lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion: d Product Rule: y d y f ( ) g ( ) f '( ) g( ) g'( ) f ( ) d Quotient Rule: hin Rule: f ( ) y g( ) f '( ) g( ) g'( ) f ( ) d ( g( )) n y ( f ( )) n n( f ( )) ( f '( )) d y sin sin cos ( ) d sin cos d sin y cos ( ) sin 6 d cos sin d y ( ) ( ) d 6( ) d
Nturl Log y ln( f ( )) f '( ) d f ( ) y ln( ) d d Power Rule: y d 5 y d onstnt with Vrile Power: f ( ) y f ( ) ln f '( ) d y ln d y ln d Vrile with Vrile power y f ( ) g( ) Tke ln of oth sides! sin y sin ln y ln ln y sin ln cos ln sin y d sin [cos ln sin ] d
Implicit Differentition: Is done when the eqution hs mied vriles: y y 5 derivtive y y d y [ ] d y y y d d y d y y Trigonometric Functions: d d sin cos d d cos sin d d tn sec d d sec sec tn Inverse Trigonometric Functions: y rcsin d y rcsin d 8 y rctn d y rctn d 6
Integrl Formuls Bsic Integrl 5 d 5 π π,where is n ritrry constnt Vrile with onstnt Power d d onstnt with Vrile Power d ln 5 d 5 ln5 Frctions d d -unless- d ln if the top is the derivtive of the ottom d ln d ln
Sustitution When integrting product in which the terms re somehow relted, we usully let u the prt in the prenthesis, the prt under the rdicl, the denomintor, the eponent, or the ngle of the trigonometric function d; u / d ( ) / u du / u / ( ) du d cos d; u du d cos d cosudu sin u sin Integrtion y Prts When tking n integrl of product, sustitute for u the term whose derivtive would eventully rech nd the other term for dv. The generl form: uv vdu (pronounced "of dove") Emple: e d u dv e d du d v e e e d e e
Emple : v d du d dv u v d du d dv u sin cos sin cos cos [ sin cos sin sin sin sin cos cos Inverse Trig Functions Formuls: rcsin rctn Emples: v rcsin ; ; 9 v rctn ; ; 6 More emples: v rcsin 9 ; ; 9 v rctn rctn 6 9 ; 6 9
Trig Functions sin d cos sin cos d tn d ln cos sec d tn sec tn sec cot d ln sin Properties of Logrithms Form logrithmic form <> eponentil form y log <> y Log properties y log > y log log log y log y log log y log (/y) hnge of Bse Lw This is useful formul to know. y log log log or ln ln Properties of Derivtives st Derivtive shows: mimum nd minimum vlues, incresing nd decresing intervls, slope of the tngent line to the curve, nd velocity nd Derivtive shows: inflection points, concvity, nd ccelertion - Emple on the net pge -
Emple: y 6 6 d 6( 6( )( ), 6 6 6) Find everything out this function st derivtive finds m, min, incresing, decresing m min (-, 6) (, -79) incresing decresing (, -] [, ) (-, ) d y 6 d 6( ) nd derivtive finds concvity nd inflection points inflection pt ( ½, -6 ½) concve up concve down (, ½) (/, )
Miscellneous Newton s Method Newton s Method is used to pproimte zero of function f ( c) c where c is the st pproimtion f '( c) Emple: If Newton s Method is used to pproimte the rel root of, then first pproimtion of would led to third pproimtion of : f ( ) f '( ) f () f '() or.75 f ( ) 59 or.686 86 f '( ) Seprting Vriles Used when you re given the derivtive nd you need to tke the integrl. We seprte vriles when the derivtive is miture of vriles Emple: If 9y nd if y when, wht is the vlue of y when d? 9y 9 d d y y d y 9 9 ontinuity/differentile Prolems f() is continuous if nd only if oth hlves of the function hve the sme nswer t the reking point. f() is differentile if nd only if the derivtive of oth hlves of the function hve the sme nswer t the reking point
