If a continuous function has derivatives at all interval points, at one or more points derivative has slope equal to slope of straight line passing through interval endpoints {mean value theorem}. For interval (a,b), line slope is (f(b) - f(a)) / (b - a). Point (x,y) has a <= x <= b and dy/dx = (f(b) - f(a)) / (b - a).
For two functions over interval, at one or more points derivative ratio equals difference ratio {extended law of the mean} {law of the mean extended}. For interval (a,b), and functions f(x) and g(x), at (x,y), (f(b) - f(a)) / (g(b) - g(a)) = f'(x) / g'(x).
If continuous function has two roots over interval and has derivatives at all interval points, first derivative equals zero at one or more points {Rolle's theorem} {Rolle theorem}.
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Date Modified: 2022.0225