From 6765fde56f3977431db0bfa83e7bd0db9669800c Mon Sep 17 00:00:00 2001 From: Anna Date: Mon, 28 May 2018 23:01:11 +0200 Subject: [PATCH 1/5] KINf-IPKult-E WS 17/18 --- KInf-IPKult-E/Capture1.PNG | Bin 0 -> 166458 bytes KInf-IPKult-E/Capture2.PNG | Bin 0 -> 96970 bytes KInf-IPKult-E/KInf-IPKult-E.tex | 90 ++++++++++++++++++++++++++++++++ 3 files changed, 90 insertions(+) create mode 100644 KInf-IPKult-E/Capture1.PNG create mode 100644 KInf-IPKult-E/Capture2.PNG create mode 100644 KInf-IPKult-E/KInf-IPKult-E.tex diff --git a/KInf-IPKult-E/Capture1.PNG b/KInf-IPKult-E/Capture1.PNG new file mode 100644 index 0000000000000000000000000000000000000000..87383ce31350e7a48f82cc8e623c130fff266d1b GIT binary patch literal 166458 zcmWh!WmuD68(+c50fK}ys30IMF~%6(j8a5W0RbgPjvUeggOQ?i{3!)#L1Gpgpp+=3 zgn)F7?$`IjdCob{^?W$jxz2Onzqn7LsfjKl-8DJ@0KllPr)3TRfVlsSztB+sd!8pt z_y6mFe&)IwfZ73`m46e6tGcl|08pPwfA)~--<;M*&)N?FfPMOZ1CqQeodJN$a(yjz z%V7JM^MDyClJ7>;>WyE1SE40)WubW)N+OBTai8(|pYhhzQ7$raa&n%dkN=)adtAAG 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Schmid, Prof. Dr. C. Schlieder, Prof. Dr. T. Gross, Prof. Dr. A. Henrich} + {Wintersemester 17/18} + {90} + {-} + \section{Aufgabenteil 1 - Prof. Schlieder} + \begin{enumerate} + \item Aufgabe 1 (KInf: Geoinformationsverarbeitung; 10P) + \begin{enumerate} + \item Schreibe die Definition von Georeferenzierung auf. Erkläre den Unterschied zwischen direkter und indirekter Georeferenzierung (2P) + \item Bestimme die kürzeste Route von Aurach (A) nach Gerolzhofen (G) durch Anwendung des Djikstra-Algorithmus und gebe die Länge der Strecke sowie alle Zwischenschritte an (8P) + \image{0.5}{A1.png}{Wandertour}{Wandertour} + + \end{enumerate} + \item Aufgabe 2 (KInf: Digitale Bibliotheken; 10P) + \begin{enumerate} + \item Das GroupLens-Empfehlungsverfahren prädiziert die Bewertung r(p,i), die eine Person p für ein Item i abgeben wird. Dabei bezeichnet $\bar{r}(p)$ die über alle Items bestimmte durchschnittliche von p abgegebene Bewertung und sim(p,q) das Ähnlichkeitsmaß, welches angibt, wie ähnlich zwei Personen p und q bewerten:\\ $r(p,i)=\bar{r}(p)+c(p,i)$\\\\ $c(p,i)=\frac{\sum_{q \in P tex\ (p)}(r(q,i)-\bar{r}(q))*sim(p,q)}{\sum_{q \in P tex\ (p))}|sim(p,q)|} $\\ + Erkläre, warum die Bestimmung von r(p,i) auf $\bar{r}$(p,i) zurückgreift. In welchem Wertebereich liegen die Werte für sim(p,q)? Welche Bedeutung haben die beiden Extremwerte? (3P) + \item Vier Websites sind wie abgebildet durch Hyperlinks verbundem (siehe Graph). Stelle die linearen Gleichungen auf, die den PageRank von A, B, C und D durch Lösen der Gleichungen. Verwende einen Dämpfungsfaktor von d=0.5 (7P) + \item Schreibe die Definition von Georeferenzierung auf. Erkläre den Unterschied zwischen direkter und indirekter Georeferenzierung (2P) + \image{0.5}{A2.png}{Hyperlinks}{Hyperlinks} + \end{enumerate} + \newpage + \section{Aufgabenteil 2 - Prof. Henrich} + +\item Aufgabe 3 (Suchen, Suchmaschinen; 10P) + \begin{enumerate} + \item Schreibe die Definition von Suchmaschinenoptimierung. In Suchmaschinenoptimierung sollte man drei Tipps (Aufgabenbereiche) adressieren, welche sind dies? (5P) + \item (Auf Basis d. Abbildung:) Wie kann man das rote Fragezeichen in einer Enterprise Search Lösung ersetzen. \\ + Anders ausgedrückt: Welche grundlegenden Ansätze gibt es um eine unternehmensweite Suchlösung zu realisieren. (5P) + \image{0.5}{A3.png}{Abbildung}{Abbildung} + \end{enumerate} + \item Aufgabe 4 (Museum Digital \& Visualisierung; 10P) + \begin{enumerate} + \item In einer Ausstellung soll ein wertvolles historisches Buch als zentrales Objekt ausgestellt werden. Was sind die Probleme? Welche digitale Unterstützung ist möglich? Welche Funktionen sollten angeboten werden? Falls nötig treffe sinnvolle Annahmen und erläutere diese (5P) + \item Die Datei test1.gnu besteht aus drei Zeilen mit folgenden Gnuplot-Anweisungen:\\ + set terminal png size 1600,1200\\ + set output 'test1.png'\\ + plot 'silver.dat' using 1:2 with lines\\ + Beschreibe möglichst genau was die Anweisungen bewirken. Treffe falls nötig sinnvolle Annahmen. (5P) + \end{enumerate} + \newpage + \section{Aufgabenteil 3 - Prof. Schmid} + \item Aufgabe 5 (Problemlösen; 10P) + \begin{enumerate} + \item Problemraum (7P)\\ + Robi ist im Arbeitszimmer und dort geht die Glühlampe in der Deckenleuchte kaputt. Robi muss erstmal genau schauen, um welchen Typ Glühlampe es sich handelt, bevor er eine neue besorgt. Da es sich um eine hohe Altbauwohnung handelt, muss Robi eine Leiter holen. Diese steht in der Küche. Zur Vereinfachung nehmen wir an, dass es nur die räumliche Position Küche und Arbeitszimmer gibt. Das heißt, Robi steht immer schon unter der Lampe, wenn er im Arbeitszimmer ist und direkt an der Leiter, wenn er in der Küche ist. Robi hat folgende Aktionen zur Verfügung. Er kann von einem Raum in den anderen gehen (go(from, to)). Er kann die Leiter von einem Raum in den anderen transportieren (move(from, to)) und er kann auf die Leiter steigen (climb). Zum Transportieren der Leiter von einem Raum zum anderen muss er im gleichen Raum sein, wie die Leiter. Zum Transportieren der Leiter und zum Gehen von einem in den anderen muss Robi sich auf dem Boden (und nicht auf der Leiter) befinden. Außerdem kann er, falls er im Arbeitszimmer ist und auf der Leiter steht, die Glühbirne nehmen (takeBulb).\\\\ + Zeichne den Problemraum-Graph für das oben beschriebene Problem (alle zulässigen Zustände als Knoten, alle Zustandsübergänge als gerichtete und beschriftete Kanten). + \item Konzepte und Algorithmen (3P)\\ + \begin{itemize} + \item Greeno (1978) unterscheidet Transformationsprobleme, Neuordnungsprobleme und Induktionsprobleme. Um welchen Problemtyp handelt es sich bei einem Anagramm (einem Wort, das aus einem anderen Wort allein durch eine Umstellung der Buchstaben gebildet werden kann)? Begründe deine Auswahl kurz. (1P) + \item Was versteht man unter Problemlösen unter einer Heuristik? (1P) + \item Nenne die allgemeine Form zur Berechnung der minimalen Zahl von Aktionen zur Lösung eines Turm von Hanoi Problems mit n Scheiben und 3 Stiften. (1P) + \end{itemize} + \end{enumerate} + \item Aufgabe 6 (Lernen; 10P) + \begin{enumerate} + \item Lernalgorithmen (2P)\\ + Um zu erkennen, ob ein Pilz giftig ist oder nicht hast du die folgenden Attribute: Farbe des Hutes mit den Ausprägungen \{rot, gelb, braun\}, die durchschnittliche Größe \{2cm, 5cm, 8cm\} und die Form des Hutes \{kopfig, schwammähnlich\}.