Given is the function $f : x \mapsto \ln ( x - 3 )$ with maximal domain $D$ and derivative function $f ^ { \prime }$. (1a) [2 marks] State $D$ and the zero of $f$. (1b) [3 marks] Determine the point $x \in D$ for which $f ^ { \prime } ( x ) = 2$ holds.
Given is the function $g : x \mapsto \frac { 1 } { x ^ { 2 } } - 1$ defined in $\mathbb { R } \backslash \{ 0 \}$. (2a) [2 marks] State an equation of the horizontal asymptote of the graph of $g$ and the range of $g$.
(2b) [3 marks] Calculate the value of the integral $\int _ { \frac { 1 } { 2 } } ^ { 2 } g ( x ) \mathrm { dx }$.
A polynomial function $f$ defined in $\mathbb { R }$, which is not linear, with first derivative function $f ^ { \prime }$ and second derivative function $f ^ { \prime \prime }$ has the following properties:

• $f$ has a zero at $x _ { 1 }$.
• It holds that $f ^ { \prime } \left( x _ { 2 } \right) = 0$ and $f ^ { \prime \prime } \left( x _ { 2 } \right) \neq 0$.
• $f ^ { \prime }$ has a local minimum at the point $x _ { 3 }$.

Figure 1 shows the positions of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. [Figure]
(3a) [2 marks] Justify that the degree of $f$ is at least 3.
(3b) [3 marks] Sketch a possible graph of $f$ in Figure 1.
Figure 2 shows the graph of the function $g$ defined in $\mathbb { R }$, whose only extreme points are $( - 1 \mid 1 )$ and $( 0 \mid 0 )$, as well as the point $P$.
[Figure]
Fig. 2
(4a) [2 marks] State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3

State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3
(1a) [2 marks] Calculate the coordinates of the intersection point of $G _ { f }$ with the y-axis and prove by calculation that $G _ { f }$ is symmetric with respect to the y-axis.
(1b) [5 marks] The point $W \left( - 2 \left\lvert \, 2 e ^ { - \frac { 1 } { 2 } } \right. \right)$ is one of the two inflection points of $G _ { f }$. The tangent to $G _ { f }$ at point $W$ is denoted by $w$. Determine an equation of $w$ and calculate the point where $w$ intersects the x-axis. (for verification: $f ^ { \prime } ( x ) = - \frac { 1 } { 2 } x \cdot e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ )
For each value $c \in \mathbb { R } ^ { + }$, consider the rectangle with vertices $P ( - c \mid 0 ) , Q ( c \mid 0 )$, $R ( c \mid f ( c ) )$ and $S$.
(1c) [1 marks] Draw the rectangle PQRS in Figure 3 for $c = 2$.
(1d) [3 marks] Calculate the value of $c$ for which $\overline { \mathrm { QR } } = 1$ holds.
(1e) [3 marks] State the side lengths of rectangle PQRS as a function of $c$ and justify that the area of the rectangle is given by the term $A ( c ) = 4 c \cdot e ^ { - \frac { 1 } { 8 } c ^ { 2 } }$.
(1f) [4 marks] There is a value of $c$ for which the area $A ( c )$ of rectangle PQRS is maximal. Calculate this value of $c$.
For $k \in \mathbb { R }$, consider the functions $f _ { k } : x \mapsto f ( x ) + k$ defined in $] - \infty ; 0 ]$. Thus $f _ { 0 } ( x ) = f ( x )$, where $f _ { 0 }$ and $f$ differ in their domain.
(1g) [4 marks] Justify using the first derivative of $f _ { k }$ that $f _ { k }$ is invertible for every value of $k$. Sketch the graph of the inverse function of $f _ { 0 }$ in Figure 3.
(1h) [2 marks] State all values of $k$ for which the graph of $f _ { k }$ and the graph of the inverse function of $f _ { k }$ have no common point.
[Figure]
Fig. 4
Figure 4 shows a house with a roof dormer, whose front is shown schematically in Figure 5. The front is described by a model as the region enclosed by the graph $G _ { f }$ of the function $f$ from Part B Subtask 1, the x-axis, and the lines with equations $x = - 4$ and $x = 4$. Here, one unit of length in the coordinate system corresponds to one meter in reality.
[Figure]
Fig. 5
(2a) [2 marks] State the width and height of the front of the roof dormer.
In the front of the roof dormer there is a window. In the model, the window corresponds to the region enclosed by the graph of the function $g$ with $g ( x ) = a x ^ { 4 } + b$ and suitable values $a , b \in \mathbb { R }$ with the x-axis (see Figure 5).
(2b) [2 marks] Justify that $a$ is negative and $b$ is positive.
To determine the area of the front of the roof dormer, an antiderivative $F$ of $f$ is considered.
(2c) [2 marks] One of the graphs I, II and III is the graph of $F$. Justify that this is Graph I by giving one reason each for why Graph II and Graph III do not apply. [Figure] [Figure] [Figure]
(2d) [5 marks] Now determine the area of the entire front of the roof dormer (including the window) using the graph of $F$ from Part B Subtask 2c. Describe, incorporating this area, the essential steps of a solution method by which the value of $a$ could be calculated so that with a window height of 1.50 m, the part of the front of the roof dormer shown shaded in Figure 5 has an area of $6 \mathrm {~m} ^ { 2 }$.
(2e) [5 marks] In order to calculate an approximate value for the length of the upper profile line of the front of the roof dormer, $G _ { f }$ in the range $- 4 \leq x \leq 4$ is approximated by four circular arcs that transition seamlessly into one another and are congruent to each other. One of these circular arcs extends in the range $0 \leq x \leq 2$ and is part of the circle with center $M ( 0 \mid - 1 )$ and radius 3. Calculate the central angle of the circular sector corresponding to this circular arc and use it to determine the desired approximate value.