Grad 1 (MCV4UE) AP Calculus Pa 1 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Dfinition of (Natural Eponntial Numr) 0 1 lim( 1 ). 7188188459... 5 1 5 5 Proprtis of and ln Rcall th loarithmic function is th invrs of th ponntial function. lo which is thinvrs of, lo which is thinvrs of. Th function lo can writtn as ln and is calld th natural loarithm function. 1 1 ln Domain is R Ran is R 0 -intrcpt at 1 ln, 0 Horizontal Asmptot 0. lo ln Domain is R 0 Ran is R -intrcpt at 1 ln, R Vrtical Asmptot 0. = ln Drivativ of f() = d Rul (1): If, thn. Rul (): If d ( ) ( ) f ( ), thn f '( ) '( ) Eampl 1: Drivativ of Eponntial functions Diffrntiat 4 53 3 7 5 a) ) c) 3t ht () 1 3t Eampl : Connctin th drivativ of an ponntial function to slop of tannt. 4 Find all points at which th tannt to th curv dfind 3 is horizontal. RHHS Mathmatics Dpartmnt
Grad 1 (MCV4UE) AP Calculus Pa of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Dfinition of Natural Loarithm Natural loarithm is th loarithm to th as. lo ln Basic Proprtis 1) lo 1 0 ln1 0 ) lo =1 ln 1 3) lo ln lo 4) ln Drivativ of = ln d 1 If = ln, thn. d d 1 du If = ln u, thn. d u d Laws of Loarithm 1) lo lo lo ln ln ln (Product Law) ) lo lo lo ln ln ln (Quotint Law) p 3) lo plo ln p pln (Powr Law) Proof : ln d d ln d 1 d d d d d 1 d d 1 Eampl 3: Drivativs of Natural Loarithmic functions Diffrntiat a) ln(5 3) ) 4 ln(5 3) c) ln(5 3) 4 Eampl 4: Drivativs of Natural Loarithmic functions Diffrntiat u ln a) h( u) ln u ) RHHS Mathmatics Dpartmnt
Grad 1 (MCV4UE) AP Calculus Pa 3 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Eampl 5: Drivativs of Natural Loarithmic functions with Laws 1 1 Diffrntiat a) f ln ) f ( ) ln (3 1)( 5) Eampl 6: Equation of tannt to th Natural loarithmic function ln 1 at th point whr 0. Find th quation of th tannt to th curv dfind Eampl 7: Equation of tannt to th Natural loarithmic function Find th quation of th tannt to th curv dfind that is prpndicular to th lin dfind 4 1. RHHS Mathmatics Dpartmnt
Grad 1 (MCV4UE) AP Calculus Pa 4 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Drivativ of Gnral Eponntial functions ( ) 1 d ' ln 0 ( ) ln ln d ln ln d ln ' d d ln ' d Drivativ of Gnral Eponntial functions For ' ln ' Eampl 8: Drivativ of nral ponntial functions Diffrntiat a) 3 5 ) 3 5 1 4 c) 4ln 4 4 ln5 4 d) f ln8 3 RHHS Mathmatics Dpartmnt
Grad 1 (MCV4UE) AP Calculus Pa 5 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Eampl 9: Solvin a prolm involvin an ponntial modl A ioloist is studin th incras in th population of a particular insct in a provincial park. Th population tripls vr wk. Assum th population continus to incras at this rat. Initiall thr ar 100 inscts. a) Writ an quation to rprsnt th numr of inscts in t wks. ) Dtrmin th numr of inscts prsnt aftr 4 wks. c) How fast is th numr of inscts incrasin i) whn th ar initiall discovrd? ii) at th nd of 4 wks? Rcall: Chan of as lo ln lo a lo a ln a Drivativ of nral loarithmic function ' For loa ' lna Basic Proprtis 1) lo 1 0 ) lo =1 3) lo lo 4) Laws of Loarithm 1) lo lo lo ) lo lo lo 3) lo plo p Proof lo ' a ' ln a ln ln a lo lo a 0 ln ln a ' ln a ' ln a Eampl 10: Drivativ of nral loarithmic functions Diffrntiat lo 6 4 7 ) lo (4 3 1) a) 5 RHHS Mathmatics Dpartmnt
Grad 1 (MCV4UE) AP Calculus Pa 6 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: lo5 ln d) lo 3 5 3 lo5 5 c) RHHS Mathmatics Dpartmnt Homwork: P. 178 #1-4, 51, 53 Cal & Vc (Optional) P. 3 # 13, 15 P. 575 #3,4,5a,6,7a,8, 9a,10-13 P. 40 # 1 6, 7a, 8 P. 578 # 1,, 3a, 4acf, 5, 7, 9a