\\ + In der Veranstaltung wurden Perceptron, Cal2 und ID3 behandelt. + Welches der Lernverfahren ist für diese Daten (ohne weitere Aufbereitung der Daten) nicht geeignet. Begründe die Antwort. + \item Perzeptron (2P)\\ + Gegeben sind folgende Daten:\\ + \begin{tabular}{ccc} + X & Y & Klasse \\ + 0 & 0 & + \\ + -1 & 0 & + \\ + 0 & 1 & + \\ + 2 & 0 & - \\ + 1 & -1 & - \\ + 0 & -2 & -\\ + \end{tabular} + + Kann ein Perzeptron trainiert werden, das alle Beispiele dieses Datensatzes korrekt klassifiziert? Begründe die Antwort mit Hilfe einer Skizze. + + \item CAL2 (6P)\\ + Führe eine Handsimulation des Entscheidungsbaum-Algorithmus CAL2 für die in der Tabelle angegebene Trainingsmenge durch. Führe die Merkmale in der vorgegebenen Reihenfolge (beginnend mit x1) in den Baum ein. Zeichne den Baum in jeder Entwicklungsstufe.\\ + + \begin{tabular}{ccccc} + Bsp & x1 & x2 & x3 & Klasse \\ + 1 & 0 & 0 & 0 & B \\ + 2 & 1 & 0 & 1 & B \\ + 3 & 0 & 1 & 1 & A \\ + 4 & 0 & 0 & 1 & B \\ + 5 & 1 & 0 & 0 & B \\ + 6 & 0 & 1 & 0 & A \\ + 7 & 1 & 1 & 0 & A \\ + 8 & 1 & 1 & 1 & B \\ + \end{tabular} + + + \end{enumerate} + \newpage + \section{Aufgabenteil 4 - Prof. Gross} + \item Aufgabe 7 (Mensch-Computer-Interaktion; 10P) + \begin{enumerate} + \item Human-Computer Interaction and Usability: bitte definiere die Begriffe Mensch-Computer-Interaktion und Gebrauchstauglichkeit (4P). + \item Consistency Maintenance of Replicated Documents: Werden zeitgleich Änderungen an einem replizierten Dokument in einem verteilten Editor vorgenommen, können verschiedene Arten von Konsistenzproblemen auftreten, die behandelt werden müssen.\\ + Gegeben ist die Ausgangssequenz "klm". NutzerInnen an zwei verteilten Orten (Sites $S_{1}$ und $S_{2}$) führen in etwa zeitgleich darauf jeweils eine Operation ($o_{1}$ und $o_{1}$) lokal aus. Danach wird die jeweils entfernt ausgeführte Operationen lokal ausgeführt. Bitte benenne jeweils welches Konsistenzproblem durch das Ausführen der Operationen I), II) und III) auftritt und erläutere es am jeweiligen Beispiel (6P).\\\\ + I) $o_{1}$ =insert['x',1]\\ + $o_{2}$ =insert['y',2]\\\\ + II) $o_{1}$ =insert['y',4]\\ + $o_{2}$ =delete[1]\\\\ + III) $o_{1}$ =insert['y',2]\\ + $o_{2}$ =insert['y',2]\\ + \end{enumerate} + \item Aufgabe 8 (Mensch-Computer-Interaktion; 10P) + \begin{enumerate} + \item Interaction: Bitte beschreibe den Verlauf der Interaktion zwischen Benutzerinnen und Benutzern einerseits und dem System andererseits (6P). + \item Dimensions: Bitte beschrifte den Sensor und Aktuator im folgenden Diagramm (2P). + \image{0.7}{A8.png}{Sensor und Aktuator}{Sensor und Aktuator} + \item Building Blocks: Bitte nenne die Funktion von Sensor und Aktuator in einer sensorbasierten Infrastruktur (2P). + \end{enumerate} + \end{enumerate} +\end{document} \ No