updates

quarto/precalc/calculator.qmd

@@ -184,6 +184,27 @@ A right triangle has sides $a=11$ and $b=12$. Find the length of the   hypotenus
 11^2 + 12^2
 ```
 
+##### Example
+
+An article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/) describes estimates for the number of ants on earth.
+
+
+They describe the number of ants two ways:
+
+* There are $20$ *trillion* ants.
+* There are $12$ megatons of ants
+
+With this how many ants make up a pound?
+
+Below we use the underscore as a separator, which is parsed as commas are to separate groups of numbers. The calculation is the number of ants divided by the number of pounds of ants (one megaton is $1$ million pounds):
+
+```{julia}
+20_000_000_000_000_000 / (1_000_000 * 12 * 2000)
+```
+
+Or not quite a million per pound.
+
+
 ## Order of operations
 
 

quarto/precalc/inversefunctions.qmd

@@ -139,12 +139,19 @@ $$
 We see that $f^{-1}(x) = 1 + (x - 2)^{1/5}$. The fact that the power $5$ is an odd power is important, as this ensures a unique (real) solution to the fifth root of a value, in the above $y-2$.
 
 
+In the section on  [polynomial roots](../precalc/polynomial_roots.html) we introduce the `solve` function of `SymPy`, which can  algebraically solve for inverse functions in easier cases.
+
+
+
 ##### Example
 
 
 The function $f(x) = x^x,  x \geq 1/e$ is strictly increasing. However, trying to algebraically solve for an inverse function will quickly run into problems (without using specially defined functions). The existence of an inverse does not imply there will always be luck in trying to find a mathematical rule defining the inverse.
 
 
+In the section on the [intermediate value theorem](../limits/intermediate_value_theorem.html#the-find_zero-function) we will see how to *numerically* solve for an inverse function.
+
+
 ## Functions which are not always invertible