INTRODUCTIONTOCALCULUS

ETIMOLOGY

ThewordcalculuscomesfromLatinandmeans«asmallpebbleorstoneusedforcounting»

BRIEFHISTORY

Moderncalculuswasdevelopedin17th-

centurybyNewtonandLeibnizindipendentlyofeachother(eveniftherewas

agreatcontroversy.

Newton:Oirsttoapplycalculustogeneral

physics.

Leibniz:developedmuchofthenotationused

incalculustoday

Whatdoescalculusstudy?

Calculusisthestudyofthechangeanditstudieschangebystudyingistantaneouschange(overatinyintervaloftime).

Example:MotionofanobjectalongaOixedpath

MotionofanobjectalongaOixedpath

•  LetusOixapointonthepath.Atanytimewecandescribetheposition(=distancefrometheOixed

point)oftheobject:positionisafunctionofthetime.

•  Whatdoesitchangeinthisexample?Thepositionvarieswithtime.

•  Andhowdoesthepositionchangewithtime?Thisdependsonthevelocityoftheobject.

Averagevelocity…

…vsinstantaneousvelocity

Twoproblems

1)  Findtheinstantaneousvelocitybyknowingthefunctionposition(calledmoreingeneralthederivativeofthefunction)

DIFFERENTIALCALCULUS

2)Findthepositionbyknowingtheistantaneousvelocityatalltime(ormoreingeneral,Oindthefunctionbyknowingitsderivative).

INTEGRALCALCULUS

CALCULUS

DIFFERENTIALCALCULUS

INTEGRALCALCULUS

FUNDAMENTALTHEOREMOFCALCULUS