From c22cf7206ec407d930bcda6a17c58d22d83ef054 Mon Sep 17 00:00:00 2001 From: h7x4 Date: Sun, 20 Sep 2020 17:22:55 +0200 Subject: [PATCH] Add exercise 4 --- Exercise 4/main.pdf | Bin 0 -> 844449 bytes Exercise 4/main.tex | 40 +++++++++++++++++++++++++++++++++++++ Exercise 4/tasks/1.tex | 37 ++++++++++++++++++++++++++++++++++ Exercise 4/tasks/2.tex | 44 +++++++++++++++++++++++++++++++++++++++++ Exercise 4/tasks/3.tex | 13 ++++++++++++ Exercise 4/tasks/4.tex | 27 +++++++++++++++++++++++++ lib/lib.tex | 2 +- 7 files changed, 162 insertions(+), 1 deletion(-) create mode 100644 Exercise 4/main.pdf create mode 100644 Exercise 4/main.tex create mode 100644 Exercise 4/tasks/1.tex create mode 100644 Exercise 4/tasks/2.tex create mode 100644 Exercise 4/tasks/3.tex create mode 100644 Exercise 4/tasks/4.tex diff --git a/Exercise 4/main.pdf b/Exercise 4/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a0e9afcb8f83272746a480d58ee7f9c2bca54f02 GIT binary patch literal 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\thispagestyle{plain} + \tittel + \tableofcontents + + \newpage + + \section{Forberedende oppgaver} + \begin{oppgaver} + + \oppg + \input{tasks/1.tex} + + \end{oppgaver} + + \newpage + + \section{Innleveringsoppgaver} + \begin{oppgaver} + \setoppg{1} + + \oppg + \input{tasks/2.tex} + + \oppg + \input{tasks/3.tex} + + \oppg + \input{tasks/4.tex} + + \end{oppgaver} + +\end{document} \ No newline at end of file diff --git a/Exercise 4/tasks/1.tex b/Exercise 4/tasks/1.tex new file mode 100644 index 0000000..596b015 --- /dev/null +++ b/Exercise 4/tasks/1.tex @@ -0,0 +1,37 @@ +\begin{deloppgaver} + \delo + \begin{align*} + a_n &= \frac{n}{n+1} \\[1ex] + a_1 &= \frac{1}{1+1} = \frac{1}{2} \\[1ex] + a_2 &= \frac{2}{2+1} = \frac{2}{3} \\[1ex] + a_3 &= \frac{3}{3+1} = \frac{3}{4} \\[1ex] + a_4 &= \frac{4}{4+1} = \frac{4}{5} \\[1ex] + a_{100} &= \frac{100}{100+1} = \frac{100}{101} \\[1ex] + \end{align*} + + Jeg gjetter at \[\lim_{n \to \infty} a_n = 1\] + + \delo + \begin{align*} + a_n &= \frac{n^2}{n+1} \\[1ex] + a_1 &= \frac{1^2}{1+1} = \frac{1}{2} \\[1ex] + a_2 &= \frac{2^2}{2+1} = \frac{4}{3} \\[1ex] + a_3 &= \frac{3^2}{3+1} = \frac{9}{4} \\[1ex] + a_4 &= \frac{4^2}{4+1} = \frac{16}{5} \\[1ex] + a_{100} &= \frac{100^2}{100+1} = \frac{10000}{101} \\[1ex] + \end{align*} + + Jeg gjetter at \[\lim_{n \to \infty} a_n = \infty\] + + \delo + \begin{align*} + a_n &= \frac{n}{n^2+1} \\[1ex] + a_1 &= \frac{1}{1^2+1} = \frac{1}{2} \\[1ex] + a_2 &= \frac{2}{2^2+1} = \frac{2}{5} \\[1ex] + a_3 &= \frac{3}{3^2+1} = \frac{3}{10} \\[1ex] + a_4 &= \frac{4}{4^2+1} = \frac{4}{17} \\[1ex] + a_{100} &= \frac{100}{100^2+1} = \frac{100}{10001} \\[1ex] + \end{align*} + + Jeg gjetter at \[\lim_{n \to \infty} a_n = 0\] +\end{deloppgaver} \ No newline at end of file diff --git a/Exercise 4/tasks/2.tex b/Exercise 4/tasks/2.tex new file mode 100644 index 0000000..d4761b5 --- /dev/null +++ b/Exercise 4/tasks/2.tex @@ -0,0 +1,44 @@ +\begin{deloppgaver} + \delo + \[\frac{1}{x^2-4}\] er definert hvor + + \begin{align*} + x^2-4 &\neq 0 \\ + (x+2)(x-2) &\neq 0 \\ + x &\notin \{-2, 2\} + \end{align*} + + Som betyr at definisjonsmengden til uttrykket blir + + \[ x \in \{ \mathbb{R}\ |\ x \neq \pm 2 \} \] + + \delo + \[g(x) = \sqrt{9 - |x-1|}\] + + $\sqrt{a}$ er definert for $a \geq 0$. Vi subsituerer $a$. + + \begin{align*} + 9- |x-1| &\geq 0 \\ + - |x-1| &\geq -9 \\ + |x-1| &\leq 9 \\ + -9 &\leq x-1\quad &&\vee &x-1 &\leq 9 \\ + -8 &\leq x \quad &&\vee &x &\leq 10 + \end{align*} + + Altså blir definisjonsmengden + + \[D_g = \left[ -8, 10\right]\] + + Verdimengden til $\sqrt{a}$ er $\left[0, \infty\right)$, som betyr at den minste verdien til $g(x)$ må være $0$. + + I tillegg vet vi at $|a|$ aldri kan bli mindre enn $0$. + + Dermed vil den største verdien av $g(x)$ være når $|x-1| = 0$ hvor + + \[g(x) = \sqrt{9-0} = \pm 3\] + + Ettersom $-3$ ikke er en del av $\left[0, \infty\right)$ blir verdimengden + + \[V_g = \left[0, 3\right]\] + +\end{deloppgaver} \ No newline at end of file diff --git a/Exercise 4/tasks/3.tex b/Exercise 4/tasks/3.tex new file mode 100644 index 0000000..700afc4 --- /dev/null +++ b/Exercise 4/tasks/3.tex @@ -0,0 +1,13 @@ +\[f(x) = \frac{10}{1+x^2}\] + +Ettersom $1+x^2$ ikke har noen reelle røtter, vil uttrykket være definert for $x \in \mathbb{R}$ + +Den minste verdien av nevneren $1+x^2$ vil være når $x = 0$ hvor $1+x^2 = 1$. + +I dette tilfellet blir \[f(0) = \frac{10}{1+0^2} = 10\] + +\[\lim_{x \to \pm \infty} 1+x^2 = \pm \infty \Leftrightarrow \lim_{x \to \pm \infty}f(x) = 0\] + +Dermed blir verdimengden + +\[V_f = \left(0, 10\right]\] \ No newline at end of file diff --git a/Exercise 4/tasks/4.tex b/Exercise 4/tasks/4.tex new file mode 100644 index 0000000..91b8da6 --- /dev/null +++ b/Exercise 4/tasks/4.tex @@ -0,0 +1,27 @@ + + $h(x)$ kan skrives som $sgn(x) \cdot x^2$ som videre kan forkortes til + \begin{align*} + h(x) &= \frac{|x|}{x} \cdot x^2 \\ + h(x) &= |x|x + \end{align*} + +\begin{deloppgaver} + \delo + Ettersom verdimengden til $x^2$ originalt var $\left[0, \infty\right)$ men at vi flipper funksjonen hvor $x < 0$, så vil verdimengden til h bli + \begin{align*} + V_h &= \left(-\infty, 0 \right) \cup \left[0, \infty\right) \\ + V_h &= \mathbb{R} + \end{align*} + + \delo + $h(x)$ er injektiv ettersom vi har vendt retningen på funksjonen akkurat ved det punktet som tidligere var ekstremalpunktet til $x^2$. Dermed har ikke funksjonen noe ekstremalpunkt lengre, og enhver $h$-verdi vil kun ha $1$ tilsvarende $x$-verdi. \newline + + $h^{-1}(x)$ kan vi skrive som kvadratroten av $|x|$ (ettersom roten av negative tall er udefinert), justert ved 0 med $sgn(x)$. Altså + + \[h^{-1}(x) = sgn(x) \cdot \sqrt{|x|}\] + + eller + + \[h^{-1}(x) = \frac{|x|\sqrt{|x|}}{x}\] + +\end{deloppgaver} \ No newline at end of file diff --git a/lib/lib.tex b/lib/lib.tex index d68cb0e..1a0f8e5 100644 --- a/lib/lib.tex +++ b/lib/lib.tex @@ -26,7 +26,7 @@ \pgfplotsset{compat=newest} \author{Øystein Tveit} -\title{MA0001 Øving 3} +\title{MA0001 Øving 4} \input{../lib/titling.tex}