newline at end of file From b7b9029f89106f55ed8fd541f0eef470c8b00451 Mon Sep 17 00:00:00 2001 From: Fabian Lamprecht Date: Mon, 28 May 2018 23:15:09 +0200 Subject: [PATCH 3/5] MfI 2 WS 17/18 --- .../WS17 MfI2.tex | 191 ++++++++++++++++++ 1 file changed, 191 insertions(+) create mode 100644 KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex diff --git a/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex new file mode 100644 index 0000000..238b93c --- /dev/null +++ b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex @@ -0,0 +1,191 @@ +\input{../settings/settings} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{multirow} + +\begin{document} + +\klausur{KTR-MfI 2} +{Prof. Dr. U. Krieger)} +{Wintersemester 17/18} +{90} +{Taschenrechner, 1 Din A4 Seite doppelseitig handbeschrieben} + +\begin{enumerate} + \item Aufgabe 1 (8+12 Punkte) Lineare Gleichungssysteme und Matlab/Octave + + Geben Sie die Ergebnisse folgender Aufgaben zu linearen Gleichungssystemen in den daf\"ur vorgegebenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie das folgende lineare Gleichungssystem: + \[\begin{array}{lllllll} + 2x_1 & + & 4x_2 & & & = 0 \\ + & & 2x_2 & + & 2x_3 & = 2 \\ + 2x_1 & + & 4x_2 & + & 4x_3 & = 1 + \end{array}\] + \begin{enumerate} + \item [1)] Geben Sie eine Darstellung dieses linearen Gleicungssystems in der Form eines Matrix-Vektor-Produkts $$A\cdot x = b$$ mit Hilfe einer geeigneten Koeffizientenmatrix $A$ und dem Vektor $x$ der Variablen sowie der rechten Seite $b$ des Gleicungssystems an. + \item[2)] Berechnen Sie die L\"osung $x = \left(\begin{array}{c}x_1\\ x_2\\ x_3\end{array}\right) \in \mathbb{R}^3$ durch die \"Uberf\"uhrung des Systems (A~~~b) in Treppennormalform und anschlie{\ss}ende L\"osung unter Verwendung des Gau{\ss}schen Algorithmus. + + + \end{enumerate} + + \item [b)] Berechnen Sie da lineare Gleichungssystem + $$L\cdot x = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\cdot\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n + \end{array}\right)=\left(\begin{array}{c} + b_1 \\b_2\\\vdots\\b_n + \end{array}\right)=b$$ + mit der unteren Dreiecksmatrix + $$L = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\in \mathbb{R}^{n\times n}, n\in\mathbb{N}, n\geq 2$$ + und nehmen Sie an, dass alle Elemente $L_{ii}\neq 0,i=1,\dots,n$, sind. + + Der Vektor $x=\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n + \end{array}\right)$ kann mit Hilfe des folgenden \textit{Forward-Substitution} Algorithmus + \begin{equation} + x_i=\left(b_i-\sum^{i-1}_{j=1}~~L_{ij}\cdot x_j\right)/ L_{ii}, i=1,\dots,n + \end{equation} + bestimmt werden, der sukzessive die Elemente $x_i,i=1,\dots,n,$ berechnet. + \newpage + \begin{enumerate} + \item[1)] Implementieren Sie den \textit{Forward-Substitution} Algorithmus als MATLAB oder Octave Funktion $ForSub(b, L, y)$, die f\"ur eine gegebene rechte Seite $b\in\mathbb{R}^n$ und eine invertierbare untere Dreiecksmatrix $L\in\mathbb{R}^{n\times n}$ der Dimension $ n \in \mathbb{N}, n\geq 2$, mit Diagonalelementen $L_{ii}\neq 0, i=1,\dots,n$, als Eingabeparametern des L\"osungsvektor $x\in\mathbb{R}^n$ in (1) berechnet und diesen w\"ahrend des L\"osungsverfahrens sukzessive in den Speicherplatz des Ausgabevektors $y\in\mathbb{R}^n$ schreibt, nachdem $y$ zu Beginn des Algorithmus mit dem Nullvektor initialisieert wurde. + \item[2)] Schreiben Sie ein MATLAB oder Octave Programm, das diese erstellte MATLAB oder Octave Funktion $ForSub$ zur L\"osung des Gleichungssystems $L \cdot x = b$ mit folgenden Eingangsdaten + $$L=\left(\begin{array}{ccc} + 2 & 0 & 0 \\ + 1 & 5 & 0 \\ + 7 & 9 & 8 + \end{array}\right), b = \left(\begin{array}{c} + 6 \\2\\5 + \end{array}\right)$$ + anwendet und dann den L\"osungsvektor $x$ komponentenweise ausgibt. + \end{enumerate} + \end{enumerate} + \item Aufgabe 2 (6+6+8 Punkte) Vektorr\"aume + Geben Sie die Ergebnisse folgender Aufgaben zur Theorie der Vektorr\"aume in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie die Vektoren + $$a=\left(\begin{array}{c} + 0 \\1\\1 + \end{array}\right),b=\left(\begin{array}{c} + 1 \\1\\1 + \end{array}\right),c=\left(\begin{array}{c} + 1 \\0\\1 + \end{array}\right)$$ + des $\mathbb{F}_2$-Vektorraums $\mathbb{F}^3_2$ auf dem K\"orper deer bin\"aren Elemente $\mathbb{F}_2={0,1}$ mit den Operationen $\{\oplus, \odot\}$ der Addition und Multiplikation modulo 2, d. h. f\"ur $x, y\in\mathbb{F}_2,x\oplus y = (x+y)~mod~2,\\ x \odot y = (x\cdot y)~mod~2$.\\ + Sind diese drei Vektoren $\{a,b,c\}\subset\mathbb{F}^3_2$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort durch geeignete algebraische Argumente. + \item [b)] Betrachten Sie die Matrizen + $$A=\left(\begin{array}{cc} + 0 & 1 \\ + 0 & 1 \\ + \end{array}\right), B= \left(\begin{array}{cc} + 0 & 0 \\ + 1 & 0 + \end{array}\right), C = \left(\begin{array}{cc} + 0 & 1 \\ + 1 & 0 + \end{array}\right)$$ + des Vektorraums $\mathbb{R}^{2\times 2}$.\\ + Sind diese drei Matrizen $\{A,B,C\}\subset\mathbb{R}^{2\times 2}$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort mit Hilfe einer geeigneten algebraischen Argumentation. + \item[c)] Weisen Sie durch eine geeignete algebraische Argumentation nach, dass die Menge + $$U=\{v=\left(\begin{array}{c} + x_1 \\x_2\\x_3 + \end{array}\right)\in \mathbb{R}^{3} | 4x_1+3x_2=0\} \subset \mathbb{R}^3$$ + ein Untervektorraum des reellen Vektorraums $V = \mathbb{R}^3$ ist. + \end{enumerate}\newpage + \item Aufgabe 3 (6+6+4+4 Punkte) Lineare Abbildungen + + Betrachten Sie die lineare Abbildung + $$ f: V \longrightarrow W $$ + $$ v=\left(\begin{array}{c}x \\ y \\ z \end{array}\right)\rightarrow f(v)=\left(\begin{array}{c} y\oplus z \\ x\oplus z \\ x\oplus y \end{array}\right) $$ + zwischen den $\mathbb{F}_2$-Vektorr\"aumen $V=\mathbb{F}^3_2$ und $W=\mathbb{F}^3_2$ \"uber dem K\"orper der bin\"aren Elemente $\mathbb{F}_2 = {0,1}$ mit den Operationen $\{\oplus,\odot\}$ der Addition und Multiplikation modulo 2, d.h. $x\oplus y = (x+y) mod 2, x\odot y = (x\cdot y) mod 2$ f\"ur $x,y\in\mathbb{F}_2$. + \begin{itemize} + \item [a)] W\"ahlen Sie jeweils die kanonische Basis $\mathcal{B} = \{e_1,e_2,e_3\}$ im $\mathbb{F}_2$-Vektorraum $V = \mathbb{F}^3_2$ bzw. $W=\mathbb{F}^3_2$ aus und geben Sie die der Abbildung $f$ zugeordnete Matrix $A_f$ bzgl. dieser beiden Basen an. + \item [b)] Bestimmen Sie mit Hilfe einer geeigneten algebraischen Argumentation oder eines geeigneten Berechnungsverfahrens den Rang $rg(f)$ der Abbildung $f$. + \item [c)] Bestimmen Sie mit Hilfe einer geeigneten Formel, die Sie explizit angeben sollen, die Dimension $k=dim(Ker(f))$ des Kerns $Ker(f)$ der Abbildung $f$. + \item [d)] Geben Sie eine Basis $\mathcal{C} = \{v_1,\dots,v_k\}$ des Kerns $Ker(f)$ der Abbildung $f$ an. + \end{itemize} + + \item Aufgabe 4 (10+4+6 Punkte) Gruppen- und Matrizenalgebra + \begin{enumerate} + \item [a)] Beantworten Sie folgende Fragen zur Algebra einfacher Gruppen $(G,\oplus,e)$ + \begin{enumerate} + \item [1)] Betrachten Sie die Gruppe $(G,\oplus,e)=(\mathbb{F}^3_2,\oplus,e)$ aller bin\"arer Tupel + $$(a,b)\in\mathbb{F}^3_2 = \{(0,0),(1,0),(0,1),(1,1)\}$$ + mit den Komponenten $a,b\in\mathbb{F}_2= \{ 0,1\}$ und der komponentenweise definierten Addition modulo 2 als Gruppenverkn\"upfung $\oplus$. + + Geben Sie das neutrale Element $e\in G$ an und berechnen Sie f\"ur $f=(1,0), g=(0,1),h=(1,1)$ die folgenden Gruppenelemente $k,l\in G$ + + $$k = g \oplus f = $$ + $$l = g \oplus h = $$ + $$e = $$ + + \item[2)] Definiert die Teilmenge + $$U=\{(0,0),(1,0),(0,1)\}\subset \mathbb{F}^2_2=G$$ + der Untergruppe von $(\mathbb{F}^2_2,\oplus,e)$? Geben Sie eine algebraische Begr\"undung an. + \item[3)] Die aus drei verschiedenen Elementen besthende Menge $G={e,a,h}$ mit der additiven Verkn\"upfung $z=x\oplus y \in G $ auf zwei Gruppenelementen $x,y \in G$ kann durch eine entsprechende Verkn\"upfungstabelle T beschrieben werden (siehe unten), z.B. ergibt sich aus der Additon der Elemente $x = b$ in der 5. Zeile und $y=a$ in der 4. Spalte das Element + $$z=x\oplus y = b \oplus a = e$$ + Erg\"anzen Sie die fehlenden Elemente der 5. Zeile hinter $x=b$ und der 5. Spalte unterhalb von $y=b$ in der Verkn\"upfungstabelle $T$ sowie das fehlende Element der 5. Zeile und 3. Spalte f\"ur $x=b$ und $y=e$ derart, dass $T$ eine kommutative Gruppe $(G,\oplus,e)$ definiert. + + \begin{tabular}{|c|c||c|c|c|}\hline + $\oplus$ & \multicolumn{4}{|c|}{}\\\hline + &y= & e & a & b \\\hline\hline + \multirow{3}{*}{b=} & e & e & a & \\\hline + & a & a & b & e \\\hline + & b & & e & \\\hline + \end{tabular} + \end{enumerate} + \item[b)] Es sei $A=\left(\begin{array}{ccc} + 2 & 1 & 4\\ + 1 & -1 & -5 + \end{array}\right), B = \left(\begin{array}{ccc} + 4 & 1 & 0\\ + 5 & 2 & 1\\ + 10& 7 & 6 + \end{array}\right)$ + \begin{enumerate} + \item[1)] Berechnen Sie das Element $C_{23}$ der Produktmatrix $C=A\cdot B$ + \item[2)] Wie lautet die transponierte Matrix $A^t$ zu $A$? + \item[3)] Es sei + $$A=\left(\begin{array}{ccc} + 1 & 0 & 0\\ + 0 & 1 & -1\\ + 2 & -4 & 3 + \end{array}\right)$$ + Berechnen Sie die inverse Matrix $C=A^{-1}$ zu $A$ + \end{enumerate} + \item Aufgabe 5(4+6+10 Punkte) Lineare Abbildungs- und Eigenwerttheorie + + Geben Sie die Ergebnisser folgender Aufgaben in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Berechnen Sie die Determinante der folgenden Matrix + $$A=\left(\begin{array}{ccc} + 0.3 & 0 & 0\\ + 0.7 & 0 & 1\\ + 0 & 2 & 0 + \end{array}\right)$$ + $det(A)=$ + \item[b)] Berechnen Sie einen Eigenvektor $v\in\mathbb{R}^2$ zum Eigenwert $\lambda = 2$ der folgenden Matrix $B$: + $$B=\left(\begin{array}{cc} + 1 & -1\\ + -1 & 1 + \end{array}\right)\in\mathbb{R}^{2\times 2}$$ + \item[c)] Bestimmen Sie das charakteristische Polinom $p(\lambda) = \lambda^3+p_2\lambda^2+p_1\lambda+p_0$ der folgenden Matrix $A$ und berechnen Sie mit dessen Hilfe alle Eigenwerte $\lambda_i\in\mathbb{R}, i\in\{1,2,3\}$, der Matrix $A$: + $$A=\left(\begin{array}{ccc} + 0.9 & 0 & 0\\ + 0 & -1 & 1\\ + 0.1 & -2 & 2 + \end{array}\right)$$ + \end{enumerate} + \end{enumerate} +\end{enumerate} +\end{document} \ No newline at end of file From ca9a90c919dea0a7676f286adcf2871fa6985a48 Mon Sep 17 00:00:00 2001 From: Anika Date: Mon, 28 May 2018 23:50:32 +0200 Subject: [PATCH 4/5] add WiMa-B-01a exam (task 12 missing?) --- WiMa-B-01a/WS17 WiMa-B-01a.tex | 154 +++++++++++++++++++++++++++++++++ 1 file changed, 154 insertions(+) create mode 100644 WiMa-B-01a/WS17 WiMa-B-01a.tex diff --git a/WiMa-B-01a/WS17 WiMa-B-01a.tex b/WiMa-B-01a/WS17 WiMa-B-01a.tex new file mode 100644 index 0000000..a8f304f --- /dev/null +++ b/WiMa-B-01a/WS17 WiMa-B-01a.tex @@ -0,0 +1,154 @@ +\input{../settings/settings} +\usepackage{amsmath, amssymb} + +\begin{document} + \klausur{WiMa-B-01a Wirtschaftsmathematik 1} + {Prof. Dr. Christian Aßmann} + {Wintersemester 17/18} + {60} + {Taschenrechner, Formelsammlung, 1 handbeschriebenes DIN-A-4-Blatt} + + \begin{enumerate} + \item Aufgabe 1 (5 Punkte)\\ + Geben sie die Lösungsmenge der Gleichung + \begin{equation*} + | x- 3 | = 2 + \end{equation*} + an. + \begin{enumerate} + \item $L={x|x<3} $ + \item $L={} $ + \item $L={1;5} $ + \item $L={x|x>3} $ + \item $L={x|x<1} $ + \end{enumerate} + + \item Aufgabe 2 (6 Punkte)\\ + Welche der folgenden Aussagen ist falsch? + \begin{enumerate} + \item Sei $f(x)=log_a(x)$. Dann gilt $f'(x)=\frac{1}{x\ln a}$ mit $a>0$ und $a\neq1$. + \item Sei $h(x)=exp\{g(x)\}$. Dann gilt $h'(x)=g'(x)exp\{g(x)\}$. + \item $\lim_{h \rightarrow 0}\frac{f(x+h-f(x)}{h} = f'(x)$, sofern der Grenzwert existiert. + \item Sei $ g(x) = f(x) + c $ für $c \in \mathbb{R}$. Dann gilt $g'(x) = f'(x)+c$. + \item Sei $ f(x)=\ln x $. Dann gilt $f'(x)=\frac{1}{x}$. + \end{enumerate} + + \item Aufgabe 3 (6 Punkte)\\ + Geben sie den natürlichen Definitionsbereich $D_f$ der Funktion + \begin{equation*} + f(x) = \ln(a-x) + \end{equation*} + für $a > 0$ an. + \begin{enumerate} + \item $D_f=\{\}$. + \item $D_f=\{x|x<0\}$. + \item $D_f=\{x|xa\}$. + \end{enumerate} + \item Aufgabe 4 (5 Punkte)\\ + Seien $x_1,x_2,\dots, x_n \in \mathbb{R}$ fest gegeben. Wie lautet die Minimalstelle $y_{min}$ der Funktion + \begin{equation*} + f(y)=\sum_{i=1}^{n}(x_1-y)^2\mathrm{?