Harold s Calculus Notes Cheat Sheet 15 December 2015
|
|
- Marybeth Payne
- 5 years ago
- Views:
Transcription
1 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 δ > 0 such tht if 0 < x < δ, the f(x) L < ε Tip : Direct sustitutio: Plug i f() see if it provies legl swer. If so the L = f(). The Existece of Limit The limit of f(x) s x pproches is L if oly if: Defiitio of Cotiuity A fuctio f is cotiuous t c if for every ε > 0 there exists δ > 0 such tht x c < δ f(x) f(c) < ε. Tip: Rerrge f(x) f(c) to hve (x c) s fctor. Sice x c < δ we c fi equtio tht reltes oth δ ε together. Two Specil Trig Limits lim f(x) = L x lim x + f(x) = L Prove tht f(x) = x is cotiuous fuctio. f(x) f(c) = (x ) (c ) = x c + = x c = (x + c)(x c) = (x + c) (x c) Sice (x + c) c f(x) f(c) c (x c) < ε So give ε > 0, we c choose δ = ε > 0 i the c Defiitio of Cotiuity. So sustitutig the chose δ for (x c) we get: f(x) f(c) c ( ε) = ε c Sice oth coitios re met, f(x) is cotiuous. si x lim = x 0 x cos x lim = 0 x 0 x Copyright 05 y Hrol Toomey, WyzAt Tutor
2 Derivtives Defiitio of Derivtive of Fuctio Slope Fuctio Nottio for Derivtives The Costt Rule The Power Rule The Geerl Power Rule The Costt Multiple Rule The Sum Differece Rule (See Lrso s -pger of commo erivtives) f f(x + h) f(x) (x) = lim h 0 h f f(x) f(c) (c) = lim x c x c f (x), f () (x), y x, y, x [f(x)], D x[y] x [c] = 0 x [x ] = x x [x] = (thik x = x x 0 = ) x [u ] = u u where u = u(x) x [cf(x)] = cf (x) x [f(x) ± g(x)] = f (x) ± g (x) Positio Fuctio s(t) = gt + v 0 t + s 0 Velocity Fuctio v(t) = s (t) = gt + v 0 Accelertio Fuctio (t) = v (t) = s (t) Jerk Fuctio j(t) = (t) = v (t) = s (3) (t) The Prouct Rule x [fg] = fg + g f The Quotiet Rule x [f g ] = gf fg g The Chi Rule x [f(g(x))] = f (g(x))g (x) y x = y u u x Expoetils (e x, x ) x [ex ] = e x, x [x ] = (l ) x Logorithms (l x, log x) Sie Cosie Tget Secet Cosecet Cotget x [l x] = x, x [log x] = (l ) x [si(x)] = cos(x) x [cos(x)] = si (x) x x [t(x)] = sec (x) [sec(x)] = sec(x) t (x) x [csc(x)] = csc(x) cot (x) x x [cot(x)] = csc (x) Copyright 05 y Hrol Toomey, WyzAt Tutor
3 Applictios of Differetitio Rolle s Theorem f is cotiuous o the close itervl [,], f is ifferetile o the ope itervl (,). Me Vlue Theorem If f meets the coitios of Rolle s Theorem, the L Hôpitl s Rule Grphig with Derivtives Test for Icresig Decresig Fuctios The First Derivtive Test The Seco Derivitive Test Let f (c)=0, f (x) exists, the Test for Cocvity Poits of Iflectio Chge i cocvity Alyzig the Grph of Fuctio If f() = f(), the there exists t lest oe umer c i (,) such tht f (c) = 0. f f() f() (c) = f() = f() + ( )f (c) Fi c. P(x) If lim f(x) = lim x c x c Q(x) = { 0 0,, 0,, 0 0, 0, }, ut ot {0 }, P(x) the lim x c Q(x) = lim x c P (x) Q (x) = lim P (x) x c Q (x) =. If f (x) > 0, the f is icresig (slope up). If f (x) < 0, the f is ecresig (slope ow) 3. If f (x) = 0, the f is costt (zero slope). If f (x) chges from to + t c, the f hs reltive miimum t (c, f(c)). If f (x) chges from + to - t c, the f hs reltive mximum t (c, f(c)) 3. If f (x), is + c + or - c -, the f(c) is either. If f (x) > 0, the f hs reltive miimum t (c, f(c)). If f (x) < 0, the f hs reltive mximum t (c, f(c)) 3. If f (x) = 0, the the test fils (See st erivtive test). If f (x) > 0 for ll x, the the grph is cocve up. If f (x) < 0 for ll x, the the grph is cocve ow If (c, f(c)) is poit of iflectio of f, the either. f (c) = 0 or. f oes ot exist t x = c. (See Hrol s Illegls Grphig Rtiols Chet Sheet) x-itercepts (Zeros or Roots) f(x) = 0 y-itercept f(0) = y Domi Vli x vlues Rge Vli y vlues Cotiuity No ivisio y 0, o egtive squre roots or logs Verticl Asymptotes (VA) x = ivisio y 0 or uefie Horizotl Asymptotes (HA) lim f(x) y lim f(x) y x x + Ifiite Limits t Ifiity lim f(x) lim x x + Differetiility Limit from oth irectios rrives t the sme slope Reltive Extrem Crete tle with omis, f(x), f (x), f (x) Cocvity If f (x) +, the cup up If f (x), the cup ow Poits of Iflectio f (x) = 0 (cocvity chges) Copyright 05 y Hrol Toomey, WyzAt Tutor 3
4 Approximtig with Differetils Newto s Metho Fis zeros of f, or fis c if f(c) = 0. Tget Lie Approximtios Fuctio Approximtios with Differetils Relte Rtes x + = x f(x ) f (x ) y = mx + y = f (c)(x c) + f(c) f(x + x) f(x) + y = f(x) + f (x) x Steps to solve:. Ietify the kow vriles rtes of chge. (x = m; y = 3 m; x = 4 m s ; y =? ). Costruct equtio reltig these qutities. (x + y = r ) 3. Differetite oth sies of the equtio. (xx + yy = 0) 4. Solve for the esire rte of chge. (y = x y x ) 5. Sustitute the kow rtes of chge qutities ito the equtio. (y = 3 4 = 8 m 3 s ) Copyright 05 y Hrol Toomey, WyzAt Tutor 4
5 Itegrtio Bsic Itegrtio Rules Itegrtio is the iverse of ifferetitio, vice vers. f(x) = 0 f(x) = k = kx 0 The Costt Multiple Rule The Sum Differece Rule The Power Rule f(x) = kx The Geerl Power Rule Reim Sum Defiitio of Defiite Itegrl Are uer curve Swp Bous Aitive Itervl Property The Fumetl Theorem of Clculus The Seco Fumetl Theorem of Clculus (See Hrol s Fumetl Theorem of Clculus Chet Sheet) Me Vlue Theorem for Itegrls The Averge Vlue for Fuctio x f (x) x = f(x) + C f(x) x = f(x) x 0 x = C k x = kx + C k f(x) x = k f(x) x [f(x) ± g(x)] x = f(x) x ± g(x) x x x = x+ + C, where + If =, the x x = l x + C If u = g(x), u = g(x) the x u u x = u+ + C, where + f(c i ) x i, where x i c i x i = x = lim f(c i) x i = f(x) x 0 f(x) x = f(x) x f(x) x = f(x) x h(x) c + f(x) x c f(x) x = F() F() x x g(x) x f(t) t = f(x) f(t) t = f(g(x))g (x) f(t) t = f(h(x))h (x) f(g(x))g (x) g(x) f(x) x = f(c)( ) Fi c. f(x) x Copyright 05 y Hrol Toomey, WyzAt Tutor 5
