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zbnyeI?jH52UvhNfN9QyANxO1iwlS%H2XgsS3x=VKK^Lh@1$pMB_@UqLYv1Q-S|!H+ zEib8Gob-nm7i^~RsUeJ4yjPPw|Kx?GOy+O$`ym2uxDrLDwYKGyd`-9-LO(~aZ4=|Q zd6jy!;IH?Cnce5^4oJI2&>!OHjsUENZmp6R+~v4@etDoUfcw7|vHzU@(_=2KM$WFO V_AoF=%p9D|ObC>e;tCQ7{{^4+jFJEV diff --git a/cheatsheets/ma1521/finals.tex b/cheatsheets/ma1521/finals.tex index f1926e4..7cc3e54 100644 --- a/cheatsheets/ma1521/finals.tex +++ b/cheatsheets/ma1521/finals.tex @@ -357,6 +357,25 @@ $S = \iint_R\sqrt{f_x^2 + f_y^2 + 1} dA$ $y' + P(x)y = Q(x)y^n$ & \makecell{$z = y^{1-n}$ \\ $\Then$ sub in Z \\ solve linear}\\ \end{tabular} +\section{Population Models} +\begin{center} + $N_{\infty} = \frac{B}{s}$, $\hat{N} = $ Population Now +\begin{multicols}{2} + \textbf{Malthus}\\ + $N(t) = \hat{N}e^{kt}$\\ + $k = B - D$ + + \columnbreak + \textbf{Logistic}\\ + $\frac{1}{N} = \frac{1}{N_{\infty}} + (\frac{1}{\hat{N}} - \frac{1}{N_{\infty}})e^{-Bt}$\\ + $N = \frac{N_{\infty}}{1+(\frac{N_{\infty}}{N} - 1)e^{-Bt}}$ +\end{multicols} +\end{center} + +\subsection{Uranium Decay into Thorium} +$U(t) = U_{0}e^{-k_ut}$, $k = \frac{\ln2}{\text{halflife}}, \frac{dU}{dt} = -k_{u}U$\\ +Thorium: $T(t) = \frac{K_{u}U_{0}}{K_{t}-K_{u}}(e^{-k_{u}t} - e^{-k_{t}t}), \frac{dT}{dt} = k_{u}U - k_{T}T$ + \end{multicols*} \end{document}