On Monday, July 7, 2014 2:56:14 PM UTC-4, iwa...@gmail.com wrote: > 1. Does the slope of a straight line ever change? >
No. Except for a vertical line, the slope is defined everywhere. Likewise the derivative of a linear function is defined everywhere -- contrary to John Gabriel's wacky definition.
(The cowardly John Gabriel snipped all my answers, so I have had to reinsert them.)
> > > No. Even a vertical line has a slope in terms of an angle.
But not in terms of rise/run between 2 points.
> The slope is defined everywhere. Likewise the slope of a straight line is defined everywhere -- in accordance with John Gabriel's New Calculus definition. >
Slope is not defined between (0,0) and (0,1).
> > > > > > 2. Is the derivative a special kind of slope? > >
This question makes no sense.
> > This question makes a lot of sense. Slope always means the inclination of a straight line with the horizontal or vertical.
> Derivative always means the slope of a special kind of straight line called a tangent line. >
Actually, the derivative f' is always means the instantaneous rate of change of a function f wrt to its dependent variable. The tangent at (x,f(x)), for example, is just the straight line through this point with slope f'(x).
So, neither your question, nor you answer makes any sense, John Gabriel.
> > > 3. Do the tangent lines to any function have slopes that ever change? >
The tangent at every point on a straight line is just that line itself.
> > No. The slopes of tangent lines never change. >
Of course they do. If f(x)=x^2, then f'(0)=0 and f'(1)=2. The tangents change.
> > > > > > 4. Is an instantaneous rate of change real? > > > > No. It is a misconception by ignorant mathematicians whom John Gabriel has corrected. >
You have never corrected anything, John Gabriel.
> > > > After all, f(x)=5x^6 - 3x^5 + 2 has not changed in the last trillion light years or before that. It's tangent lines have always had the same predictable slopes. >
This argument makes no sense.
> > > This argument makes perfect sense. If we plotted f(x) one trillion light years ago and we plotted it today, it would have exactly the same tangent lines whose slopes are always the same, because ALL tangent lines are straight lines.
Again, neither your question, nor your answer makes any sense, John Gabriel.