em dash; sentence case

quarto/precalc/transformations.qmd

@@ -22,7 +22,7 @@ nothing
 ---
 
 
-Thinking of functions as objects themselves that can be manipulated - rather than just blackboxes for evaluation - is a major abstraction of calculus. The main operations to come: the limit *of a function*, the derivative *of a function*, and the integral *of a function* all operate on functions. Hence the idea of an [operator](http://tinyurl.com/n5gp6mf). Here we discuss manipulations of functions from pre-calculus that have proven to be useful abstractions.
+Thinking of functions as objects themselves that can be manipulated---rather than just blackboxes for evaluation---is a major abstraction of calculus. The main operations to come: the limit *of a function*, the derivative *of a function*, and the integral *of a function* all operate on functions. Hence the idea of an [operator](http://tinyurl.com/n5gp6mf). Here we discuss manipulations of functions from pre-calculus that have proven to be useful abstractions.
 
 
 ## The algebra of functions
@@ -141,7 +141,7 @@ The real value of composition is to break down more complicated things into a se
 ### Shifting and scaling graphs
 
 
-It is very useful to mentally categorize functions within families. The difference between $f(x) = \cos(x)$ and $g(x) = 12\cos(2(x - \pi/4))$ is not that much - both are cosine functions, one is just a simple enough transformation of the other. As such, we expect bounded, oscillatory behaviour with the details of how large and how fast the oscillations are to depend on the specifics of the function. Similarly, both these functions $f(x) = 2^x$ and $g(x)=e^x$ behave like exponential growth, the difference being only in the rate of growth. There are families of functions that are qualitatively similar, but quantitatively different, linked together by a few basic transformations.
+It is very useful to mentally categorize functions within families. The difference between $f(x) = \cos(x)$ and $g(x) = 12\cos(2(x - \pi/4))$ is not that much---both are cosine functions, one is just a simple enough transformation of the other. As such, we expect bounded, oscillatory behaviour with the details of how large and how fast the oscillations are to depend on the specifics of the function. Similarly, both these functions $f(x) = 2^x$ and $g(x)=e^x$ behave like exponential growth, the difference being only in the rate of growth. There are families of functions that are qualitatively similar, but quantitatively different, linked together by a few basic transformations.
 
 
 There is a set of operations of functions, which does not really change the type of function. Rather, it basically moves and stretches how the functions are graphed. We discuss these four main transformations of $f$:
@@ -322,7 +322,7 @@ datetime = 12 + 10/60 + 38/60/60
 delta = (newyork(266) - datetime) * 60
 ```
 
-This is off by a fair amount - almost $8$ minutes. Clearly a trigonometric model, based on the assumption of circular motion of the earth around the sun, is not accurate enough for precise work, but it does help one understand how summer days are longer than winter days and how the length of a day changes fastest at the spring and fall equinoxes.
+This is off by a fair amount---almost $8$ minutes. Clearly a trigonometric model, based on the assumption of circular motion of the earth around the sun, is not accurate enough for precise work, but it does help one understand how summer days are longer than winter days and how the length of a day changes fastest at the spring and fall equinoxes.
 
 
 ##### Example: the pipeline operator
@@ -358,7 +358,7 @@ Suppose we have a data set like the following:^[Which comes from the "Palmer Pen
 | 48.8          | 18.4        | 3733      | male   | Chinstrap |
 | 47.5          | 15.0        | 5076      | male   | Gentoo |
 
-We might want to plot on an $x-y$ axis flipper length versus bill length but also indicate body size with a large size marker for bigger sizes.
+We might want to plot on an $x$-$y$ axis flipper length versus bill length but also indicate body size with a large size marker for bigger sizes.
 
 We could do so by transforming a marker: scaling by size, then shifting it to an `x-y` position; then plotting. Something like this:
 
@@ -473,7 +473,7 @@ S(D(f))(15),  f(15) - f(0)
 That is the accumulation of differences is just the difference of the end values.
 
 
-These two operations are discrete versions of the two main operations of calculus - the derivative and the integral. This relationship will be known as the "fundamental theorem of calculus."
+These two operations are discrete versions of the two main operations of calculus---the derivative and the integral. This relationship will be known as the "fundamental theorem of calculus."
 
 
 ## Questions