[size=100][b]Interpretation:[/b][br]Suppose [math]P[/math] is a continuous function on [math][a,b][/math]. [br]Then the function [math]S[/math] defined by [math]S(t)=\int_a^tP(x)dx[/math]is interpreted as [br][b]the measure of the change for a quantity[/b] [math]y[/math] [b]during the interval [/b][br][math][a,t][/math], [b]when[/b]  [math]\frac{dy}{dx}=P(x)[/math] [b]for the interval [/b][math][a,b][/math]. [br]The theorem's interpretation of the derivative with mapping diagrams says that [b]the rate of change of[/b] [math]S[/math] [b]at time[/b] [math]x=c[/math] [b]is the value of [br]the derivative of [/b][math]y[/math]; namely, [math]\frac{dy}{dx}|_{x=c}=S'\left(c\right)=P(c)[/math] . [/size]