Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$. (for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ ) )$} The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$. Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin. (1a) [3 marks] State the equations of all vertical asymptotes of $G _ { f }$. Justify that $G _ { f }$ has the x-axis as a horizontal asymptote. (1b) [5 marks] Determine the monotonicity behavior of $f$ in each of the three subintervals $] - \infty ; - 2 [$, $] - 2 ; 2 [$ and $] 2 ; + \infty [$ of the domain. Also calculate the slope of the tangent to $G _ { f }$ at the point $( 0 \mid f ( 0 ) )$. (for verification: $\left. f ^ { \prime } ( x ) = - \frac { 6 \cdot \left( x ^ { 2 } + 4 \right) } { \left( x ^ { 2 } - 4 \right) ^ { 2 } } \right)$ The points $A ( 3 \mid 3,6 )$ and $B ( 8 \mid 0,8 )$ lie on $G _ { f }$; between these two points $G _ { f }$ runs below the line segment [AB]. (1c) [4 marks] Sketch $G _ { f }$ in the range $- 10 \leq x \leq 10$ using the information obtained so far in a coordinate system. (1d) [5 marks] Calculate the area enclosed by $G _ { f }$ and the line segment $[ A B ]$. Consider the family of functions $f _ { a , b , c } : x \mapsto \frac { a x + b } { x ^ { 2 } + c }$ with $a , b , c \in \mathbb { R }$ and maximal domain $D _ { a , b , c }$. (2a) [1 marks] The function $f$ from Task 1 is a function of this family. State the corresponding values of $a , b$ and $c$. (2b) [2 marks] Justify: If $a = 0$ and $b \neq 0$, then the graph of $f _ { a , b , c }$ is symmetric with respect to the y-axis and does not intersect the x-axis. (2c) [3 marks] State all values for $a , b$ and $c$ such that both $D _ { a , b , c } = \mathbb { R }$ holds and the graph of $f _ { a , b , c }$ is symmetric with respect to the origin, but is not identical to the x-axis. (2d) [4 marks] For the first derivative of $f _ { a , b , c }$ it holds: $f _ { a , b , c } ^ { \prime } ( x ) = - \frac { a x ^ { 2 } + 2 b x - a c } { \left( x ^ { 2 } + c \right) ^ { 2 } }$. Show: If $a \neq 0$ and $c > 0$, then the graph of $f _ { a , b , c }$ has exactly two extreme points. Consider the function $p : x \mapsto \frac { 40 } { ( x - 12 ) ^ { 2 } + 4 }$ defined on $\mathbb { R }$; the figure shows the graph $G _ { p }$ of $p$. [Figure] (3a) [4 marks] Describe how $G _ { p }$ is obtained step by step from the graph of the function $h : x \mapsto \frac { 5 } { x ^ { 2 } + 4 }$ defined on $\mathbb { R }$, and justify thereby that $G _ { p }$ is symmetric with respect to the line with equation $x = 12$. A photovoltaic system installed on a house roof converts light energy into electrical energy. For $4 \leq x \leq 20$, the function $p$ describes the temporal development of the power output of the system on a particular day. Here $x$ is the time elapsed since midnight in hours and $p ( x )$ is the power in kW (kilowatts). (3b) [4 marks] Determine computationally the time in the afternoon to the nearest minute from which the power output of the system is less than $40 \%$ of its daily maximum of 10 kW. (3c) [2 marks] The function $p$ has an inflection point in the interval [4;12]. State the meaning of this inflection point in the context of the problem. (3d) [3 marks] The electrical energy produced by the system is completely fed into the power grid. The homeowner receives a remuneration of 10 cents per kilowatt-hour (kWh) for the electrical energy fed in. The function $x \mapsto E ( x )$ defined on [4;20] gives the electrical energy in kWh that the system feeds into the power grid on the day in question from 4:00 a.m. until $x$ hours after midnight. It holds that $E ^ { \prime } ( x ) = p ( x )$ for $x \in [ 4 ; 20 ]$. Use the figure to determine an approximate value for the remuneration that the homeowner receives for the electrical energy fed into the power grid from 10:00 a.m. to 2:00 p.m.
Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$.\\
(for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ )
)$}
The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$.
Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin.\\
\textbf{(1a)} [3 marks] State the equations of all vertical asymptotes of $G _ { f }$.\\
Justify that $G _ { f }$ has the x-axis as a horizontal asymptote.
\textbf{(1b)} [5 marks] Determine the monotonicity behavior of $f$ in each of the three subintervals $] - \infty ; - 2 [$, $] - 2 ; 2 [$ and $] 2 ; + \infty [$ of the domain. Also calculate the slope of the tangent to $G _ { f }$ at the point $( 0 \mid f ( 0 ) )$.\\
(for verification: $\left. f ^ { \prime } ( x ) = - \frac { 6 \cdot \left( x ^ { 2 } + 4 \right) } { \left( x ^ { 2 } - 4 \right) ^ { 2 } } \right)$
The points $A ( 3 \mid 3,6 )$ and $B ( 8 \mid 0,8 )$ lie on $G _ { f }$; between these two points $G _ { f }$ runs below the line segment [AB].
\textbf{(1c)} [4 marks] Sketch $G _ { f }$ in the range $- 10 \leq x \leq 10$ using the information obtained so far in a coordinate system.
\textbf{(1d)} [5 marks] Calculate the area enclosed by $G _ { f }$ and the line segment $[ A B ]$.
Consider the family of functions $f _ { a , b , c } : x \mapsto \frac { a x + b } { x ^ { 2 } + c }$ with $a , b , c \in \mathbb { R }$ and maximal domain $D _ { a , b , c }$.
\textbf{(2a)} [1 marks] The function $f$ from Task 1 is a function of this family. State the corresponding values of $a , b$ and $c$.
\textbf{(2b)} [2 marks] Justify: If $a = 0$ and $b \neq 0$, then the graph of $f _ { a , b , c }$ is symmetric with respect to the y-axis and does not intersect the x-axis.
\textbf{(2c)} [3 marks] State all values for $a , b$ and $c$ such that both $D _ { a , b , c } = \mathbb { R }$ holds and the graph of $f _ { a , b , c }$ is symmetric with respect to the origin, but is not identical to the x-axis.
\textbf{(2d)} [4 marks] For the first derivative of $f _ { a , b , c }$ it holds: $f _ { a , b , c } ^ { \prime } ( x ) = - \frac { a x ^ { 2 } + 2 b x - a c } { \left( x ^ { 2 } + c \right) ^ { 2 } }$.\\
Show: If $a \neq 0$ and $c > 0$, then the graph of $f _ { a , b , c }$ has exactly two extreme points.
Consider the function $p : x \mapsto \frac { 40 } { ( x - 12 ) ^ { 2 } + 4 }$ defined on $\mathbb { R }$; the figure shows the graph $G _ { p }$ of $p$.\\
\textit{[Figure]}
\textbf{(3a)} [4 marks] Describe how $G _ { p }$ is obtained step by step from the graph of the function $h : x \mapsto \frac { 5 } { x ^ { 2 } + 4 }$ defined on $\mathbb { R }$, and justify thereby that $G _ { p }$ is symmetric with respect to the line with equation $x = 12$.
A photovoltaic system installed on a house roof converts light energy into electrical energy. For $4 \leq x \leq 20$, the function $p$ describes the temporal development of the power output of the system on a particular day. Here $x$ is the time elapsed since midnight in hours and $p ( x )$ is the power in kW (kilowatts).
\textbf{(3b)} [4 marks] Determine computationally the time in the afternoon to the nearest minute from which the power output of the system is less than $40 \%$ of its daily maximum of 10 kW.
\textbf{(3c)} [2 marks] The function $p$ has an inflection point in the interval [4;12]. State the meaning of this inflection point in the context of the problem.
\textbf{(3d)} [3 marks] The electrical energy produced by the system is completely fed into the power grid. The homeowner receives a remuneration of 10 cents per kilowatt-hour (kWh) for the electrical energy fed in.
The function $x \mapsto E ( x )$ defined on [4;20] gives the electrical energy in kWh that the system feeds into the power grid on the day in question from 4:00 a.m. until $x$ hours after midnight.
It holds that $E ^ { \prime } ( x ) = p ( x )$ for $x \in [ 4 ; 20 ]$.\\
Use the figure to determine an approximate value for the remuneration that the homeowner receives for the electrical energy fed into the power grid from 10:00 a.m. to 2:00 p.m.