10.5: Summary

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We reviewed exponential functions of the form $y=a^{x}$, where $a>0$, the base, is constant.
The function $y=e^{x}$ is its own derivative, that is $\frac{d y}{d x}=e^{x}$. This function satisfies $\frac{d y}{d x}=y$, which is an example of a differential equation.
If $y=f(x)$, its inverse function is denoted $f^{-1}(x)$ and satisfies $f\left(f^{-1}(x)\right)=$ $x$ and $f^{-1}(f(x))=x$. The graph of $f^{-1}$ is the same as the graph of $f$ reflected across the line $y=x$. The domain of a function may have to be restricted so that its inverse function exists.
Let $f(x)=e^{x}$. The inverse of this function is $f^{-1}(x)=\ln (x)$. The derivative of $\ln (x)$ is $\frac{1}{x}$.
We can transform exponential relationships into linear relationships using logarithms. Such transformations allow for more meaningful plots, and can aid us in finding unknown constants in exponential relationships.
The applications in this chapter included:
1. the Andromeda strain of E. coli (a bacterium) and its doubling
2. the Ricker equation for fish population growth from one year to the next
3. chemical reactions: the fraction which result in a successful reaction
4. how the advantage and disadvantage of plants growing near a tree depend on distance from the tree
5. allometry: the relationship between body weight and basal metaboloci rate

Instead of 1 E. coli cell, suppose we began with 2 which also doubled every 20 min. How long would it take for the population to grow to the size of the earth?
Given $\sqrt{3} \approx 1.74205$, compute without taking square roots: (a) $3^{3 / 2}$ (b) $3^{5 / 2}$
Let $x=e^{\rho a}$. Determine $\frac{d x}{d a}$.
Consider the following log-log plot

$clipboard_e711460a967a640ca85752d83f814efce.png$

Let $Y=\log (y)$ and $X=\log (x)$. Find constants $A$ and $B$ such that $Y=A X+B$.
Determine constants $a$ and $b$ such that $y=a x^{b}$.

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