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Comment by baby | Hacker News Reader

I'm trying to explain what he meant. Which step don't you understand specifically in my comment?

I'm trying to understand why there's f(a) and f(b), why is b being subtracted from x, why it's divided by a-b, why there's a similar thing for f(a), why the two are being added, why the zero values are important, and what the whole thing actually means.

It's demonstrating how to find the line that passes through two points x=a, x=b on the function y=f(x).

Without having read the actual page, I assume he's going to move a and b close to eachother to approximate the tangent of the function at a point.

edit: In higher level math, you use circles/spheres or parabolas/paraboloids to approximate functions, but in high school level calculus you stick to using a straight line to approximate a function

Unfortunately, he doesn't do that. There's an even more complicated formula under that which is supposed to be related to the slope of some line and a ratio between b and a (I think?) and then I'm guessing he reduces it, but doesn't explain how he did it.

I've tried learning calculus 3-4 times during my life, using materials for an "absolute beginner", and this has always been my experience, as if one were teaching programming by going from "This is a variable. It can store data." directly to "A monad is a monoid in the category of endofunctors." To this day I have no idea what calculus even is, or what it's for.

You want a linear function f(x) such that f(a)=A and f(b)=B given a, b, A, B. A valid f(x) (actually it's the only one) is defined as

             x-b     x-a
    f(x) = A --- + B ---
             a-b     b-a

You can calculate f(a) and f(b) to confirm that it works.

However, I don't find this kind of guide any good for learning math...

It’s simpler if you use 0 and 1 instead of a and b. If we want a linear function such that

f(0) = A, f(1) = B,

then the unique solution is

f(t) = (1-t)A + tB

The more general version with a and b is just composing this with a transformation of [a, b] into [0, 1]

Relevant keywords: linear interpolation, affine combination, barycentric coordinates.

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