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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

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.

Xeanort 7y ago
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...
llamaz 7y ago
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
- kstenerud 7y ago
  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.
jacobolus 7y ago
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.

Xeanort 7y ago

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...

llamaz 7y ago

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

kstenerud 7y ago
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.

jacobolus 7y ago

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.