If you have a series of (nx + m) terms multiplied together (where each n and m can be different), just count the number of times x comes up (let’s call that z), and the order of the function will be x^z. For z = 3, that’s cubic. None of the other terms matter for the expansion. If there’s an x^3 and no higher powers of x, it will always be cubic and the coefficients of any of the terms are irrelevant.
If you’re studying computer science, you can think of it like big O notation.
And for the other part, one trick for questions like that is to look at what happens when you substitute an equality they are talking about, specifically looking for terms that result in 0s or 1s and can be cancelled out of the equation. If you’re not given actual values for any coefficients, there’s not much else interesting you can do to a polynomial like that, other than maybe take the derivative or integrate.
It’s not too bad, once you consider that everything in each term is a constant value, except for “x” itself.
So the numerator of each term is the product of three linear factors, like (x-4)(x-2)(x-6), which should produce a cubic, like x³ - 12x² + 44x - 48. Then the denominator of each term is a pure constant, so it would be like taking that cubic and dividing it by 4, getting 0.25x³ - 3x² + 11x - 12. Then the yₙ terms are also constants, so no different than doing something like multiplying by 2, getting you something like 0.5x³ - 6x² + 22x - 24, if I take that example a bit too far. And at that point it’s just the sum of four cubics, which will be cubic as long as the x³ terms don’t perfectly cancel out - which I believe would only happen if the four pairs of points used to make the function were all perfectly laying on the same line or parabola.
The construction’s also pretty clever: OP said the point was to fit the function to the four points (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄). Let’s say we set x = x₂, then. Because (x-x₂) appears in the numerator of every term but the second term, every term but the second term will have a 0 in its numerator and cancel out - so we only need to consider the second term. Its numerator is then (x₂-x₁)(x₂-x₃)(x₂-x₄) - exactly the same as its denominator. So they both cancel out, leaving only y₂ - meaning we get P₃(x₂) = y₂, as desired.
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.
Guess I’ll just come out and say it. I’m a mixed fraction fan. 23+2/3 instantly tells you it’s “23 and a bit”, unlike 74/3, and it’s more accurate than 23.67.
For me the problem is notation, putting a number in front of a fraction usually means multiplication and when giving a solution in anything but maths, the needed accuracy can be achieved with decimals
If a recipe calls for 3 and 3/4 cups flour, I know right away I need three 1 cup scoops of flour and one 3/4 cup scoop.
If it calls for 15/4 cups, now I need to calculate how many one cup scoops it is and also what the additional remaining fraction is in addition to how much I’ve actually measured out so far.
The more numbers you need to keep in your head when following a recipe, the more likely you are to make a mistake.
This is a great example of why volumetric recipes are inferior. With grams it’s just a single weight standard across the board. I’d much rather just use a scale, when a recipe call for 50g I know I need… a scale. When a recipe calls for 75g I know I need… a scale. No need for dirtying a bunch of inaccurate measuring implements.
The first 10 minutes cover the sort of problems people have when baking cookies, and - not much of a spoiler - ultimately reaches the conclusion that measuring ingredients by weight is better.
I haven’t seen it, but I’ve been baking for a long time and came to the same conclusion. One of my exes couldn’t get her mom’s pupusa recipe right and kept saying her mom does it differently every time, a pinch here and a splash there. So I just stuck her bag of Masa and everything else on a scale and copied her recipe, even extra dustings and splashes of water, to the gram. They all looked at me like I was an idiot. Guess who made identical pupusas every single time?
This is from my probability class in an actuarial sciences mba. Xj is a random variable of the value of a claim, N is another random variable of the number of claims in a year, S is another random variable of the total sum of claims in a year.
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