My historic geometric theorem which proves that the mainstream formulation of calculus is flawed.

Link to the article explaining the theorem:

https://drive.google.com/open?id=1RDulODvgncItTe7qNI1d8KTN5bl0aTXj

Link to article explaining how the theorem fixes the flawed mainstream formulation by removing all ill-formed concepts such as infinity, infinitesimals and the rot of limit theory:

https://drive.google.com/open?id=1uIBgJ1ObroIbkt0V2YFQEpPdd8l-xK6y

Link to download the applet:

https://drive.google.com/open?id=1ON1GQ7b6UNpZSEEsbG14eAFCPv8p03pv

An easy proof of the theorem (you need to refer to the geometry to see what f1 and f2 mean):

We can prove that f'(x) = [f(x+h)-f(x)]/h - Q(x,h) as follows.

Let t(x) be the equation of the tangent line which we don't yet know.

Then [t(x+h)-t(x)]/h = f2/h = f'(x) from the geometry theorem.

This means that f'(x) contains no terms in h because t(x) is a straight line and we know that h is a factor of every term in t(x+h)-t(x).

But f1/h = [f(x+h)-f(x)-f2]/h and so f2/h = [f(x+h)-f(x)]/h - f1/h

Thus, f'(x)= [f(x+h)-f(x)]/h - f1/h which implies that f1/h = Q(x,h).

So, Q(x,h)=[f(x+h)-f(x)]/h - f'(x).

Since the secant line slope [f(x+h)-f(x)]/h contains the sum of f'(x) and Q(x,h), it follows that Q(x,h) has terms with factors of h because f'(x) consists of terms that don't contain h.