MTH213 Calculus. Trigonometry: Unit Circle ( ) ( ) ( )

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1 MTH3 Clculus Formuls from Geometry: Trigle A = h Pythgore: + = c Prllelogrm A = h Trpezoi A = h ( + ) Circle A = π r C = π r = π Trigoometry: Uit Circle 0, π 3,, 3 35, 3π 4,, 50, 5π 6, 3, 90 π 0, Coe V = 3 A se h = 3 π r h Lterl Surfce: A = π r r + h Right Circulr Cylier V = A se h = π r h Lterl Surfce: A = π r h Sphere V = 4 3 π r3 Surfce Are = 4π r 60, π 3,, 3 45, π 4,, 30, π 6, 3, Defiitios: 80, π, (, 0) 0, 0,, 0 360, π,, 0 0, 7π 330, π 6, 3, 6, 3, 5, 5π 4,, 35, 7π 4,, 300, 5π 3,, 3 40, 4π 3,, π siθ = opp. hyp. cscθ = siθ 0, cosθ = j. hyp. secθ = cosθ tθ = opp. j. = siθ cosθ cotθ = tθ = cosθ siθ

2 Ietities Pythgore: si u + cos u = + t u = sec u + cot u = csc u Doule Agle Formuls: siu = siu cosu cosu = cos u si u = cos u = si u tu = tu t u Lw of Sies: si A = si B = c sic Power-Reucig Formuls: si u = ( cosu ) cos u = ( + cosu ) cosu t u = + cosu Lw of Cosies: = + c cos B Chpter Solutio: Ay ll vlues of the vrile which will mke sttemet TRUE. Grph: A picture, rw o pproprite coorite system, showig ll solutios to give sttemet. Itercepts: Where grph crosses oe of the xes. Fi the x-itercept(s): Set y = 0 solve for x Fi the y-itercept(s): Set x = 0 solve for y Symmetry of Grph - A grph is symmetric to: y-xis if replcig x with -x gives equivlet equtio x-xis if replcig y with -y gives equivlet equtio origi if replcig x with -x y with -y gives equivlet equtio Itersectios of Grphs Solve oth equtios for the sme vrile (e.g. oth for y) Set expressios i other vrile equl (e.g. sustitute for y to get equtio with x) Solve for the vrile (e.g. solve for ll vlues of x) Use solutios to fi coorites of itersectio(s) (e.g. use x vlues to fi (x, y)) Slope: Compriso of verticl chge to horizotl chge i grph. Slope is rise over ru Slope, m, of o-verticl lie through poits ( x, y ) ( x, y ): m = chge i y chge i x = y y = Δy x x Δx Lier Equtios Slope-Itercept Form y = mx + Poit-Slope Form y y = m x x Geerl Form Ax + By + C = 0 Verticl Lie x = Horizotl Lie y = Prllel Lies m = m Perpeiculr Lies m m = Grph Trsformtios (c > 0) Origil grph: y = f ( x) θ c

3 Horizotl shift c uits right: y = f x c Verticl shift ow: Reflectio out x-xis: Reflectio out origi: Rtiol Fuctio: Composite Fuctio: O-Eve Fuctio Tests Eve: f x Horizotl shift left: y = f ( x + c) y = f ( x) c Verticl shift up: y = f ( x) + c y = f ( x) Reflectio out y: y = f ( x) y = f ( x) f ( x) = p( x) q( x) for q( x) 0 ( f! g) ( x) = f ( g( x) ) = f ( x) Thik: f ( x) = x = f ( x) Thik: f ( x) = x O: f x Qurtic Equtio - Str Form: x + x + c = 0 Completig the Squre x + = 4c 4 eve power o power Qurtic Formul x = ± 4c Iverse Fuctios f (re: f iverse ) is the iverse of f if: ( ) = for i omi of f ] f ( f ( ) ) = for i omi of f ] f f Fuctio hs iverse oly if it is oe-to-oe (psses the horizotl lie test). Fuctios c e me to hve iverse y restrictig omi /or rge. Iverse Trig Fuctios: Domi Rge rcsie x x π y π rccosie x x 0 x π rctget x x π y π Expoetil Logrithmic Fuctios Logrithmic fuctio is iverse of expoetil: log x = log x = x Commo Logrithm: log x log 0 x = 0 = x Nturl Logrithm: l x log e x = e = x Ietities: log = 0 log = Properties: log xy = log x + log y x log y = log x log y log x = log x Gretest Iteger Fuctio: x]] = gretest iteger such tht x. E.g. 5.3]] = 5 -.3]] = -3

4 Chpter Slope of Sect Lie of Fuctio: m sec = f ( x + Δx) f ( x) Δx Commo Resos why lim f x. f x. f x (Note: No limit. This is slope etwee two poits.) oes ot exist: pproches ifferet limit from the right s from the left icreses or ecreses without ou s x c oscilltes etwee two fixe vlues s x c 3. f x Defiitio of Limit (Mie): Let f e fuctio efie o ope itervl which cotis c. (Vlue f ( c) my or my ot exist.) lim f x = L mes tht for every ε > 0, there exists δ > 0 such tht if 0 < x c < δ, the 0 < f ( x) L < ε. Gettig close to the limit, L, of fuctio t vlue, c, mes fuctio vlue f x withi istce ±ε of the limit, L. If the limit exists, there must e vlue δ, istce from poit c, so if we choose our x withi tht istce from c, f x withi istce ±ε of L. If limit L exists, we c lwys fi smll eough umer δ so tht f x + δ will e withi the istce ±ε of the limit, L. Bsic Limits for Polyomil-Type Terms Limit of costt: lim = Limit of vrile: lim x = c Limit of power : x = c lim Properties of Limits (for lim Scler Multiple lim Sum or Differece Prouct Quotiet Power = L lim g( x) = K f ( x) = L f ( x) ± g( x) = L ± K f ( x) g( x) = L K f x lim lim f x lim = L g x K (If K is ot zero.) lim f x = L (Iclues ricls s frctiol powers.) f ( x)! g( x) f g( x) Composite Fuctio lim = lim = f K Limits of Trsceetl Fuctios: (All ehve s expecte for vlues c i omi) lim si x = sic lim sec x = secc lim cos x = cosc lim csc x = cscc lim t x = tc lim x = c ( > 0) lim cot x = cot c lim l x = l c is will e

5 Fuctios which gree t ll ut oe poit: If f x = g( x) for ll poits except for x = c, if lim lim = lim f x g x Three Specil Limits: (Note x 0 i ll cses) si x cos x lim = lim x 0 x x 0 x Defiitio of Cotiuity: ( Fuctio f is cotiuous t c if...). f ( x) is efie. lim f x g x exists, the = 0 lim ( + x) x = e x 0 exists 3. lim f x = f ( c) Existece of Limit: lim f ( x) = lim + f ( x) Properties of Cotiuity: If f ( x) g( x) re cotiuous t x = c, the the followig re lso cotiuous t c:. Scler multiple f c. Sum Differece f ( c) ± g( c) f ( c) g( c) if g( c) 0 ( c) = f ( g( c) ) 3. Prouct f ( c) g( c) 4. Quotiet 5. Composite f! g 6. Therefore Polyomil Fuctios, Rtiol Fuctios, Ricl Fuctios, Trigoometric Fuctios, Expoetil, Logrithmic Fuctios re cotiuous for ll vlues withi their omis. Itermeite Vlue Theorem: If f is cotiuous o the close itervl, ] k is y umer etwee f ( ), the there is t lest oe umer c, somewhere i,] such = k. f tht f c Verticl Asymptotes: Let f g e fuctios with o commo fctors cotiuous o ope itervl cotiig c. If f c 0 whe x c), the f ( x) g( x) g c Properties of Ifiite Limits: For lim. Sum or Differece lim f x 0 g( c) = 0 (BUT hs verticl symptote t x = c. = lim ± g( x) = g( x) g( x) g x. Prouct lim f x = + if L > 0 lim f x = if L < 0 g x 3. Quotiet lim = L if L 0 f ( x) = L

6 Chpter 3 Defiitio of the Derivtive of Fuctio: f ( x) = lim Δx 0 f ( x) f x + Δx Δx f x or f ( c) = lim x c f ( c) x c Differetile Implies Cotiuity, ut NOT Vice Vers: If f is ifferetile t x = c, the f is cotiuous t x = c. However, f might e cotiuous, ut ot ifferetile, if f hs cusp t x = c. Geerl Differetitio Rules Let u v e ifferetile fuctios of x. The vlue, c, is costt Derivtive of Costt: x c ] = 0 Simple Power Rule: Costt Multiple Rule: Sum or Differece Rule: Prouct Rule: Quotiet Rule: Chi Rule: Geerl Power Rule (from Chi): ] = x x x x cu ] = c u x u ± v ] = u ± v x u v ] = u v + v u u x v = v u u v v x f ( u )] = f ( u) u = f u u x ] x u = u u x x ] = Derivtives of Trigoometric Fuctios x siu] = cosu x u] x cosu] = siu x u] x tu ] = sec u x u ] x cscu x secu ] = cscu cotu x u] ] = secu tu x u] x cotu ] = csc u x u ] Derivtives of Expoetil Logrithmic Fuctios = e u x u] x eu x lu ] = u x u ] x u x log u = ( l) u x u] ] = ( l)u x u ]

