Initial commit - Task 1
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\author{Øystein Tveit}
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||||
\title{MA0001 Øving 1}
|
||||
|
||||
\input{lib/titling.tex}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\thispagestyle{plain}
|
||||
\tittel
|
||||
\tableofcontents
|
||||
|
||||
\newpage
|
||||
|
||||
\section{Forberedende oppgaver}
|
||||
|
||||
\begin{oppgaver}
|
||||
|
||||
\oppg
|
||||
\input{tasks/1.tex}
|
||||
|
||||
\oppg
|
||||
\input{tasks/2.tex}
|
||||
|
||||
\end{oppgaver}
|
||||
|
||||
\section{Innleveringsoppgaver}
|
||||
|
||||
\begin{oppgaver}
|
||||
\setoppg{2}
|
||||
|
||||
\oppg
|
||||
\input{tasks/3.tex}
|
||||
|
||||
\oppg
|
||||
\input{tasks/4.tex}
|
||||
|
||||
\break
|
||||
|
||||
\oppg
|
||||
\input{tasks/5.tex}
|
||||
|
||||
\end{oppgaver}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,22 @@
|
|||
\babel@toc {norsk}{}
|
||||
\contentsline {section}{\numberline {1}Forberedende oppgaver}{2}{section.1}%
|
||||
\contentsline {subsection}{Oppgave 1}{2}{Item.1}%
|
||||
\contentsline {subsubsection}{a)}{2}{Item.2}%
|
||||
\contentsline {subsubsection}{b)}{2}{Item.3}%
|
||||
\contentsline {subsubsection}{c)}{2}{Item.4}%
|
||||
\contentsline {subsection}{Oppgave 2}{2}{Item.5}%
|
||||
\contentsline {section}{\numberline {2}Innleveringsoppgaver}{2}{section.2}%
|
||||
\contentsline {subsection}{Oppgave 3}{2}{Item.6}%
|
||||
\contentsline {subsubsection}{a)}{2}{Item.7}%
|
||||
\contentsline {subsubsection}{b)}{2}{Item.8}%
|
||||
\contentsline {subsubsection}{c)}{3}{Item.9}%
|
||||
\contentsline {subsubsection}{d)}{3}{Item.10}%
|
||||
\contentsline {subsubsection}{e)}{3}{Item.11}%
|
||||
\contentsline {subsection}{Oppgave 4}{3}{Item.12}%
|
||||
\contentsline {subsubsection}{a)}{3}{Item.13}%
|
||||
\contentsline {subsubsection}{b)}{3}{Item.14}%
|
||||
\contentsline {subsubsection}{c)}{4}{Item.15}%
|
||||
\contentsline {subsubsection}{d)}{4}{Item.16}%
|
||||
\contentsline {subsection}{Oppgave 5}{6}{Item.17}%
|
||||
\contentsline {subsubsection}{a)}{6}{Item.18}%
|
||||
\contentsline {subsubsection}{b)}{6}{Item.19}%
|
||||
|
|
@ -0,0 +1,14 @@
|
|||
|
||||
\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}
|
||||
\begin{deloppgaver}
|
||||
|
||||
\delo
|
||||
\[ -135\degree \cdot \frac{\pi}{180\degree} \approx -2.35 \]
|
||||
|
||||
\delo
|
||||
\[ 1,5\pi \cdot \frac{180\degree}{\pi} = 270\deg \]
|
||||
|
||||
\delo
|
||||
\[ sin\left( \frac{-\pi}{2} \right) = -1\]
|
||||
|
||||
\end{deloppgaver}
|
||||
|
|
@ -0,0 +1,43 @@
|
|||
\begin{deloppgaver}
|
||||
|
||||
\delo
|
||||
\begin{alignat*}{4}
|
||||
& 1^{-1} &&+ 2^{-1} &&+ 3^{-1} \\[1ex]
|
||||
& \frac{1}{1} &&+ \frac{1}{2} &&+ \frac{1}{3} \\[2ex]
|
||||
