diff --git a/Exercise 6/main.pdf b/Exercise 6/main.pdf new file mode 100644 index 0000000..5ce50f0 Binary files /dev/null and b/Exercise 6/main.pdf differ diff --git a/Exercise 6/main.tex b/Exercise 6/main.tex new file mode 100644 index 0000000..169b468 --- /dev/null +++ b/Exercise 6/main.tex @@ -0,0 +1,41 @@ +\documentclass{article} + +\input{../lib/lib.tex} + +\begin{document} + + \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} + + \oppg + \input{tasks/5.tex} + + \end{oppgaver} + +\end{document} \ No newline at end of file diff --git a/Exercise 6/tasks/1.tex b/Exercise 6/tasks/1.tex new file mode 100644 index 0000000..8e00793 --- /dev/null +++ b/Exercise 6/tasks/1.tex @@ -0,0 +1,10 @@ +\begin{deloppgaver} + \delo + \[D(x^5) = 5x^{5-1} = 5x^4\] + + \delo + \[f'(x) = '(\frac{1}{4}x^2) = \frac{1}{2}x\] + + \delo + \[\frac{d}{dx}5 = 0\] +\end{deloppgaver} \ No newline at end of file diff --git a/Exercise 6/tasks/2.tex b/Exercise 6/tasks/2.tex new file mode 100644 index 0000000..3d44b36 --- /dev/null +++ b/Exercise 6/tasks/2.tex @@ -0,0 +1,5 @@ +\begin{align*} + \lim_{x \to 7} \left[ \frac{(x-7)^2 + 5(x-7)}{(x-7)(x-4)} \right] \\ + \lim_{x \to 7} \left[ \frac{x-7 + 5}{x-4} \right] &= \frac{7-7+5}{7-4} \\ + &= \frac{5}{3}\\ +\end{align*} \ No newline at end of file diff --git a/Exercise 6/tasks/3.tex b/Exercise 6/tasks/3.tex new file mode 100644 index 0000000..7b3c1ba --- /dev/null +++ b/Exercise 6/tasks/3.tex @@ -0,0 +1,12 @@ +\[h(x) = \begin{cases} + x^2,\quad &hvis\ x>2 \\ + 3|x|, &hvis\ x \le 2 +\end{cases}\] + +Ettersom +\[3|2| = 6\] +og +\[\lim_{x \to 2} x^2 = 4\] + +så er funksjonen ikke kontinuerlig. Den gjør et hopp fra $6$ til $4$ ved $x=2$ + diff --git a/Exercise 6/tasks/4.tex b/Exercise 6/tasks/4.tex new file mode 100644 index 0000000..6f39e7a --- /dev/null +++ b/Exercise 6/tasks/4.tex @@ -0,0 +1,15 @@ + +\[f(x) = \begin{cases} + \sqrt{-x}, \qquad &hvis\ x < -1 \\ + 1, &hvis\ x = -1 \\ + (x+1)^2+1, &hvis\ x > -1 \\ +\end{cases}\] + + +\begin{align*} + \lim_{x \to -1} \sqrt{-x} &= \sqrt{-(-1)} = \sqrt{1} = 1 \\ + 1 &= 1 \\ + \lim_{x \to -1} (x+1)^2 + 1 &= (-1+1)^2 + 1 = 0^2 + 1 = 1 +\end{align*} + +Ettersom alle grenseverdiene og verdiene blir til $1$ ved $x=-1$ må funksjonen være kontinuerlig. \ No newline at end of file diff --git a/Exercise 6/tasks/5.tex b/Exercise 6/tasks/5.tex new file mode 100644 index 0000000..e099cf6 --- /dev/null +++ b/Exercise 6/tasks/5.tex @@ -0,0 +1,7 @@ +\[g(x) = log(x) + x^2\] + +\[g(0.1) \approx -0.99\] + +\[g(1) = 1\] + +Ettersom funksjonen er kontinuerlig og har minst en negativt og en positiv y-verdi så må det bety at funksjonen har et nullpunkt \ No newline at end of file diff --git a/lib/lib.tex b/lib/lib.tex index e798f0d..398ceb3 100644 --- a/lib/lib.tex +++ b/lib/lib.tex @@ -26,7 +26,7 @@ \pgfplotsset{compat=newest} \author{Øystein Tveit} -\title{MA0001 Øving 5} +\title{MA0001 Øving 6} \input{../lib/titling.tex}