Noticethat the points(t 0 ,x 0 ) and (t 1 ,x 1 ) lie on the position-versus-timecurve,as Figure1 shows.This
expressionis also the expressionfor the slopeof a secantline connectingthe two points.Thuswe conclude
that the averagevelocityof an objectbetweentimet 0 andt 1 is representedgeometricallyby the slopeof
the secantline connectingthe two points(t 0 ,x 0 ) and (t 1 ,x 1 ). If we chooset 1 closetot 0 , then the average
velocitywill closelyapproximatethe instantaneousvelocityat timet 0.
Geometrically, the averagerate of changeis representedby the slopeof a secantline and the instantaneous
rate of changeis representedby the slopeof the tangentline (Figures2 and 3).
AverageRateof Change(suchas theaveragevelocity) The averagerate of
changeofx=f(t) over the time interval[t 0 ,t 1 ] is the slopemsec of the secantline
to the points(t 0 ,f(t 0 )) and (t 1 ,f(t 1 )) on the graph(Figure2).