I. Exponential Function

Transcription

1 MATH & STAT Ch. Eoetil Fuctios JCCSS I. Eoetil Fuctio A. Defiitio f () =, whee ( > 0 ) d is the bse d the ideedet vible is the eoet. [ = L 4 ] ties (Resf () = is owe fuctio i which the bse is the vible d the eoet is the costt) B. Ghs > 0 0 < < 1 Poeties of Eoetil Fuctios. The doi of eoetil fuctio is ll el ubes. b. The ge is ll ositive el ubes. c. Ech gh hs y-itecet (0, 1). Thee is o -itecet. d. The gh of y = is the eflectio, with esect to y-is, of the gh of e. the gh hs two bsic shes, deedig o whethe 1 o 01. y =. II. Lws of eoets (Cetificte Level). b. c. d. = = ( ) = ( b) = + e. ( ) = b b f. 0 = 1 g. 1 = h. i. 1 = q = q b THOMAS / 001 / NOTE / P.1

2 MATH & STAT Ch. Eoetil Fuctios JCCSS III. Eoetil Seies Coside the seies (-1)!, ( -1)( -)! ! ! ( -1)( -) L( - + 1)!! !! ! Whe teds to ifiity, 1 + 1!!! li e o =0!! +! + +! + Ele Ed - d e e i scedig owes of s f s the te i 4. [ANS: & ] THOMAS / 001 / NOTE / P.

3 MATH & STAT Ch. Eoetil Fuctios JCCSS IV. Alictios A. Cooud Iteest 1. Coouded Aully If iteest te is %.., d coouds iteest oce e ye, icil out is P, the At the ed of 1st ye, P(1) = P + P = ( 1 + ) P Iteest At the ed of d ye, P() = ( 1 + ) P + ( 1 + ) P = ( 1 + ) P At the ed of d ye, P() = ( 1 + ) P The, Afte yes, the out o deosit is P() = ( 1 + ) P How bout the followig cses. Coouded Mothly. Coouded Qutely 4. Coouded Dily THOMAS / 001 / NOTE / P.

4 MATH & STAT Ch. Eoetil Fuctios JCCSS. Cotiuously coouded iteest If iteest te is %, d coouds iteest ties e ye, icil out is P, the Fo iteest is coouded qutely ( = 4 ) 1 P ( ) = (1 + ) P P( ) = (1 + ) P P( ) = (1 + ) P At the ed of 1st ye, 4 P(1) = (1+ ) P 4 I geel, if cooudig ties e ye t iteest te is %, the At the ed of 1st ye, P(1) = (1+ ) P At the ed of d ye, P() = ( 1+ ) (1+ ) P = (1+ ) P The out o deosit fte yes is P() = (1 + ) P Coo vlues of : = 1 ul = 4 qutely = 1 othly = 65 dily THOMAS / 001 / NOTE / P.4

5 MATH & STAT Ch. Eoetil Fuctios JCCSS. Cotiuously coouded iteest (cot.) Fo cooudig cotiuously, P(1) = li (1 + ) P ( ) = li (1 + ) P = 1 li (1 ) + P = e P Aout o deosit fte yes, P() = e [e P] = e P Aout o deosit fte yes, P() = e P Eecise A citl of $ is ivested t te of 8% e u fo 5 yes. Fid the out ccuulted if iteest is coouded () yely, [$ ] (b) othly, [$ ] (c) cotiuously. [$ ] THOMAS / 001 / NOTE / P.5

6 MATH & STAT Ch. Eoetil Fuctios JCCSS C. Eoetil Gowth & Decy 1. Odiy Diffeetil Equtio: THOMAS / 001 / NOTE / P.6

7 MATH & STAT Ch. Eoetil Fuctios JCCSS. Eoetil Models (i) Eoetil Gowth (ii) Eoetil Decy Q(t) = Q o e t Q(t) = Q o e -t (iii) Leig Cuves (iv) Logistic Cuve Q(t) = BA e -t Q(t) = 1+ B -Bt Ae. Eles () It is ojected tht t yes fo ow, the oultio of ceti couty will be P(t) = 50 e 0.0t illio. (i) Wht is the cuet oultio? [50,000,000] (ii) Wht will the oultio be 0 yes fo ow? [91,105,940] (b) A ceti idustil chie deecites so tht its vlue fte t yes is give by fuctio of the fo Q(t) = Q o e -0.04t. Afte 0 yes, the chie is woth $8, Wht ws its oigil vlue? [0,000] (c) The te t which ostl cle c sot il is fuctio of the cle's eeiece. Suose the ostste of lge city estites tht fte t oths o the job, the vege cle c sot Q(t) = e -0.5 t lettes e hou? (i) How y lettes c ew eloyee sot e hou? [00] (ii) How y lettes c cle with 6 oths' eeiece sot e hou? [680] (d) Public helth ecods idicte tht t wees fte the outbe of ceti fo of ifluez, oitely Q(t) = 0 t 1+ 19e -1. thousd eole hd cught the disese. (i) How y eole hd the disese whe it fist boe out? [1] (ii) How y hd cught the disese by the ed of the secod wee? [7.4] (iii) If the ted cotiues, oitely how y eole i ll will cotct the disese? [0] THOMAS / 001 / NOTE / P.7