The Derivative of the Natural Logarithmic Function. Derivative of the Natural Exponential Function. Let u be a differentiable function of x.

Transcription

1 Th Ntrl Logrithmic n Eponntil Fnctions: : Diffrntition n Intgrtion Objctiv: Fin rivtivs of fnctions involving th ntrl logrithmic fnction. Th Drivtiv of th Ntrl Logrithmic Fnction Lt b iffrntibl fnction of. [ ln ], > 0, [ ln ], 0 >, [ ] ln, 0 Drivtiv of th Ntrl Eponntil Fnction Lt b iffrntibl fnction of., E: Diffrntit :. ln(5 + ) b. ( ) f( ) ln c. ( 3), ln( ln) f. 3 Ln Rl for Intgrtion Lt b iffrntibl fnction of ln +, Intgrtion Rls for Eponntil Fnctions Lt b iffrntibl fnction of. +, ln + + E: Evlt b. csc c. cot

2 E: Fin th r of th rgion bon b th grph, th -is, th lin E: Evlt. / b. sin cos c Intgrls of th Si Bsic Trig Fnctions. sin cos+. cos sin+ 3. tn ln cos + 4. cot ln sin + 5. sc ln sc+ tn + 6. csc ln csc+ cot + Bss Othr thn n Applictions Objctiv: Dfin ponntil fnctions tht hv bss othr thn. Diffrntit n intgrt ponntil fnctions tht hv bss othr thn. Us ponntil fnctions to mol compon intrst n ponntil growth. Dfinition of Eponntil Fnction to Bs : If is positiv rl nmbr ( ) n is n rl nmbr, thn th ponntil fnction to th bs is not b n is fin b (ln ) Dfinition of Logrithmic Fnction to Bs : If is positiv rl nmbr ( ) n is n positiv rl nmbr, thn th logrithmic fnction to th bs is not b log n is fin s log ln ln Drivtivs for Bss othr thn : Lt b positiv rl nmbr ( ) n lt b iffrntibl fnction of.. (ln ). (ln ) (ln ) 3. [ log ] 4. [ log ] E: Fin th rivtiv of ch fnction (ln )

3 . 5 b. 3 + c. 5log34 E: Evlt 5. Prviosl th powr rl rqir n to b rtionl nmbr. Howvr, now th rl cn b tn to covr n rl vl nmbr. Th Powr Rl for Rl Eponnts Lt n b n rl nmbr n lt b iffrntibl fnction of. n n n n n E: Fin th rivtiv of ch fnction. b. Invrs Trigonomtric Fnctions: Diffrntition Objctiv: Dvlop proprtis of th si invrs trigonomtric fnctions. Diffrntit n invrs trigonomtric fnction. Rviw th bsic iffrntition formls for lmntr fnctions. Non of th si bsic trigonomtric t fnctions hs n invrs fnction! rcsin D: R: π/ π/ rccos Domin: Rng: 0 π rctn D: R: π/ < < π/ rccot D: R:0< < π rcsc D: R:0 < < π, π/ rccsc D: R: π/ π/, 0 E: Evlt:. rcsin(- ½ ) b. rcos(0) c. rctn( 3)

4 Proprtis of Invrs Trigonomtric Fnctions. If n π π, thn. If n 0 π, thn. sin(sin - ) n sin - (sin ). cos(cos - ) n cos - (cos ). 3. If is rl nmbr n π π, thn tn(tn - ) n tn - (tn ). similr proprtis hol for othr invrs trigonomtric fnctions E: Solv rctn( 3) π/4 Drivtivs of Invrs Trigonomtric Fnctions Lt b iffrntibl fnction of. [ rcsin] [ rccos] + [ rctn] + [ rccot] [ rcsc] [ rccsc] To riv ths formls, o cn s implicit iffrntition E: Diffrntit:. rcsi n( ) b. f( ) rctn(3 + ) c. rccos. rcsc E: Diffrntit rcsin+

5 Invrs Trigonomtric Fnctions: Intgrtion Objctiv: Intgrt fnctions whos ntirivtivs involv invrs trigonomtric fnctions. Us complting th sqr to intgrt fnction. Rviw th bsic intgrtion formls involving lmntr fnctions. Intgrls Involving Invrs Trigonomtric Fnctions Lt b iffrntibl fnction of, n lt > 0. rcsin +, rc t n + +, rcsc + Ths rls ll com from th prcing rivtiv rls of invrs fnctions. Sinc th rcsin n rccos rivtivs r th ngtiv of ch othr o onl n on rivtiv for th pir. E: Evlt. b c Rmmbr complting th sqr!?!?!? Wll, o n it for lcls too! E: Evlt b c b b b c b b + + c

6 E: Fin th r of th rgion bon b th grph of f( ) 3 th -is, n th lins 3/ n 9/4. E: Fin s mn of th following intgrls s o cn sing th formls n tchniqs sti so fr.. b. c.. ln ln. f. f. ln