Exponential Functions

Transcription

1 Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,. and. 8., Dining or irrational is too diicult now. Tis grap rprsnts ponntial growt sinc it is incrasing as incrass. 0 MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd y y

2 Eponntial Growt: All unctions a wr a >, ibit ponntial growt. y y y 0 9 y y y. 0 y. MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

3 Eponntial Dcay: All unctions a wr 0 < a <, ibit ponntial dcay. y 0. 6 y 0. y y y y 0. 6 y 0. 8 y MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

4 THE Eponntial Function Tr is only on ponntial unction tat as a slop o at t point 0,. Tis is calld t ponntial unction and w dnot t bas wit t lttr. W will ind out tat ~ y m MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

5 Important rlatd unctions ar k I k > 0, ponntial growt unctions. k or ral k. 0 y k I k < 0, ponntial dcay unctions * 0.* MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

6 Drivativs o Eponntial Functions As it turns out: Rcall tat ponntial unctions y a or a > look as ollows: MATH 80 Lctur 6 o Ronald Brnt 08 All rigts rsrvd.

7 MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

8 Rcall tat T ponntial unction as a slop qual to at 0. A slop o at 0, mans 0, wic w can writ as 0 lim 0 0 Wit, 0 0, 0 lim 0 wic givs lim 0 Tis quation is t rquirmnt or t unction to av a slop o at t point 0,, wic is usd nt: MATH 80 Lctur 8 o Ronald Brnt 08 All rigts rsrvd.

9 MATH 80 Lctur 9 o Ronald Brnt 08 All rigts rsrvd. T gnral drivativ is writtn as lim 0 Wr so lim lim lim Factoring out an, and rmoving it rom t limit givs lim 0 And so d d Using t cain rul u u u d d in particular k k k d d

10 MATH 80 Lctur 0 o Ronald Brnt 08 All rigts rsrvd. Eampls: a. 0 b c. d. 8

11 MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.. g sin cos sin cos sin.

12 MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd. i j tan sc tan sc tan k

13 Eampl: grap it. Lt. Find out vryting about tis unction and tn Domain:, Intrcpts: y-intrcpt is 0, 0 intrcpts: Solv 0, so 0, sinc > 0 Symmtry: No Symmtry Asymptots: It can b sown lim 0, so tr is a orizontal asymptot as approacs ininity.,6 Intrvals o Incras and Dcras. Rl. Etrma Sinc, is a stationary point. Also, > 0 or <, and < 0 or >. So incrass on <, and dcrass on >. T is a rlativ ma at, MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

14 ,8 Sinc, is an inlction point, bcaus > 0 or >, and < 0 or <. So is concav up on >, and concav down on <. T unction looks lik: y MATH 80 - Lctur o Ronald Brnt 08 All rigts rsrvd.

15 Eampl: Lt. tn grap it. Find out vryting about tis unction and Domain:, Intrcpts: y-intrcpt is 0, 0 intrcpts com rom solving 0, so 0. Symmtry: No Symmtry Asymptots: It can b sown lim 0, so tr is a orizontal asymptot as approacs ininity.,6 Intrvals o Incras and Dcras. Rl. Etrma Sinc, 0, ar critical points. MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.

16 Eamining t irst drivativ, > 0 or 0 < <, and < 0 or < 0, and >. So incrass on 0 < <, and dcrass on < 0 and >. Tr is a rlativ ma at,,8 Sinc, Concavity is dtrmind by 0 wic as solutions ±, Lt s call a and b Sinc > 0 or < < a, and b < <, t unction is concav up tr. Sinc < 0 or a < < b is concav down tr MATH 80 Lctur 6 o Ronald Brnt 08 All rigts rsrvd.

17 a a, b givs t inlction points. a, a T unction looks lik: y b and b, b MATH 80 Lctur o Ronald Brnt 08 All rigts rsrvd.