} + \end{equation*} + \begin{enumerate} + \item $y_{min}=0$. + \item $y_{min}=x_n-x_1$. + \item $y_{min}=x_1$. + \item $y_{min}=\frac{1}{n}\sum_{i=1}^{n}x_i$. + \item $y_{min}=\sum_{i=1}^{n}x_i$. + \end{enumerate} + + \item Aufgabe 5 (5 Punkte)\\ + Bilden Sie die erste Ableitung der Funktion + \begin{equation*} + f(x)=(1+e^{-2x})^{-3} + \end{equation*} + und berechnen Sie die Ableitung an der Stelle $x_0=1$. + \begin{enumerate} + \item $f'(x_0)=-0,3887$. + \item $f'(x_0)=0,3887$. + \item $f'(x_0)=0,4887$. + \item $f'(x_0)=0,5887$. + \item $f'(x_0)=-0,4887$. + \end{enumerate} + + \item Aufgabe 6 (5 Punkte)\\ + Wie lautet die Maximalstelle $x_{max}$ der Funktion + \begin{equation*} + f(x)=exp\{-\frac{1}{2}(x-\mu)^2\} \mathrm{?} + \end{equation*} + Welchen Wert nimmt die Funktion an der Maximalstelle an? + \begin{enumerate} + \item $x_{max}=\inf$, $f(x_{max}=0)$.\item $x_{max}=\mu$, $f(x_{max}=0)$. + \item $x_{max}=0$, $f(x_{max}=1)$. + \item $x_{max}=\mu$, $f(x_{max}=1)$. + \item $x_{max}=1$, $f(x_{max}=\mu)$. + \end{enumerate} + + \item Aufgabe 7 (5 Punkte)\\ + Sei $D_f\subset \mathbb{R}$ und $f : D_f \rightarrow \mathbb{R}$ zweimal differenzierbar in einer offenen Umgebung von $x_0 \in D_f$. Ferner sei $f(x_0)=0$. Welche der folgenden Aussagen ist falsch? + \begin{enumerate} + \item $f''(x)<0 \Rightarrow f$ hat ein striktes lokales Maximum in $x_0$. + \item $f''(x)>0 \Rightarrow f$ hat ein striktes lokales Minimum in $x_0$. + \item $f''(x)<0$ für alle $x \in D_f \Rightarrow f$ hat ein striktes globales Maximum in $x_0$. + \item $f''(x)>0$ für alle $x \in D_f \Rightarrow f$ hat ein striktes globales Minimum in $x_0$. + \item $x_0$ ist ein striktes lokales Minimum $\Rightarrow x_0$ ist auch ein striktes globales Minimum. + \end{enumerate} + \item Aufgabe 8 (5 Punkte)\\ + Bestimmen sie den Grenzwert + \begin{equation*} + \lim_{n \rightarrow \inf}1-\left[1-\frac{y}{c}\right]^n + \end{equation*} + für $00$. + \end{enumerate} + + \item Aufgabe 11 (6 Punkte)\\ + Bestimmen Sie mit der Lagrangemethode die kritische Stelle der Kostenfunktion + \begin{equation*} + K : \mathbb{R}^2_+ \rightarrow \mathbb{R}, \quad K(x,y)=x^2-y^2 + \end{equation*} + unter der Nebenbedingung + \begin{equation*} + g: \mathbb{R}^2_+ \rightarrow \mathbb{R}, \quad g(x,y)=x^3+y^3-16=0. + \end{equation*} + Die hinreichenden Bedingungen müssen nicht geprüft werden. + \begin{enumerate} + \item Kritische Stelle $ (x,y)=(2,2)$. + \item Kritische Stelle $ (x,y)=(0,0)$. + \item Kritische Stelle $ (x,y)=(12,4)$. + \item Kritische Stelle $ (x,y)=(0,4)$. + \item Kritische Stelle $ (x,y)=(4,0)$. + \end{enumerate} + \end{enumerate} +\end{document} From e577707b0f8a2d84c89e6067a79ccadefedaf257 Mon Sep 17 00:00:00 2001 From: Anika Date: Tue, 29 May 2018 00:30:24 +0200 Subject: [PATCH 5/5] correct some mistakes --- WiMa-B-01a/WS17 WiMa-B-01a.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/WiMa-B-01a/WS17 WiMa-B-01a.tex b/WiMa-B-01a/WS17 WiMa-B-01a.tex index a8f304f..88c99ec 100644 --- a/WiMa-B-01a/WS17 WiMa-B-01a.tex +++ b/WiMa-B-01a/WS17 WiMa-B-01a.tex @@ -17,7 +17,7 @@ an. \begin{enumerate} \item $L={x|x<3} $ - \item $L={} $ + \item $L=\{\} $ \item $L={1;5} $ \item $L={x|x>3} $ \item $L={x|x<1} $

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