Reading the Comics, October 27, 2018: Surprise Rerun Edition

While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th is a joking reminder that educational texts, including in mathematics, don’t have to be boring. We can have narrative thrust and energy. It’s a good reminder.

Caption: 'I wish all educational texts were written like Epictetus wrote.' Textbook: 'What, hapless wretch? Do you suppose int(sqrt(x^2 + x)dx = (x^2 + x)^{-1/2}? And when you eat, do you carry the food to your mouth or to your eyes? Slave! — Zach Weinersmith’s **Saturday Morning Breakfast Cereal** for the 26th of October, 2018.

As fits the joke, the bit of calculus in this textbook paragraph is wrong. $\int \sqrt{x^2 + x} dx$ does not equal $\left(x^2 + x\right)^{-\frac12}$ . This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of $\sqrt{x^2 + x}$ is not $\left(x^2 + x\right)^{-\frac12}$ . You can confirm that by differentiating $\left(x^2 + x\right)^{-\frac12}$ . The result is nothing like $\sqrt{x^2 + x}$ . Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.

As I make it out, a correct indefinite integral would be:

$\int{\sqrt{x^2 + x} dx} = \frac{1}{4}\left( \left(2x + 1\right)\sqrt{x^2 + x} + \log \left|\sqrt{x} + \sqrt{x + 1} \right| \right)$

Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.

Berle: 'It was just a happy stroll through the gloomy graveyard when suddenly ... ' Spivak's Calculus, 3rd Edition appears. Berle: 'Math jumped out of nowhere!' Harvey: 'Drink it off.' — Stephen Beals’s **Adult Children** rerun for the 27th of October, 2018. It also appeared the 31st of March, 2018. I’m surprised it was that recent. I can’t blame Beals if he needed a break. This is a Halloween-ready example of the comic.

Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.

Manfred: 'How do you want to divide up the bill?' Wink: 'Lemme see it .. add that and that ... carry the two ... let's just split it htree ways.' Manfred: 'Nice try.' Dusty: 'Mr Surf 'n' Turf needs a calculator!' — Rob Harrell’s **Big Top** rerun for the 27th of October, 2018. It originally appeared the 13th of December, 2008.

Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)