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 $G _ { f }$.\\
\textit{[Figure]}

\textbf{(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.

\textbf{(1b)} [4 marks] Investigate by calculation the monotonicity behavior of $G _ { f }$.\\
$\left( \right.$ for verification: $\left. f ^ { \prime } ( x ) = \frac { 4 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } \right)$

\textbf{(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 $G _ { F }$.\\

\textbf{(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.

\textbf{(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 $F ( 1 )$.

\textbf{(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 $g$.

\begin{figure}[h]
\begin{center}
  \textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 2\\
(Part B)}
\end{center}
\end{figure}

(for verification: $F ( 1 ) \approx - \frac { 2 } { \pi }$ )

\textbf{(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 $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.

\textbf{(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 $A ( k ) = \frac { 2 k } { k ^ { 2 } + 1 }$.\\
\textit{[Figure]}

\textbf{(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 $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.