Emple: f ( ) Useful Informtion, 6 9, > f '( ) 6( plug in ) 6 6 - At, oth hlves 9, therefore, f() is continuous - At, oth hlves of the derivtive 6, therefore, f() is differentile - We designte position s (t) or s(t) - The derivtive of position (t) is v(t), or velocity - The derivtive of velocity, v (t), equls ccelertion, (t). - We often tlk out position, velocity, nd ccelertion when we re discussing prticles moving long the -is. - A prticle is t rest when v(t). - A prticle is moving to the right when v(t) > nd to the left when v(t) < - To find the verge velocity of prticle: v( t) dt Averge Vlue Use this formul when sked to find the verge of something f ( ) d Men Vlue Theorem NOT the sme verge vlue. According to the Men vlue Theorem, there is numer, c, etween nd, such tht the slope of the tngent line t c is the sme s the slope etween the points (, f()) nd (, f()). f '( c) f ( ) f ( )
Growth Formuls Doule Life Formul: y y Hlf Life Formul: y y kt Growth Formul: y e y (/ ) () t / h t / d y ending mount y initil mount t time k growth constnt d doule life time h hlf life time Useful Trig. Stuff Doule Angle Formuls: sin sin cos cos cos Identities: sin tn cot cos sec csc sin secθ cosθ sinθ tnθ cosθ cscθ sinθ cosθ cotθ sinθ Integrtion Properties Are [ f ( ) g( )] d f() is the eqution on top Volume f() lwys denotes the eqution on top Aout the -is: π [ f ( )] π [( f ( )) d ( g( )) ] d Aout the y-is: π π [ f ( )] d [ f ( ) g( )] d out line y - out the line - π [ f ( ) ] d π ( )[ f ( )] d Emples:
Trpeziodl Rule f ( ) -is: [,] y-is: ) π ( d π d π [ ] d π d out y - In this formul f() or y is the rdius of the shded region. When we rotte out the line y -, we hve to increse the rdius y. Tht is why we dd to the rdius ] d π ( π [ ) d out - In this formul, is the rdius of the shded region. When we rotte out the line -, we hve the incresed rdius y. π ( )[ ] d π ( ) d Used to pproimte re under curve using trpezoids. n where n is the numer of sudivisions Are [ f ( ) f ( ) f ( )... f ( ) f ( )] n n Emple: f(). Approimte the re under the curve from [,] using trpezoidl rule with sudivisions n A 8 [ f () f (.5) f () f (.5) f ()] [ (5 / ) () ( / ) 5] 76 6 [(76 / ) ]. 75 Riemnn Sums
Reding Grph Used to pproimte re under the curve using rectngles. ) Inscried rectngles: ll of the rectngles re elow the curve Emple: f() from [,] using sudivisions (Find the re of ech rectngle nd dd together) I.5() II.5(5/) III.5() IV.5(/) Totl Are.75 ) ircumscried Rectngles: ll rectngles rech ove the curve Emple: f() from [,] using sudivisions When Given the Grph of f () I.5(5/) II.5() III.5(/) IV.5(5) Totl Are 5.75 Mke numer line ecuse you re more fmilir with numer line. This is the grph of f (). Mke numer line. - Where f () (-int) is where there re possile m nd mins. - Signs re sed on if the grph is ove or elow the -is (determines incresing nd decresing) min m -, - cont d on net pge - incresing decresing (-,) (,6] [-6,-] (,)
To red the f () nd figure out inflection points nd concvity, you red f () the sme wy you look t f() (the originl eqution) to figure out m, min, incresing nd decresing. For the grph on the previous pge: inflection pt -,,, 5 Signs re determined y if f () is incresing () nd decresing (-) concve up concve down (-6,-) (,) (5,6) (-,) (,5) Originl Source: http://www.geocities.com/are5/strgte/587/inde.html