Dmis o Mrie Biologil Resores A FUNCTION * * * REVIEW OF SOME MATHEMATICS * * * z () z g(,) A tio is rle or orml whih estlishes reltioshi etwee deedet vrile (z) d oe or more ideedet vriles (,) sh tht there is sigle vle o the deedet vrile or eh set o vles or the ideedet vriles. LIMIT OF A FUNCTION ( ) A or ( ) A s I () is siietl lose to A whe is lose to, the A is sid to e the it o () s rohes. A tio m ot hve iite it t oe or more oits. Sh oits re lled siglrities. For emle, the tio ( ) / hs siglrit t the oit 0. Some Bsi Rles or Tkig Limits Give tht is ostt, [ ( ) ] A ( ) A, d g( ) B, the: [ ( ) ± g( ) ] A ± B [ ( ) g( ) ] A B [ ( ) g( ) ] A B, rovided B 0. Right Hd d Let Hd Limits ( ) Here rohes rom the right (rom ove). Iiite Limits ( ) ( ) Here rohes rom the let (rom elow). Here rohes ositive iiit. ( ) Here rohes egtive iiit. Note tht iiit ( ) is ot mer. Positive iiit is lrger th ositive mer; egtive iiit is smller th egtive mer. Review o Some Mthemtis -
Dmis o Mrie Biologil Resores Some Usel Limits [ / ] 0 or > 0 [ / e( ) ] 0 or > 0 { NOTE: e() is the sme s e } CONTINUOUS FUNCTION A tio () is sid to e otios t the oit i d ol i ( ) A d ( ) A d ( ) A. For emle, the tio ( ) /( ) is disotios t the oit ese the right hd d let hd its re dieret. EXPONENTS I the eressio ower. Some Bsi Rles or Eoets t. the mer is lled the se d the mer is lled the eoet or 0 or 0 q q q ( ) q / / q / q LOGARITHMS The eressio ( ) log deotes the logrithm o to the se. I The mer is the tilogrithm o to the se. Some Bsi Rles or Logrithms, the ( ) log. ( ) 0 log ( ) log ( ) log ( ) log log log ( / ) log ( ) log ( ) log ( ) log ( ) ( ) log ( ) log ( ) Review o Some Mthemtis -
Dmis o Mrie Biologil Resores Ntrl Logrithms The eressios log ( ) or ( ) e l deote the trl logrithm o. The ostt e is tht mer X or whih the re der the rve ( ) /, over the rge rom to X, is eql to oe. It is lso eql to the ollowig: e.788k e L L 3 3 4! Bse 0 logrithms re the other oes tht re ommol sed. These two kids o logrithms re relted the ollowig, l ( ) l( 0) log ( ).303 ( ) 0 log0 DERIVATIVE OF A FUNCTION. B deiitio, the derivtive o the tio ( ) ( ) ( ) 0 t the oit is eql to rovided this it eists (i.e., is ot iiite) d is ideedet o whether the roh to the it is rom the let or the right. The derivtive is jst the sloe o the tio s grh. The d derivtive m e deoted ', '( ), or. I the tio () hs derivtive or eh o the itervl, the the tio is sid to e dieretile o this itervl. I () is dieretile over itervl, the it is lso otios over tht itervl. However, tio m e otios over itervl d et ot e dieretile. Some Bsi Rles or Dieretitio Give tht,, d v re ll dieretile tios o, the: d d / ( d) d d { The Chi Rle } d, ostt ' 0 ' v K ' ' v' K Review o Some Mthemtis - 3