7 Positio, Velocity, Accelertio: If s( t) is the positio fuctio of oject (for fllig oy: s( t) = gt + v 0 t + s 0 ) = = = s ( t) The velocity fuctio is v t s t The ccelertio fuctio is t v t Explicit - Implicit forms of Fuctio: Explicit: y is give s fuctio i x ( y = x ) Implicit: x y re oth ivolve i the equtio together ( xy =) Guielies for Implicit Differetitio:. Differetite oth sies with respect to x usig the chi rule o y o-x vriles.. Collect y x 3. Solve for y x Derivtive of Iverse Fuctio f ] x = terms o left sie fctor out y x f f x y iviig y fctor multiplie times the y x. ( ) Derivtives of the Iverse Trig Fuctios x rcsiu u ] = u x rctu] = u + u rc cot u x rc sec u x ] = u rc sec u u u x x rccosu ] = u u u + u u ] = ] = u u Relte Rtes. Fi equtio (formul) which reltes the esire qutities.. Differetite with respect to time to fi the relte rtes of chge of the qutities. Newto s Metho of Approximtig the Zero of Fuctio. Mke iitil estimte, x, close to the root.. Clculte ew estimte, x, usig the formul: x = x f x f x 3. Repet s ecessry util esire ccurcy is ttie.

8 Chpter 4 Criticl Numer: c is criticl umer if: ) f ( c) = 0 ) f is ot ifferetile t c. Guielies for fiig extrem i the itervl,]:. Fi the criticl umer of f i (,).. Evlute f t ech criticl umer. 3. Evlute f t ech e poit. Rolle s Theorem Let f e cotiuous o the close itervl, (,). If f ( ) = f ( ), the there is t lest oe c i, Me Vlue Theorem Let f e cotiuous o the close itervl, (,). There exists umer c i, ] ifferetile o the ope itervl where f ( c) = 0. ] ifferetile o the ope itervl where f ( c) = f ( ) f ( ). The erivtive of f evlute t c will equl the slope of the sect lie etwee the epoits.] Icresig Fuctio Decresig Fuctio. If x < x implies f ( x ) < f ( x ).. If x < x implies f ( x ) > f ( x ). > 0. < 0.. If f x. If f x Test for Reltive Miimum If c is criticl umer:. If f ( x) chges from egtive to positive t c.. If f ( c) > 0 (Seco Derivtive Test) Test for Reltive Mximum If c is criticl umer:. If f ( x) chges from positive to egtive t c.. If f ( c) < 0 (Seco Derivtive Test) Cocvity If f ( x) > 0, grph is cocve upwrs. Alt. f ( x) is icresig i the itervl. If f ( x) < 0, grph is cocve owwrs. Alt. f x Poit of Iflectio Poit t which grph chges from cocve up to cocve ow or vice vers. Is the poit ( c, f ( c) ) if f ( c) = 0 or if f ( c) oes ot exist. Domi of fuctio f(x): Vlues of x for which fuctio f exists. Look for:. Deomitor = 0?. Negtive umer uer the ricl for eve-umer root? is icresig i the itervl. 3. Other vlues where f is ot efie (E.g. t π, l( ), etc.)? Verticl Asymptotes i Rtiol Fuctios At vlues of x which mke eomitor = 0 Exceptio: Commo fctors etwee umertor eomitor crete holes, ot symptotes.

9 Horizotl Asymptote Lie y = L is horizotl symptote if lim f x x Limits t Ifiity If r is positive rtiol c is rel umer: Rtiol Fuctios lim x c x = L or lim r = 0 lim x + x + c x r = 0 Expoetil Fuctios lim e x = 0 lim e x = 0 x x + Horizotl Olique Asymptotes of Rtiol Fuctios f ( x) = L Rtiol fuctio is rtio of polyomils: R( x) = x + x + x +... m x m + m x m + m x m +... Degree of umertor = egree of eomitor = m. If - m >, there is o symptote.. Fctor x from ech term i umertor x m from ech term i eomitor. Ccel x with x m. 3. Tke the limit s x goes to ifiity of the tht expressio. 4. Result is the equtio of the symptote: If - m =, the equtio will e lier equtio i x olique (slt) symptote. If = m, it will e the horizotl lie m If < m, it will e the x xis. Optimizig (Mximum or Miimum) Prolems. Write primry equtio: (Qutity to e optimize) = (formul to clculte).. Cosier the omi of the primry fuctio i the cotext of the prolem. For wht vlues oes the equtio mke sese?. Ietify ecessry equtios which c e sustitute ito primry equtio to yiel: (Qutity to e optimize) = (expressio i sigle, iepeet vrile) 3. Differetite with respect to the iepeet vrile 4. Set erivtive equl to zero. 5. Solve for vlue(s) of iepeet vrile tht mke sese for the prolem. x Differetils: Differetil of y is y = f x Error Propgtio Uses tget lie to fuctio to estimte the fuctio vlue er give poit. y = f ( x) x Where y is error i y crete y (propgte) error of mesuremet i x, x. Error of mesurmet Rememer: Percet Error = Vlue of Mesuremet Estimtig Fuctio Vlue t Poit x + Δx f x + Δx f x f x f ( x) + x or ifferece i y is f ( x + Δx) f ( x) y = x