6 Summtio Formuls Sum of Powers Misc. Summtio Formuls c = c i = i = ( + ) i 3 = ( i) i 4 i 5 i 6 i 7 = + ( + )( + ) 6 = ( + ) 4 = = = ( + )( + )(3 + 3 ) 30 = ( + ) ( + ) = = = ( + )( + )( ) 4 = ( + ) ( ) 4 S k () = i k ( + )k+ = k + k + (k + r ) S r() i(i + ) = i + i = i(i + ) = + = i(i + )(i + ) ( + 3) 4( + )( + ) k r=0 ( + )( + ) 3 Copyright 05 y Hrol Toomey, WyzAt Tutor 6
7 Itegrtio Methos. Memorize See Lrso s -pger of commo itegrls. U-Sustitutio f(g(x))g (x)x = F(g(x)) + C Set u = g(x), the u = g (x) x f(u) u = F(u) + C u = u = x u v = uv v u u = u = v = v = 3. Itegrtio y Prts 4. Prtil Frctios 5. Trig Sustitutio for x Pick u usig the LIATED Rule: L Logrithmic : l x, log x, etc. I Iverse Trig.: t x, sec x, etc. A Algeric: x, 3x 60, etc. T Trigoometric: si x, t x, etc. E Expoetil: e x, 9 x, etc. D Derivtive of: y x P(x) Q(x) x where P(x) Q(x) re polyomils Cse : If egree of P(x) Q(x) the o log ivisio first Cse : If egree of P(x) < Q(x) the o prtil frctio expsio x x Sustututio: x = si θ Ietity: si θ = cos θ 5. Trig Sustitutio for x x x Sustututio: x = sec θ Ietity: sec θ = t θ x + x 5c. Trig Sustitutio for x + Sustututio: x = t θ Ietity: t θ + = sec θ 6. Tle of Itegrls CRC Str Mthemticl Tles ook 7. Computer Alger Systems (CAS) TI-Nspire CX CAS Grphig Clcultor TI Nspire CAS ip pp 8. Numericl Methos Riem Sum, Mipoit Rule, Trpezoil Rule, Simpso s Rule, TI WolfrmAlph Google of mthemtics. Shows steps. Free. Copyright 05 y Hrol Toomey, WyzAt Tutor 7
8 Riem Sum Mipoit Rule Trpezoil Rule Simpso s Rule TI-84 Plus TI-Nspire CAS Numericl Methos P 0 (x) = f(x) x = lim f(x i ) x i P 0 where = x 0 < x < x < < x = x i = x i x i P = mx{ x i } Types: Left Sum (LHS) Mile Sum (MHS) Right Sum (RHS) P 0 (x) = f(x) x f(x i) x = x[f(x ) + f(x ) + f(x 3) + + f(x )] where x = x i = (x i + x i ) = mipoit of [x i, x i ] Error Bous: E M K( )3 4 P (x) = f(x) x x [f(x 0) + f(x ) + f(x 3 ) + + f(x ) + f(x )] where x = x i = + i x Error Bous: E T K( )3 P (x) = f(x)x x 3 [f(x 0) + 4f(x ) + f(x ) + 4f(x 3 ) + + f(x ) + 4f(x ) + f(x )] Where is eve x = x i = + i x Error Bous: E S K( ) [MATH] fit(f(x),x,,), [MATH] [] [ENTER] Exmple: [MATH] fit(x^,x,0,) x x = 0 3 [MENU] [4] Clculus [3] Itegrl [TAB] [TAB] [X] [^] [] [TAB] [TAB] [X] [ENTER] Copyright 05 y Hrol Toomey, WyzAt Tutor 8
9 ( Prtil Frctios mpositio) f(x) = P(x) Coitio Q(x) where P(x) Q(x) re polyomils egree of P(x) < Q(x) Cse I: Simple lier ( st A egree) (x + ) Cse II: Multiple egree lier ( st A egree) (x + ) + B (x + ) + C (x + ) 3 Cse III: Simple qurtic ( Ax + B egree) (x + x + c) Cse IV: Multiple egree qurtic ( Ax + B egree) (x + x + c) + Cx + D (x + x + c) + Ex + F (x + x + c) 3 Typicl Solutio for Cses I & II x = l x + + C x + Typicl Solutio for Cses