Substituting ,
Thusthe slopeof the tangentline at for the curve is To find the equationof the
tangentline, we simplyuse the point-slopeformula,
where
whichis the equationof the tangentline.
AverageRatesof Change
The primaryconceptof calculusinvolvescalculatingthe rate of changeof a quantitywith respectto another.
For example,speedis definedas the rate of changeof the distancetravelledwith respectto time.If a person
travels120 milesin four hours,his speedis 120/4= 30 mi/hr. This speedis calledtheaveragespeedor the
averagerate of changeof distancewith respectto time.Of coursethe personwho travels120 milesat a
rate of 30 mi/hrfor four hoursdoesnot do so continuously. He musthavesloweddownor spedup during
the four-hourperiod.But it doessuffice to say that he traveledfor four hoursat an averagerate of 30 miles
per hour. However, if the driverstrikesa tree, it wouldnot be his averagespeedthat determineshis survival
but his speedat theinstantof the collision. Similarly, whena bulletstrikesa target,it is not the average
speedthat is significantbut itsinstantaneousspeedat the momentit strikes.So here we havetwo distinct
kindsof speeds,averagespeedand instantaneousspeed.
The averagespeedof an objectis definedas the object’s displacement∆x dividedby the time interval∆t
duringwhichthe displacementoccurs: