ETIMOLOGY - a IEZZIaiezzi.it/teaching/calculus-spring2018/slides/intro... · 2020-05-12 ·...

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INTRODUCTION TO CALCULUS

Transcript of ETIMOLOGY - a IEZZIaiezzi.it/teaching/calculus-spring2018/slides/intro... · 2020-05-12 ·...

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INTRODUCTIONTOCALCULUS

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ETIMOLOGY

ThewordcalculuscomesfromLatinandmeans«asmallpebbleorstoneusedforcounting»

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BRIEFHISTORY

Moderncalculuswasdevelopedin17th-

centurybyNewtonandLeibnizindipendentlyofeachother(eveniftherewas

agreatcontroversy.

Newton:Oirsttoapplycalculustogeneral

physics.

Leibniz:developedmuchofthenotationused

incalculustoday

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Whatdoescalculusstudy?

Calculusisthestudyofthechangeanditstudieschangebystudyingistantaneouschange(overatinyintervaloftime).

Example:MotionofanobjectalongaOixedpath

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MotionofanobjectalongaOixedpath

•  LetusOixapointonthepath.Atanytimewecandescribetheposition(=distancefrometheOixed

point)oftheobject:positionisafunctionofthetime.

•  Whatdoesitchangeinthisexample?Thepositionvarieswithtime.

•  Andhowdoesthepositionchangewithtime?Thisdependsonthevelocityoftheobject.

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Averagevelocity…

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…vsinstantaneousvelocity

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Twoproblems

1)  Findtheinstantaneousvelocitybyknowingthefunctionposition(calledmoreingeneralthederivativeofthefunction)

DIFFERENTIALCALCULUS

2)Findthepositionbyknowingtheistantaneousvelocityatalltime(ormoreingeneral,Oindthefunctionbyknowingitsderivative).

INTEGRALCALCULUS

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CALCULUS

DIFFERENTIALCALCULUS

INTEGRALCALCULUS

FUNDAMENTALTHEOREMOFCALCULUS


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