Derivatives as the Rate of Change
If a variable quantity y is some function of time t i. e., y = f( t), then small change in Δt time At have a corresponding change Δy in y.
Thus, the average rate of change =( Δy / Δt)
When limit At Δt→ 0 is applied, the rate of change becomes instantaneous and we get the rate of change with respect to at the instant x.
So, the differential coefficient of y with respect to x i. e.,( dy / dx) is nothing but the rate of increase of y relative to x.
Rolle’ s Theorem Let f be a real-valued function defined in the closed interval [ a, b ], such that
1. f is continuous in the closed interval [ a, b ]. 2. f( x) is differentiable in the open interval( a, b). 3. f( a)= f( b)
Then, there is some point c in the open interval( a, b), such that f‘( c) = 0.
Geometrically Under the assumptions of Rolle‘ s theorem, the graph of f( x) starts at point( a, 0) and ends at point( b, 0) as shown in figures.
The conclusion is that there is at least one point c between a and b, such that the tangent to the graph at( c, f( c)) is parallel to the x-axis.
Algebraic Interpretation of Rolle’ s Theorem
Between any two roots of a polynomial f( x), there is always a root of its derivative f‘( x).