& \frac{6}{6} &&+ \frac{3}{6} &&+ \frac{2}{6} & \qquad \text{Største felles divisor er 6} \\[2ex]
|
||||
& \frac{11}{6} \\
|
||||
\end{alignat*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
& x^{-7} x^{6} \\
|
||||
& x^{-7+6} \\
|
||||
& x^{-1} \\
|
||||
& \frac{1}{x}
|
||||
\end{align*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
& \frac{ \left ( x^2 y z^3 \right )^2 }{ x^4 y^3 z^5 } \\[2ex]
|
||||
& \frac{ x^4 y^2 z^6 }{ x^4 y^3 z^5 } \\[2ex]
|
||||
& x^{4-4} y^{2-3} z^{6-5} \\
|
||||
& x^0 y^{-1} z^1 \\[2ex]
|
||||
& \frac{z}{y} \\
|
||||
\end{align*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
& \frac{ e^{z-x} }{ e^x e^z} \\[2ex]
|
||||
& \frac{ \cancel{e^z} }{ e^{x+x} \cancel{e^z} } \\[2ex]
|
||||
& \frac{1}{e^{2x}} \\
|
||||
\end{align*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
& \frac{ z^2 - y^2 }{ z+y } \\[2ex]
|
||||
& \frac{ (z+y)(z-y) }{ z+y } && \text{3. Kvadratsetning} \\[2ex]
|
||||
& \frac{ \cancel{(z+y)}(z-y) }{ \cancel{z+y} } \\[2ex]
|
||||
& z-y
|
||||
\end{align*}
|
||||
|
||||
\end{deloppgaver}
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
\relax
|
||||
\@setckpt{tasks/3.tex}{
|
||||
\setcounter{page}{3}
|
||||
\setcounter{equation}{0}
|
||||
\setcounter{enumi}{3}
|
||||
\setcounter{enumii}{0}
|
||||
\setcounter{enumiii}{0}
|
||||
\setcounter{enumiv}{0}
|
||||
\setcounter{footnote}{0}
|
||||
\setcounter{mpfootnote}{0}
|
||||
\setcounter{part}{0}
|
||||
\setcounter{section}{0}
|
||||
\setcounter{subsection}{0}
|
||||
\setcounter{subsubsection}{0}
|
||||
\setcounter{paragraph}{0}
|
||||
\setcounter{subparagraph}{0}
|
||||
\setcounter{figure}{0}
|
||||
\setcounter{table}{0}
|
||||
\setcounter{parentequation}{0}
|
||||
}
|
||||
|
|
@ -0,0 +1,70 @@
|
|||
\begin{deloppgaver}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
3+2x &= 2-x\\
|
||||
\cancel{3} +2x -\cancel{3} +x &= 2 -\cancel{x} -3 +\cancel{x} \\
|
||||
3x &= -1\\
|
||||
x &= -\frac{1}{3}\\
|
||||
\end{align*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
x^2 + x &= 3 \\
|
||||
x^2 + x - 3 &= 0 \\
|
||||
\end{align*}
|
||||
|
||||
Vi bruker andregradsformelen:
|
||||
\begin{equation*}
|
||||
x = \frac{-b \pm \sqrt{ b^2 - 4ac }}{ 2a }
|
||||
\end{equation*}
|
||||
|
||||
\begin{align*}
|
||||
x &= \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -3}}{ 2 \cdot 1} \\[2ex]
|
||||
x &= \frac{-1 \pm \sqrt{1 - (-12)}}{2} \\[2ex]
|
||||
x &= -\frac{1}{2} \pm \frac{\sqrt{13}}{2}
|
||||
\end{align*}
|
||||
|
||||
\begin{equation*}
|
||||
x = -\frac{1}{2} + \frac{\sqrt{13}}{2} \quad \vee \quad x = -\frac{1}{2} - \frac{\sqrt{13}}{2}
|
||||
\end{equation*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
-x \left( x+2 \right) \left( 5x-4 \right) = 0 \\
|
||||
\end{align*}
|
||||
|
||||
Etter nullfaktorregelen må en av faktorene være 0 for at produktet skal bli 0.