10 Chpter 5 Bsic Itegrl (ti-erivtive) Formule Powers Lier Comitios: k u = k u + C k f ( u) u = k f ( u) u u u = ± g( u) ]u = f ( u) u ± g u u + + C, f u + Trig Fuctios: siu u = cosu + C cosu u tu u = lcosu + C cot u u sec u u = lsec u + tu + C csc u u = siu + C = lsiu + C = lcsc u + cot u + C Specil Trig Fuctios: sec u u = tu + C csc u u = cot u + C sec utu u = sec u + C csc ucot u u = csc u + C Logrithmic Expoetil Fuctios: e x x = e x + C x x = x + C l x x = x x = l x + C Itegrls Usig Logrithmic Itegrtio: u u u = l u + C lterte: u x = l u + C Itegrls Usig Iverse Trigoometric Fuctios: u = rcsi u u + C u u u = rcsec u + C u u + = rct u + C u Summtios: Properties: Formule: i = !+ i= k i = k i i= i= c = c i= i i= ( + ) = + 6 i ± i = i i= i= i = + i= i 3 i= ± = + 4 i= i

11 Are oue y grph of f, the x-xis, verticl lies x = x = Are = lim f ( c )Δx, Δx = i= Are = f x x x i c i x i Properties of the Defiite Itegrl: f ( x) x = 0 f ( x) x = f ( x) x k f ( x) x = k f ( x) x f ( x) ± g( x) x = f ( x) x ± g( x) x c f ( x) x = f ( x) x + f ( x) x for < c < c Averge Vlue of Fuctio o Itervl, ]: f ( x) x Fumetl Theorem of Clculus (I): f x x = F ( x)] = F ( ) F ( ) Trpezoil Rule for Numericl Itegrtio: f ( x) x f ( x 0 ) + f ( x ) + f x ( Error: E ) 3 mx f ( x) ], x Fumetl Theorem of Clculus (II): x f ( t) t x = f ( x ) +!+ f ( x ) + f ( x )] Simpso's Rule for Numericl Itegrtio ( must e eve): f ( x) x f ( x 0 ) + 4 f ( x ) + f ( x ) + 4 f x 3 3 ( Error: E ) mx f ( 4) ( x ), x +!+ 4 f ( x ) + f ( x )] Hyperolic Fuctios Defiitios: sih x = e x e x cosh x = e x + e x th x = sih x cosh x csch x = sih x, x 0 sech x = cosh x coth x = th x, x 0

12 Hyperolic Fuctio Ietities: cosh x sih x = th x + sech x = coth x csch x = sih + coshx x = sihx = sih x cosh x = sih x cosh y + cosh x sih y = sih x cosh y cosh x sih y = cosh x cosh y + sih x sih y = cosh x cosh y sih x sih y sih x + y sih x y cosh x + y cosh x y cosh x = + coshx coshx = cosh x + sih x Derivtives Itegrls of Hyperolic Fuctios: x sihu x cosh u x thu x cothu x sechu ] = ( cosh u) u ] = ( sihu) u u u ] = sech u ] = csch u ] = ( sech uthu) u x cschu ] = ( csch ucothu) u coshu u = sihu + C sihu u = cosh u + C sech u u = thu + C csch u u = cothu + C sechuthu u = sechu + C csch ucoth u u = cschu + C Iverse Hyperolic Fuctios: sih x = l x + x + Domi:, + cosh x = l x + x Domi:, ) th x = x + l x Domi: (, ) coth x = x + l x sech x = l + x x csch x = l x + + x x Domi: (, ) (, + ) Domi: ( 0,] Domi: (, 0) ( 0, + )

Harold s Calculus Notes Cheat Sheet 15 December 2015

Harold s Calculus Notes Cheat Sheet 15 December 2015 Hrol s Clculus Notes Chet Sheet 5 Decemer 05 AP Clculus Limits Defiitio of Limit Let f e fuctio efie o ope itervl cotiig c let L e rel umer. The sttemet: lim x f(x) = L mes tht for ech ε > 0 there exists