- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive s we wt) if is close eough to The defiitio c lso e geerlized for i similr mer) The it f ( ) does ot eist if f( ) does ot pproch uique vlue (umer) whe pproches (for emple whe the fuctio pproches differet vlues from left th from right) Note lso tht f ( ) L does ot sy ythig out the vlue of f( ) t : f( ) ( f( ) c e differet from L eve if f ( ) L ): f ( ) L oly sys wht hppes whe pproches I strogly suggest the followig recipe for clcultig its: Due to lgreric properties of its, whe you clculte it f ( ), we usully strt y evlutig f( ) This will either give you the vlue of the it (if f( ) evlutes to rel umer), or ide o how to procede et For polyomils (d ll other elemetry fuctios ) it c e show, usig the defiitio, tht f ( ) f ( ) Grphig the fuctio d tkig vlues for closer d closer to re lso gret strt i clcultig its Algericlly, for its of the form from ottom to eite the idetermitio m m m m we usully fctor m from the top d We hve lert simple its,,, which we will use, comied with opertios with its, d sometimes we use the defiitio of the derivtive to clculte it
Emples: Clculte the followig its: ) f) 9 9 ) 6 4 5 g) d) h) 5 4 si( ) e) i) 5 cos( ) h) Other prolems i Chpter The derivtive of fuctio : Defiitio: Cosider cotiuous fuctio f : D R, the the derivtive of the fuctio f t poit D, deoted y f '( ), is defied s : () f f ( ) f ( ) '( ) (the istteous rte of chge of f t ), which c lso e regrded s the slope of the tget lie to the grph of f( ) t Altertively, ( ) Emples: f f ( h) f ( ) '( ) h h ) Cosider f ( ) The for y, f '( ) (y usig the formul () ove) This fuctio f hs costt rte of chge of t ech poit Altertively, the slope of the tget lie t ech poit is costt: t(45 ) Grph f( ) d clculte the rte of chge d the slope of the tget lie t differet poits o the grph ) Cosider f ( ) The, for ritrry, f '( ) (y usig gi () or () ove) This time the derivtive depeds o Iterpret this s the slope of the tget lie t vrious poits (-/,, / ) o the grph This lso gives: ' This wy (y usig the formul ()) we c fid the derivtives of my fuctios (see tle of derivtives memorize) Rememer the derivtives of vrious fuctios sed o formul, rememer the sum rule, the
product rule, the quotiet rule, the chi rule, d clculte : ', cos cos ' usig the chi rule Emples: Work prolems from Sectio, Chpter d Chpter 8 Ati-derivtives: Sometimes we re iterested i the iverse prolem: If f( ) is see s rte of chge of some ukow fuctio, the rte of chge of wht fuctio is it (the derivtive of which fuctio gives f( ))? Defiitio: Cosider cotiuous fuctio f : D R A ti-derivtive of f( ) is fuctio F ( ) such tht () F '( ) f ( ) NOTE: F ( ) is lso deoted y f ( ) d Bsed o this defiitio d o the tle of derivtives, we c fid the ti-derivtive of my fuctios Emples: Fid d, d, r d, si( ) d NOTE: A ti-derivtive is ot uique, ut uique up to ritrry costt If F ( ) is ti-derivtive of f( ), the F( ) C (where C is ritrry costt) represets the etire fmily of ti-derivtives of f( )) Opertios: f ( ) g( ) d f ( ) d g( ) d f ( ) d f ( ) d f ( ) g( ) d f ( ) d g( ) d (true?) See tle of itegrls, memorize, check d do eercises See homework t the ed of Sectio 8
4 The defiite itegrl Cosider cotiuous fuctio Defiitio: The defiite itegrl f ( ) d is defied s f ( i) i (tht is the it of the Riem i sum), where {,, } represets prtitio of the itervl [,], i, i i, d ritrry poit i [ i, i] (my times the left, right or midpoit i this itervl) As such, the defiite itegrl is (for positive cotiuous fuctios) the re uder the grph of f( ) etwee d This re (sy A) is pproimted y the sum of the res of the rectgles of height i is f d width i, which pproch A s ( i ) Emple: Let f ( ) d Approimte y Riem sum with 6 equl suitervls d s the right edpoit i ech suitervl i chose Solutio: d f f f 6 6 ( ) ( ) ( ) 4 6 6 6 6 6 Note tht the ect vlue of d is the right edpoits i icresig fuctio ) Altertively, d (this vlue is overestimted y the sum ove, sice we tke ( )( ) f f f qed 6 Some properties of the defiite itegrl: f ( ) d f ( ) d c, where c (, ) f ( ) d f ( ) d f ( ) d c 4
Emples: Homework, some prolems from 7, t the ed of sectio 4 4 The first fudmetl theorem of clculus It c e show tht: Theorem (the first fudmetl theorem of clculus): If f cotiuous fuctio f :[, ], d let e ritrry poit i [,, ] the d () f ( t) dt f ( ) d Proof: This c e show ituitively usig the defiitio of the derivtive, d pproimtig h f ( t) dt hf ( ) A more rigorous proof uses tht m( ) f ( ) d M ( ), where m d M re the solute miimum d the solute mimum of f o [,, ] d the tht mh F( h) F( ) Mh (see lso ook, d tke emple similr with prolem 7 t the ed of sectio 4) This is very importt result, which reltes derivtives d itegrls Emples: Prolems t the ed of the sectio 44 The secod fudmetl theorem of clculus d the Method of Sustitutio The secod fudmetl theorem of clculus is the etremely useful tool tht we hve for evlutig defiite itegrls (s clcultig its of Riem sums c e tedious ) Theorem (the secod fudmetl theorem of clculus): If f cotiuous fuctio f :[, ], the f ( ) d = F( ) F( ), where F '( ) f ( ), tht is F ( ) is y ti-derivtive of f( ) Proof: 5
Defie G( ) f ( ) d From the first F T C, it follows tht Gis ( ) ti-derivtive of f( ) Therefore, we c show tht G( ) G( ) F( ) F( ), where F ( ) is y ti-derivtive of f( ), Sice G( ) G( ) f ( ) d, it follows clerly tht F( ) F( ) f ( ) d f( ) Theorem (Sustitutio Rule for Idefiite Itegrls):, where ( ) Let g e differetile fuctio, d let f e cotiuous fuctio (o their domis), the f ( g( )) g '( ) d F( g( )) C, where F '( ) f ( ) F is y ti-derivtive of Proof: Use the chi rule for derivtives to show tht F( g( )) is ti-derivtive of f ( g( )) g '( ) Theorem (Sustitutio Rule for Defiite Itegrls): Let g e differetile fuctio, d let f e cotiuous fuctio (o their domis), the g ( ) f ( g( )) g '( ) d f ( u) du F( g( )) F( g( )), where F '( ) f ( ) g( ) Proof: Use Theorem d Theorem Emples: Prolems 78 t the ed of the sectio 44, Review Prolems t ed of Chpter 4 5 The re of ple regio Sice f ( ) d represets the re uder the grph of f( ) etwee d, the if we wt to compute the re of ple regio etwee two curves (ssume oth positive, oe elow the other), tht c e clculted s the re uder the first curve mius the re uder the secod, tht is ( f ( ) g( )) d This formul c e pplied i similr mer if the curves re log the is: f ( y) d g( y) Emples: Prolems t the ed of the sectio 5 6