a 1and x is any real number.

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1 Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars as th bas in many appli ponntial fnctions. This irrational nmbr is approimatly qal to.7. Mor accratly, Th nmbr is call th natral bas. Th Natral Logarithm fnction has th following proprtis:. Th omain is 0, an th rang is,. Th fnction is continos, incrasing, an on-to-on.. Th graph is concav ownwar. Logarithm Proprtis: If a an b positiv nmbrs an n is rational, thn th following ar tr.. ln 0. lnab ln a ln b. lna n nln a a 4. ln ln a ln b b Dfinition of th Natral Logarithm Fnction: ln t, 0 t

2 Epaning Logarithms: E: ln 5 E: ln Drivativ of th Natral Logarithmic Fnction ln, 0 ' ln, 0 Fin th rivativ of ach of th following: E: y ln5 E: y ln sin E: y ln E: y ln E: 5 y ln 6 E: y lnln Logarithmic Diffrntiation: E: y 4 E: Fin all rlativ trma an inflction points for y ln

3 Intgration Th Natral Logarithm ln C ln C E: 4 E: 5 E: 4 0 E: 5 4 sc E: tan E: E: E: sc 4 E: Fin th avrag val. Long Division Bfor Intgrating E: / 4 E: 0 tan

4 Warm Up. Fin th rivativ of 4 y (Do not simplify). Fin th intgral of 8 4

5 Invrs Fnctions: Lt f an g b two fnctions sch that f(g()) = for vry in th omain of g an g(f()) = for vry in th omain of f. Th fnction g is th invrs of th fnction f, an is not by f (ra f-invrs ). Ths, f f an f f Th omain of f is qal to th rang of f, an vic vrsa. Th graph of an invrs is th rflction of th original fnction ovr th lin y =. I. Vrify that th following ar invrss: Show that f f an f f.. f. f 5 5 g g 0. To hav an invrs fnction, a fnction mst b on-to-on, which mans no two lmnts in th omain corrspon to th sam lmnt in th rang of f. Yo can s th horizontal lin tst to trmin if a fnction is on-to-on.. If f is strictly monotonic on its ntir omain, thn it is on-to-on an thrfor has an invrs fnction. A fnction is monotonic if it is ithr strictly incrasing or crasing on its ntir omain. Do th following fnctions hav an invrs fnction? Us th rivativ. 6 E: f, E: g 5 4 0, E: h sin, E: p

6 Discss f an th g. E: f Fin ' f E: f ln Fin f '

7 Invrss Cont: Lt f fin th invrs. Labl th invrs as a fnction nam g. What is th slop of th tangnt lin to f at = 7? W know that f ln an g ar invrss. Namly that ln an ln. E: A girl invsts $500 in a bank with an intrst of % componing continosly. How long bfor hr invstmnt obls? E: Solv 9 E: Solv ln 5 4 E: Solv ln Drivativ of th Natral Eponntial Fnction Fin th rivativ of ach of th following: E: y E: y 5 E: y E: y cos E: y

8 Intgration Th Natral Eponntial Fnction C C E: 5 E: E: 7 E: sc tan E: E: sc sctan

9 Diffrntation & Intgration Bass othr than Dfinition of Eponntial Fnction to Bas a: If a is a positiv ral nmbr ( a ) an is any nmbr, thn th ponntial fnction to th bas a is not by a an is fin as a = (ln a ), if a =, thn y = (th constant fnction). Dfinition of Logarithmic Fnction to Bas a: If a is a positiv ral nmbr ( a ) an is any nmbr, ln thn th logarithmic fnction to th bas a is not by log a an is fin as log a log a. ln a Drivativ for Bass Othr than. a ln aa a ln a a ln log a a a log ln a E: f 4 E: f 4 E: f 4 E: f log cos E: f E: f tan log sin E: f log a a a a ln a C a a ln a C E: 5 E: 5 E: 4

10 Powr Rl vrss Logarithmic Diffrntiation Show how thy both work with a polynomial: n f Diffrntial Eqations: Growth & Dcay A iffrntial qation in an y is an qation that involvs, y, an rivativs of y. Th stratgy is to rwrit th qation so that ach variabl occrs on only on si of th qation. This is call sparation of variabls. y 4. 0 y y. 8 y t Gnral Soltion: If y If y 0 6 y. tan ' 4. ln y y ty 0 Gnral Soltion: If y ln 4

11 y 5. y 6. y y ' ' y 7. y y 0 Gnral Soltion: If y 0 Rat of Growth of a poplation: Th following scribs a poplation whos growth will b ponntial. P t This poplation grows at a rat proportional to th amont prsnt at any tim (Assm poplation 0) kp P t kp Lt P( 0) P0

12 Diffrntial Eqations Sparation of Variabls If y is a iffrntiabl fnction of t sch that y > 0 an y ' ky, for som constant k, thn kt y C. C is th initial val of y, an k is th proportionality constant. Eponntial growth occrs whn k > 0, an cay occrs whn k < 0. E: Carbon (4) has a half-lif of 5,70 yars. If th initial val of y is,000. How mch of th sbstanc will b lft aftr 00 yars, 4,000 yars? E: Th rat of chang of N(t) nmbr of bars in a poplation is proportional to 00 N t, t 0 is tim in yars. Us that N(4)= 600. A) Writ th iffrntial qation. B) Solv th qation. C) What is N(8)? D) Fin th limit as t gos to infinity of N(t). E: Sppos an primntal poplation of flis incrass accoring to th ponntial of growth. Thr wr 00 flis aftr th scon ay of th primnt an 00 flis aftr th forth ay. Approimatly how many flis wr in th original poplation? E: A crtain typ of bactria incrass continosly at a rat proportional to th nmbr prsnt at tim t. If thr ar 500 at a givn tim an 000 hors latr, how many hors will it tak for thr to b 500?

13 E: Nwton s Law of Cooling stats that th rat of chang in th tmpratr of an objct is proportional to th iffrnc btwn th objct s tmpratr an th tmpratr of th srroning mim. Lt y rprsnt th tmpratr of an objct in a room whos tmpratr is kpt at a constant 70 grs. If th objcts cools from 0 grs to 00 grs in 5 mints, how mch longr will it tak for its tmpratr to cras to 80 grs? E: An orthogonal trajctory of a family of crvs is a crv that intrscts ach crv of th family orthogonally, at right angls. Fin th orthogonal trajctoris of th family of crvs ky, whr k is an arbitrary constant. E: Elctro-static fils an stramlins

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b) 4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y