commit ac1507e6c474eb824c958321b480a31e6c3a1780
parent df18b8bae3a90497de2b6aa32fc2f7c5ff449563
Author: thing1 <thing1@seacrossedlovers.xyz>
Date: Thu, 9 Oct 2025 12:07:54 +0100
made tex notes of past few lectures
Diffstat:
3 files changed, 135 insertions(+), 0 deletions(-)
diff --git a/.gitignore b/.gitignore
@@ -2,3 +2,7 @@
*.html
*.o
*.swp
+*.log
+*.aux
+*.auxlock
+*.md5
diff --git a/MP10610/tex/Makefile b/MP10610/tex/Makefile
@@ -0,0 +1,6 @@
+all: mp10610
+
+mp10610: mp10610.tex
+ pdflatex mp10610.tex
+clean:
+ rm -rf *.log *.pdf *.aux *.auxlock *.md5
diff --git a/MP10610/tex/mp10610.tex b/MP10610/tex/mp10610.tex
@@ -0,0 +1,125 @@
+\documentclass{article}
+\usepackage[margin=0.25in]{geometry}
+\usepackage{pgfplots}
+\pgfplotsset{width=10cm,compat=1.9}
+
+\newcommand{\Plot}[2] {
+ \begin{center}
+ \textbf{#2}
+ \end{center}
+
+ \begin{center}
+ \begin{tikzpicture}
+ \begin{axis}[
+ axis lines = middle,
+ xmin = -5,
+ xmax = 5,
+ ymin = -5,
+ ymax = 5,
+ ]
+
+ \addplot[color=red]{ #1 };
+ \end{axis}
+ \end{tikzpicture}
+ \end{center}
+}
+\newcommand{\Cmpplot}[3] {
+ \begin{center}
+ \textbf{#3}
+ \end{center}
+
+ \begin{center}
+ \begin{tikzpicture}
+ \begin{axis}[
+ axis lines = middle,
+ xmin = -5,
+ xmax = 5,
+ ymin = -5,
+ ymax = 5,
+ ]
+
+ \addplot[color=red, samples=100]{ #1 };
+ \addplot[color=blue, samples=100]{ #2 };
+ \end{axis}
+ \end{tikzpicture}
+ \end{center}
+}
+
+
+
+\begin{document}
+\begin{center}
+
+ \section{Exponential functions}
+ Exponential functions are functions that in some way involve \( e^x \).
+ \Plot{exp(x)}{A graph of \(e^x\)}
+ Note the fact that they never cross the X axis. They can be manipulated just
+ like any other graph.
+ \Plot{(exp(x)) + 1}{A graph of \(e^x + 1\)}
+ \(e^x\) is a special function, this is because \(\frac{dy}{dx}\) is still
+ \(e^x\)
+ \newpage
+
+ \section{Logarithmic functions}
+ Logarithmic functions involve log in some way. Log is defined such that
+ \\
+ \begin{math}
+ \log_2 8 = 3
+ \end{math}
+ \\
+ is true.
+
+ ln is a special function that defines the log of base e, it can be written
+ like so
+ \\
+ \begin{math}
+ \log_e x = ln(x)
+ \end{math}
+ \\
+ The functions \(exp(x)\) and \(ln(x)\) are linked such that \(ln(x)\)
+ is a reflection of \(exp(x)\) on the line \(y = x\)
+ \Cmpplot{exp(x)}{ln(x)}{A graph comparing \(exp(x)\) and \(ln(x)\)}
+
+ \section{Even and odd functions}
+ An even function is a function that satisfies the equation
+ \\
+ \begin{math}
+ f(x) = f(-x)
+ \end{math}
+ \\
+ An example of this is the function \(x^2\), as a value of
+ \(-x\) has the same output as its corresponding \(x\).
+
+ An odd function is a function that satisfies the equation
+ \\
+ \begin{math}
+ f(-x) = -f(x)
+ \end{math}
+ \\
+ An example of this is \(y = x\), lets say \(x = 1\),
+ this means that \(-x = -1\); if we substitute \(x\) into \(y = f(-x)\)
+ then \(y = -1\). If we then substitute \(x\) into \(y = -f(x)\) then
+ \(y = -1\); we can see that y is the same either way and is thus an odd
+ function.
+
+ Some functions may not be odd or even.
+
+ A function can be split into its odd and even components using the following
+ formula
+ \\
+ \begin{math}
+ f_o (x) = \frac{f(x) - f(x)}{2}
+ \end{math} \\
+ \begin{math}
+ f_e (x) = \frac{f(x) + f(x)}{2}
+ \end{math}
+ \\
+ The expression
+ \begin{math}
+ f_e (x) + f_o (x) = f(x)
+ \end{math}
+ \\
+ is always true.
+
+\end{center}
+\end{document}
