No, I got that part, but I don't think I understand the significance of the indexed y values and their relationships to the indexed x values. The criterion seems to suggest that P3(xn)=yn for each, but that strikes me as something that is defined as a constraint rather than something that is to be proved. Also, I woke up then and now so that might be playing a factor in my confusion.
OK, you got it then, I believe.
P3 is specifically built so that P3(xn)=yn for n from 1 to 4. The proof lies in its construction. I guess the sentence can be understood as "we know it works because we built it like that, however you may verify it yourself"
I feel like the sentence also means "it's kinda obvious when you think about it, so we won't explain, but it's actually important, so you probably should make sure you agree".
The function should be cubic, so you should be able to write it in the form "f(x) = ax^3 + bx^2 + cx + d". You could work out the entire thing to put it in that form, but you don't need to.
Since there are no weird operations, roots, divisions by x, or anything like that, you can just count how many times x might get multiplied with itself. At the top of each division, there are 3 terms with x, so you can quite easily see that the maximum will be x^3.
It's useful to know what the values x_i and x_y are though. They describe the 3 points through which the function should go: (x_1, y_1) to (x_3, y_3).
That also makes the second part of the statement ready to check. Take (x_1, y_1) for example. You want to be sure that f(x_1) = y_1. If you replace all of the "x" in the formula by x_1, you'll see that everything starts cancelling each other out. Eventually you'll get "1 * y_1 + 0 * y_2 + 0 * y_3", thus f(x_1) is indeed y_1.
They could have explained this a bit better in the book, it also took me a little while to figure it out.
I know that, but @Urist said that most of us here are bad at basic algebra. Not that I'm great at it (I had an 8 average in all 4 maths in uni), but I do believe that most of here were correct. This is unsolvable using classic algebra.
Just to be clear: the comment was meant in a lighthearted way and not meant to be taken personally, especially toward those who are confident with their math skills.
There are many such ways to memorize conversion ratios. Admittedly, this one is particularly cool, since you can construct it from the fairly trivial fibonacci series. But I still feel, it's no replacement for the actual solution; get rid of imperial and adopt metric.
Fun fact: If you have a scientific calculator (literal or app) but no other conversion tool available, the conversion factor between miles and kilometres is almost exactly ln 5. Disturbingly close in fact.
That's fewer keypresses than generating the Golden ratio or working out Fibonacci numbers. But if all you have is your head then, yeah, the Fibonacci trick is good enough in a pinch.
Yes but not in terms of market share, unfortunately. Monopolies suck. The best solution would be a standard open source operating system that would run on ARM hardware with a wide range of RAM and display support, like Java apps on midrange phones in the 2000s. And a QWERTY keyboard when rotated (we need the law changed for that to be legal in US schools). And an OK, as well as a BACK or EXIT button like CASIO. On the TI, I've seen programs that use MENU, DEL, CLEAR, ALPHA, SHIFT+MENU (QUIT), ON or a menu item confirmed by SHIFT (the usual OK key in games) to exit.
There are some open source graphic calculator projects but they aren't mass-produced and none are exam-approved. Software support is lacking but TI-84 Plus emulators exist so if someone rewrites TI-OS and the ROM (or requires people to dump it like the current ones), it could run that.
Did you know the humble Nokia 3410, the first Java phone, would dither games meant for higher-resolution color screens? Of course, programs for these calculators could use semi-vector graphics like the Web, or detect screen resolution and support the common ones natively, in a single or multiple executables like PocketPC software.
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