One reason: Exponential trends show up a LOT in environmental science (the proportional change is the same over each time span)
Math reason: Turns out it’s a very useful value for calculus
A real-world example:
But populations can’t grow exponentially forever
Gause, G. F. 1934. The Struggle for Existence. Baltimore: Williams and Wilkins.
Logistic growth
\[N_t=\frac{K}{1+[\frac{K-N_0}{N_0}]e^{-rt}}\]
Where \(N_t\) is the population size at time \(t\), \(K\) is the carrying capacity, \(N_0\) is the initial population size, and \(r\) is a growth rate.
✏️ What should be the value of the formula when \(t=0\)? Confirm your answer by substituting.
We should always understand the components of formulas and equations - both conceptually and mathematically.
Logistic growth
\[N_t=\frac{K}{1+[\frac{K-N_0}{N_0}]e^{-rt}}\]
Why might we expect logistic growth for many populations?
What variables besides time would influence the actual population?
Logarithms
Logarithms
\(\log_a(b)\) is the exponent you need to raise \(a\) to get a value of \(b\)
For example:
\(\log_2(8)=x\) asks “to what power do I have to raise 2, to get a value of 8?”
\(\log_{105}(1)=x\) asks “to what power do I have to raise 105, to get a value of 1?”
Natural logarithms
✏️
Natural logarithms
If \(y>0\), then \(\ln(x)\) is the exponent you need to raise \(e\) to in order to get \(y\).
This means:
\[
\begin{align}
\ln(e^x) &= x \\
e^{\ln(y)} &= y
\end{align}
\]
In other words, \(\ln(x)\) is the inverse of the exponential function \(e^x\).
✏️ Examples
\(\ln(e^3)\)
\(\ln(\frac{1}{e})\)
\(\ln(1)\)
\(\ln(0)\)
\(\ln(-7)\)
Natural logarithms
If \(y>0\), then \(\ln(x)\) is the exponent you need to raise \(e\) to in order to get \(y\).
This means:
\[
\begin{align}
\ln(e^x) &= x \\
e^{\ln(y)} &= y
\end{align}
\]
In other words, \(\ln(x)\) is the inverse of the exponential function \(e^x\).
Examples
\(\ln(e^3) = 3\) because 3 is the power of \(e\) needed to get \(e^3\).
\(\ln(\frac{1}{e}) = \ln(e^{-1}) = -1\)
\(\ln(1) = 0\) because 0 is the exponent we need to raise \(e\) to to get 1
\(\ln(0)\) does not exist, because there’s no number we can raise \(e\) to to get 0
\(\ln(-7)\) does not exist, because \(e\) to any number always be positive
Natural log properties
✏️
Natural log properties
Properties
\(\ln (e^x)=x\)
\(\ln(xy)=\ln x+\ln y\)
\(\ln(\frac{x}{y})=\ln x-\ln y\)
\(\ln(x^y)=y \ln x\)
Using logarithms to solve equations
✏️ Solve for \(t\) in the equation \(Pe^{rt} = A\).
Properties
\(\ln (e^x)=x\)
\(\ln(xy)=\ln x+\ln y\)
\(\ln(\frac{x}{y})=\ln x-\ln y\)
\(\ln(x^y)=y \ln x\)
Using logarithms to solve equations
✏️ Solve for \(t\) in the equation \(Pe^{rt} = A\).
\[
\begin{align}
Pe^{rt} &= A \\
e^{rt} & = \frac{A}{P} \\
rt &= \ln\left(\frac{A}{P}\right) \\
t &= \frac{\ln(A)-\ln(P)}{r}
\end{align}
\]
Limits
Limits
Definition
We say a number \(L\) is the limit of a function \(f(x)\) as its variable \(x\) approaches a number \(c\), if the function’s output values \(f(x)\) approach \(L\) when \(x\) get closer to \(c\).
We write this symbolically as:
\[
\large
\lim_{x\to c} f(x)=L
\]
We read this as: “The limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\).”
Limits example: numerical
What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?
We need to examine the values of \(f(x)\) as \(x\) gets closer to 2 from both sides.
From both directions it looks like \(f(x)\) converges to 4.
So, we say that “The limit of \(2x^2-4\) as \(x\) approaches 2 is 4”. We can write it like this:
\[
\lim_{x\to 2}(2x^2-4)=4
\]
Limits example: graph
What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?
From both directions it looks like \(f(x)\) converges to 4.
So, \(\lim_{x\to 2}(2x^2-4)=4\).
Do limits always exist?
The function \(|x|\) is the absolute value function. It is such that
\[
|x| =
\begin{cases}
x, & \text{ if } x\geq 0 \\
-x, & \text{ if } x <0
\end{cases}
.
\]