|
||||
|
||||
\begin{align*}
|
||||
-x &= 0 &&\vee& x+2 &= 0 &&\vee& 5x-4 &=0 \\
|
||||
x &= 0 &&\vee& x &= -2 &&\vee& x &= \frac{4}{5} \\
|
||||
\end{align*}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
\frac{x}{x+1} &= \frac{1}{3} + \frac{x-1}{3} \\[2ex]
|
||||
\frac{x}{x+1} &= \frac{x}{3} \\[2ex]
|
||||
\frac{3 \cdot x}{3 \cdot (x+1)} &= \frac{(x+1) \cdot x}{(x+1) \cdot 3} \\[2ex]
|
||||
\frac{3x - x(x+1)}{3(x+1)} &= 0 \\[2ex]
|
||||
\frac{x(3 - (x+1))}{3(x+1)} &= 0 \\[2ex]
|
||||
\frac{x(-x+2)}{3(x+1)} &= 0 \\[2ex]
|
||||
\end{align*}
|
||||
|
||||
Telleren må være 0 for at hele uttrykket skal bli 0, men x-verdien er ikke en gyldig løsning når nevner også blir 0. Vi deler opp telleren etter nullfaktorregelen:
|
||||
|
||||
\begin{align*}
|
||||
x &= 0 &&\vee& -x+2 &= 0 \\
|
||||
x &= 0 &&\vee& x &= 2 \\
|
||||
\end{align*}
|
||||
|
||||
Om vi løser for nevneren:
|
||||
|
||||
\begin{equation*}
|
||||
3(x+1) = 0 \rightarrow x=-1
|
||||
\end{equation*}
|
||||
|
||||
så ser vi at begge løsningene er gyldige.
|
||||
|
||||
|
||||
\end{deloppgaver}
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
\relax
|
||||
\@setckpt{tasks/4.tex}{
|
||||
\setcounter{page}{4}
|
||||
\setcounter{equation}{0}
|
||||
\setcounter{enumi}{4}
|
||||
\setcounter{enumii}{0}
|
||||
\setcounter{enumiii}{0}
|
||||
\setcounter{enumiv}{0}
|
||||
\setcounter{footnote}{0}
|
||||
\setcounter{mpfootnote}{0}
|
||||
\setcounter{part}{0}
|
||||
\setcounter{section}{0}
|
||||
\setcounter{subsection}{0}
|
||||
\setcounter{subsubsection}{0}
|
||||
\setcounter{paragraph}{0}
|
||||
\setcounter{subparagraph}{0}
|
||||
\setcounter{figure}{0}
|
||||
\setcounter{table}{0}
|
||||
\setcounter{parentequation}{0}
|
||||
}
|
||||
|
|
@ -0,0 +1,47 @@
|
|||
\begin{deloppgaver}
|
||||
|
||||
\delo
|
||||
\begin{align*}
|
||||
y&=3-x \\
|
||||
y&=x-1
|
||||
\end{align*}
|
||||
|
||||
Ettersom utrykkene er rette linjer, vil de ha samme y-verdi på kun ett punkt:
|
||||
|
||||
\begin{align*}
|
||||
3-x &= x-1 \\
|
||||
2x &= 4 \\
|
||||
x = 2
|
||||
\end{align*}
|
||||
|
||||
Vi setter x-verdien i ett av uttrykkene og får at
|
||||
|
||||
\begin{align*}
|
||||
y &= 3-2 = 1
|
||||
\end{align*}
|
||||
|
||||
Linjene møtes i $(2, 1)$
|
||||
|
||||
Linjene har stigningstallene 1 og -1, og vi kan skissere dem ut ifra skjæringspunktet.
|
||||
|
||||
\begin{graphbox}
|
||||
\input{figures/5a.tex}
|
||||
\end{graphbox}
|
||||
|
||||
\delo
|
||||
Ettersom absoluttverdien av et uttrykk ikke kan bli mindre enn null, vil $|x-2|+1$ aldri være mindre enn 1. Dette skjer når
|
||||
|
||||
\begin{align*}
|
||||
|x-2| &= 0 \\
|
||||
x-2 &= \pm 0 \\
|
||||
x &= 2
|
||||
\end{align*}
|
||||
|
||||
Dermed vender stigningsfarta fra -1 til 1 i punktet $(2, 1)$.
|
||||
|
||||
Vi kan skissere grafen fra dette punktet.
|
||||
\begin{graphbox}
|
||||
\input{figures/5b.tex}
|
||||
\end{graphbox}
|
||||
|
||||
\end{deloppgaver}
|
||||
Loading…
Reference in New Issue