A hiccup in my calculus quest

I’ve gotten back into my quest to learn calculus, but I’ve run into a little hiccup in the form of what appears to be a fundamental error in the book I’m using (Silvanus P. Thompson’s Calculus Made Easy, Second Edition (the PDF version at that link).

On page 18, we see this paragraph:

What does (dx)^2 mean? Remember that dx meant a bit – a little bit – of x. Then (dx)^2 will mean a little bit of a little bit of x; that is, as explained above (p. 4), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.

Emphasis mine. This is the part that really made me go “Huh?”, but I figured that the author knew more than I do, and decided to leave it.

Then later, on pages 19 & 20, we are given an example of differentiating y=x^3, and just to make sure I was understanding things correctly I tried to work it out. I decided I would have x=3 and dx=1. Here is the example from the text:

This ends with dy/dx = 3x^2, but that part won’t copy for some reason.

But, there’s a problem. If, as I did, you set x=3 and dx=1, then you end up with y=27 and y+dy=64, making dy=37. But, if you take (3x^2)(dx) when x=3 and dx=1, you end up with dy=27.

So, I tried working it out without discarding (dx)^2 and (dx)^3, and got:

y + dy = x3 + 3x2(dx) + 3x(dx)2 + (dx)3

y = x3, so they cancel and give:

dy = 3x2(dx) + 3x(dx)2 + (dx)3

dy = 3(32)(1) + 3(3)(12) + (1)3

dy = 27 + 9 + 1 = 37

The right answer.

A 10 out of 37 error is not negligible as the book asserts. So, can anyone who knows calculus tell me who is wrong here? Me, or the book? And why?

Help!

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An Android App for Geeks!

I was poking around the Android Market Google Play and found… a slide rule app! Why use your phone’s built in calculator when you can bust out an electronic slide rule? There are even several ebooks available for you to learn how to use it!

Shiny!

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Mmmmm, venison – Part II

Well, my experimental dish finished this morning, and I brought some to work for lunch today. Here’s what I started with.

Ingredients

And this, which I forgot to put in the original picture (and almost forgot to add to the pot!).

Stewed tomatoes

The actual “recipe” (which was more of an improvisation than an actual recipe)?

  • 1 lb venison steak, cubed
  • 1 can (14 oz.) low sodium beef broth
  • 1 can (14.5 oz.) Hunt’s Stewed Tomatoes
  • About 1/2 cup chopped onion (I used a medium yellow onion)
  • 1 cup rice (I used medium grain white rice)
  • 1/2 cup water
  • 1 can (5 oz.) condensed milk (optional)
  • 1/2 Tbsp. ground mustard
  • 1/2 Tbsp. minced garlic (Christopher Ranch minced garlic in olive oil)
  • Salt and pepper to taste

Mix the broth, stewed tomatoes, condensed milk, water, rice, and ground mustard in the slow cooker. Brown the venison in a skillet and add to the pot, then sauté the onions and add them. Deglaze the skillet and add that to the pot as well. Add salt and pepper to taste. Cook on “low” for about 6 hours (adjust to fit your slow cooker).

What did I end up with? This.

Yummy! I hope.

Yummy! (I hope.)

Mine cooked for seven hours, which looks to be a little bit too long – there’s a little crust of rice along the side of the pot, and the rice in the dish looks a little overdone. I probably should have gone for six hours instead, but that would have meant either staying up an extra hour last night or getting up an hour early this morning, and since I’ve never done this before I wasn’t really sure how long it would need.

I haven’t actually tasted it yet, but I brought some to work for lunch, so we’ll see how it turned out later.

—–

Update: I couldn’t wait and snuck a bite early. It turned out pretty good!

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Exercises in situational awareness

Required equipment: One (1) kitten. One (1) kitchen.

Exercise 1: While kitten is between roughly 8 and 12 weeks old, cook dinner. Awareness of the kitten’s position and expected movements must be maintained at all times. Stepping on the kitten indicates a failure to maintain proper situational awareness. If you are using the oven, additional care must be taken to ensure that the kitten does not jump or reach up to hot parts of the oven that are within its range (the open door, etc.).

Exercise 2: Around approximately 12 weeks old, the kitten will be able to jump onto the countertops unassisted. Cook again. In addition to the precautions from Exercise 1, note that the stovetop is at counter height, and that the burners are close enough to the edge that a leap from floor to stovetop will result in the kitten landing on a hot burner. Also note that most cooking involves using dangerously sharp implements such as kitchen knives while handling kitten bait (i.e., most types of food, especially anything involving dairy or meat). As in Exercise 1, any injury to the kitten indicates a failure to maintain proper situational awareness.

Apply the lessons learned in these exercises to your daily life, and to firearms safety.

(Guess what D’Artagnan started being able to do over the weekend.)

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Self-education: Calculus

One of the things I regret not having done yet is learning calculus. I know I’m perfectly capable of it – math has always been pretty easy for me, if I actually put my mind to it and focus – but I was psychologically unprepared for college. When I did take calculus, not only was I heavily distracted from school by some personal issues, but I also got shafted with a professor who was unintelligible ( a “visiting professor” from India – she spoke English well, but her accent was so thick she was barely intelligible in normal conversation, much less when rattling off (at warp speed) mathematical terms and formulae to a bunch of freshthings). Needless to say, I didn’t learn squat. I ended up changing majors right after that and not needing to take calculus anymore.

So, I recently decided to teach myself. I’m starting off with Calculus Made Easy, available free at Project Gutenberg. It seems to be a good text, and comes highly recommended. I’m also considering supplementing it with the Khan Academy, which I’ve also heard good things about.

I’ve already run into one problem, though. Since it’s been almost 15 years since I’ve had to do any kind of higher math (very basic algebra, at most), I’m finding myself, even in the first couple of chapters, having to go back and look up or review a lot of what I learned in high school – even some basic geometry.

As you might imagine, for a proudly self-professed geek like me, this is both irritating and somewhat embarrassing.

So, does anyone have any recommendations for a good review source for someone in my position?

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[Also, I added a new category: Self-education. I'll try to post occasional updates, in case anyone is interested.]

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