Andrew's git - notes.git/blob - methods/stuff.tex

   1\section{Exponentials \& Logarithms}
   2
   3\subsubsection*{Logarithmic identities}
   4
   5\begin{align*}
   6  \log_b (xy) &= \log_b x + \log_b y \\
   7  \log_b x^n &= n \log_b x \\
   8  \log_b y^{x^n} &= x^n \log_b y \\
   9  \log_a(\frac{m}{n}) &= \log_am - \log_a \\
  10  \log_a(m^{-1}) & = -\log_am \\
  11  \log_b c &= \frac{\log_a c}{\log_a b}
  12\end{align*}
  13
  14\subsubsection*{Index identities}
  15
  16\begin{align*}
  17  b^{m+n} &= b^m \cdot b^n \\
  18  (b^m)^n &= b^{m \cdot n} \\
  19  (b \cdot c)^n &= b^n \cdot c^n \\
  20  {b^m \div a^n} &= {b^{m-n}}
  21\end{align*}
  22
  23\subsection*{Inverse functions}
  24
  25For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is:
  26
  27\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\]
  28
  29\subsection*{Euler's number \(e\)}
  30
  31\[e= \lim_{n \rightarrow \infty} \left(1 + \dfrac{1}{n}\right)^n\]
  32
  33\subsection*{Modelling}
  34
  35\[A = A_0 e^{kt}\]
  36
  37\begin{itemize}
  38\tightlist
  39\item
  40  \(A_0\) is initial value
  41\item
  42  \(t\) is time taken
  43\item
  44  \(k\) is a constant
  45\item
  46  For continuous growth, \(k > 0\)
  47\item
  48  For continuous decay, \(k < 0\)
  49\end{itemize}
  50
  51\subsection*{Graphing exponential functions}
  52
  53\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\]
  54
  55\begin{itemize}
  56\tightlist
  57\item
  58  \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as
  59  \(x \rightarrow \infty\)
  60\item
  61  \textbf{horizontal asymptote} at \(y=c\)
  62\item
  63  \textbf{domain} is \(\mathbb{R}\)
  64\item
  65  \textbf{range} is \((c, \infty)\)
  66\item
  67  dilation of factor \(|A|\) from \(x\)-axis
  68\item
  69  dilation of factor \(1 \over k\) from \(y\)-axis
  70\end{itemize}
  71
  72\begin{tikzpicture}
  73  \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none]
  74    \addplot[red, thick, smooth, samples=100]  plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)};
  75    \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)};
  76    \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)};
  77    \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ;
  78    \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)};
  79  \end{axis}
  80\end{tikzpicture}
  81
  82\subsection*{Graphing logarithmic functions}
  83
  84\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\))
  85
  86\[f(x)=A \log_a k(x-b) + c\]
  87
  88where
  89
  90\begin{itemize}
  91\tightlist
  92\item
  93  \textbf{domain} is \((b, \infty)\)
  94\item
  95  \textbf{range} is \(\mathbb{R}\)
  96\item
  97  \textbf{vertical asymptote} at \(x=b\)
  98\item
  99  \(y\)-intercept exists if \(b<0\)
 100\item
 101  dilation of factor \(|A|\) from \(x\)-axis
 102\item
 103  dilation of factor \(1 \over k\) from \(y\)-axis
 104\end{itemize}
 105\begin{tikzpicture}
 106  \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none]
 107    \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)};
 108    \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)};
 109    \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)};
 110    \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)};
 111    \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ;
 112  \end{axis}
 113\end{tikzpicture}
 114
 115\subsection*{Finding equations}
 116
 117\colorbox{cas}{On CAS:}
 118\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png}