Dmis o Mrie Biologil Resores ' ' v ' v' v ' / v ' v ' v' / v ( ),,,3,... ' ' ' l( ) ' ( / ) ' ( ) ' log ( e) ( / ) ' log ( ) e e e ' e ' e ' Higher Derivtives Prtil Derivtives ( ) l e ' l( ) d d Seod derivtive ''( ) ' ' Third derivtive ( ) ' '' d 3 d d d 3 B deiitio, the rtil derivtive o tio (,) with reset to t the oit 0 (, ) (, ) (3) is d the rtil derivtive o (,) with reset to t the oit is 0 (, ) (, ) Totl Dieretil Sose z (, ). The totl dieretil o tio z is give dz d Review o Some Mthemtis - 4
Dmis o Mrie Biologil Resores Imliit Dieretitio Sose z (, ) 0, the dz d 0 d d / MAXIMA AND MINIMA OF A FUNCTION I the tio () is dieretile o the itervl d i () hs reltive mimm or miimm t the oit 0 o the itervl 0, the '( 0 ) 0. At mimm or miimm, the sloe o the tio is lws zero. Those vles o or whih '( ) 0 re lled the sttior, or ritil oits, o the tio. I 0 is sttior oit, so tht '( 0 ) 0, the the tio (): hs mimm vle t 0 i '( ) hges rom ositive to egtive s ireses throgh 0, whih is eqivlet to hvig egtive seod derivtive, ''( 0 ) < 0 ; hs miimm vle t 0 i '( ) hges rom egtive to ositive t 0, whih is eqivlet to hvig ositive seod derivtive, ''( 0 ) > 0. hs iletio oit t 0 i '( ) does ot hge sig t 0, whih is eqivlet to hvig seod derivtive o zero, ''( 0 ) 0. SERIES Arithmeti Series Geometri Series ( ) / 3 K X X X X K X K X K X K X ( X ) Tlor Series M tios e writte s iiite sm (or series) o terms ivolvig the derivtives o the tio. The Tlor series esio o () rod the oit is ( ) ( ) ( ) '( ) ( ) ( ) ( ) ( ) ( ) K '' K!! Review o Some Mthemtis - 5
Dmis o Mrie Biologil Resores The series m ot overge or ll vles o. The esio with 0 is lso kow s Mlri series. I m ses we roimte tio its trted Tlor series. For tio o two vriles, the Tlor series is give (, ) (, ) ( ) (, ) ( ) (, ) [( ) (, ) ( ) (, ) ( ) ( ) (, ) ] K! where deotes the seod rtil derivtive o with reset to d deotes the rtil derivtive o with reset to d. Some Series Esios For For < < e e( ) ( ) ( ) < l( ) l( ) K 3 3! 3 3! 3 K INDEFINITE INTEGRALS Sose F() is tio d () is the derivtive o F(). The F() is lled the ideiite itegrl, or tiderivtive, o (). Note tht, ese the derivtive o F() is lso eql to the derivtive o F() ls ostt, the ideiite itegrl F() does ot iqel deie (). Itegrtio d dieretitio re relted the ollowig Fdmetl Rle o Clls, d ( ) F( ) C [ F( ) C] ( ) where C is ritrr ostt, reerred to s the ostt o itegrtio. Some Bsi Rles or Ideiite Itegrls d [ ( ) ] ( ) C Sose d re ostts, d d g re tios o. C [ ( ) g( ) ] ( ) g( ) C C e e C or Review o Some Mthemtis - 6
Dmis o Mrie Biologil Resores e e ( ) ( ) l CLi > 0 l CLi < 0 l ( ) C C Sometimes itegrl e simliied sig rorite trsormtio. dg g gd C { Itegrtio Prts } ( ) ( g) dg C where g DEFINITE INTEGRALS Sose F() is ideiite itegrl o () d () is otios o the itervl the the deiite itegrl o () is give ( ) F( ) F( ) The deiite itegrl is the re der the grh o () o the itervl., Some Bsi Rles or Deiite Itegrls ( ) ± g( ) ( ) g( ) ± ( ) 0 ( ) ( ) Imroer Itegrls I () otios o (, ), the ( ) ( ) F( F( ) ) I () otios o (, ), the ( ) ( ) F( ) F( ) I () otios o eet or siglrit t, the ε ( ) ( ) ( ) ε 0 ε ' 0 ε ' Review o Some Mthemtis - 7