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 ).