $f$ is denoted by $G _ { f }$ and the graph of $g _ { m }$ by $G _ { m }$. (3a) [3 marks] Sketch $G _ { f }$ in a coordinate system. Calculate the coordinates of the common point of the graphs $G _ { f }$ and $G _ { 4 }$.
(3b) [2 marks] There are values of $m$ for which the graphs $G _ { f }$ and $G _ { m }$ have no common point. State these values of $m$.
The function $g$ is given by $g ( x ) = 0,7 \cdot e ^ { 0,5 x } - 0,7$ with $x \in \mathbb { R }$. The function $g$ is invertible. Figure 2 shows the graph $G _ { g }$ of $g$ and part of the graph $G _ { h }$ of the inverse function $h$ of $g$. [Figure]
(4a) [2 marks] Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$.
Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$.
The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]

Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$.
Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$.
The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]
(1a) [5 marks] Verify by calculation that $G _ { f }$ is symmetric with respect to the $y$-axis, and investigate the behavior of $f$ for $x \rightarrow + \infty$ using the function term. Determine those $x$-values for which $f ( x ) = 0,96$ holds.
(1b) [4 marks] Investigate by calculation the monotonicity behavior of $G _ { f }$. $\left( \right.$ for verification: $\left. f ^ { \prime } ( x ) = \frac { 4 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } \right)$
(1c) [4 marks] Determine by calculation an equation of the tangent $t$ to $G _ { f }$ at the point ( $3 \mid f ( 3 )$ ). Calculate the angle at which $t$ intersects the $x$-axis, and draw $t$ in Figure 1 (Part B).
Now consider the integral function $F : x \mapsto \int _ { 0 } ^ { x } f ( t ) \mathrm { dt }$ defined on $\mathbb { R }$; its graph is denoted by $G _ { F }$. (2a) [5 marks] Justify that $F$ has a zero at $x = 0$, and use the course of $G _ { f }$ to make plausible that there is another zero of $F$ in the interval [ $1 ; 3$ ]. State what special property $G _ { F }$ has at the point $( - 1 \mid F ( - 1 ) )$, and justify your statement.
(2b) [2 marks] The line with equation $y = x - 1$ bounds a triangle together with the coordinate axes. State the area of this triangle and the resulting approximate value for $F ( 1 )$.
(2c) [5 marks] Figure 2 (Part B) shows the graph $G _ { f }$ and the graph $G _ { g }$ of the function $g : x \mapsto - \cos \left( \frac { \pi } { 2 } x \right)$ defined on $\mathbb { R }$. Describe how $G _ { g }$ is obtained from the graph of the function $x \mapsto \cos x$ defined on $\mathbb { R }$, and calculate another approximate value for $F ( 1 )$ by integrating $g$.
[Figure]
Fig. 2 (Part B)
(for verification: $F ( 1 ) \approx - \frac { 2 } { \pi }$ )
(2d) [4 marks] Calculate the arithmetic mean of the two approximate values computed in tasks 2b and 2c. Sketch the graph of $F$ for $0 \leq x \leq 3$ taking into account the previous results in Figure 1 (Part B).
For each value $k > 0$, the points $P _ { k } ( - k \mid f ( - k ) )$ and $Q _ { k } ( k \mid f ( k ) )$ lying on $G _ { f }$ together with the point $R ( 0 \mid 1 )$ determine an isosceles triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.
(3a) [5 marks] Calculate the area of the corresponding triangle $\mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } \mathrm { R }$ for $k = 2$ (see Figure 3). Then show that the area of the triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$ in general can be described by the term $A ( k ) = \frac { 2 k } { k ^ { 2 } + 1 }$. [Figure]
(3b) [6 marks] Show that there is a value of $k > 0$ for which $A ( k )$ is maximal. Calculate this value of $k$ and the area of the corresponding triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.