III & IV x + x = t ( x ) + C Sequece Summtio Nottio Series Arithmetic Geometric S = k k= lim = L (Limit) Exmple: (, +, +, ) S = 0 r k = 0 r k k=0 k= Summtio Expe S = (Prtil Sum) S = r + 0 r r Sum of Terms (fiite series) Sum of Terms (ifiite series) S = ( + ) S = ( + ( )) S S = 0 ( r ) r S = lim ( r ) r = r oly if r < where r is the rius of covergece ( r, r) is the itervl of covergece Recursive th Term = + = r Explicit th Term = + ( ) = 0 r Copyright 05 y Hrol Toomey, WyzAt Tutor 9
10 Covergece Tests Telescopig Series Power Series Covergece Tests Tylor Series Power Series Aout Zero Mcluri Series Tylor series out zero Mcluri Series with Remier Tylor Series (See Hrol s Series Covergece Tests Chet Sheet). th Term. Geometric Series 3. p-series 4. Altertig Series 5. Itegrl 6. Rtio 7. Root 8. Direct Compriso 9. Limit Compriso 0. Telescopig ( + ) = Coverges if lim = L Diverges if N/A Sum: S = L (x c) = 0 + (x c) + (x c) + =0 x = 0 + x + x + =0 f(x) P (x) = f() (0)! = f() (0)! =0 =0 f(x) = P (x) + R (x) x + f(+) (x ) ( + )! where x x mx lim R (x) = 0 x x x + f(x) P (x) = f() (c) (x c)! =0 f(x) = P (x) + R (x) Tylor Series with Remier = f() (c)! =0 (x c) + f(+) (x ) ( + )! where x x c lim R (x) = 0 x (x c) + Copyright 05 y Hrol Toomey, WyzAt Tutor 0
11 Expoetil Fuctios Commo Series e x = x! =0 x = e x l () = Nturl Logrithms =0 = for ll x (x l())! for ll x l ( x) = x for x < l (x) = ( ) (x ) = l ( + x) = ( ) Geometric Series = x = ( ) (x ) =0 + x = ( ) x =0 x = x =0 ( x) = x = ( ) = ( x) 3 Biomil Series = x + x + x! + x3 3! + x4 4! + (x l()) (x l())3 + x l() + + +! 3! x + x + x3 3 + x4 4 + x5 5 + (x ) (x )3 (x )4 for x < (x ) for x < for 0 < x < for x < for x < for x < x for x < x x + x3 3 x4 4 + x5 5 (x ) + (x ) (x ) 3 + (x ) 4 + x + x x 3 + x 4 + x + x + x 3 + x x + 3x + 4x 3 + 5x x + 6x + 0x 3 + 5x 4 + ( + x) r = ( r ) x =0 for x < ll complex r where ( r r k + ) = k k= r(r )(r ) (r + ) =! Trigoometric Fuctios si (x) = ( ) x + for ll x ( + )! =0 cos (x) = ( ) ()! x =0 for ll x r(r ) + rx + x r(r )(r ) + x 3 +! 3! x x3 3! + x5 5! x7 7! + x9 9! x! + x4 4! x6 6! + x8 8! Copyright 05 y Hrol Toomey, WyzAt Tutor
12 t (x) = B ( 4) ( 4 ) ()! = for x < π Beroulli Numers: x B 0 =, B =, B = 6, B 4 = 30, B 6 = 4, B 8 = 30, B 0 = 5 66, B = , B 4 = 7 6 sec (x) = ( ) E ()! =0 x for x < π Euler Numers: E 0 =, E =, E 4 = 5, E 6 = 6, E 8 =,385, E 0 = 50,5, E =,70,765 rcsi (x) = ()! (!) ( + ) =0 for x rccos (x) = π rcsi (x) for x rct (x) = ( ) ( + ) x+ =0 for x <, x ±i Hyperolic Fuctios sih (x) = ex e x = x+ ( + )! =0 cosh (x) = ex + e x = x ()! =0 th (x) = B 4 (4 ) ()! = for x < π rcsih (x) = ( ) ()! (!) ( + ) =0 for x x + for ll x for ll x x rccosh (x) = π i i! ( + ) =0 for x x+ rcth (x) = ( + ) =0 for x <, x ± x + x + x + x3 x5 x7 x ! 5! 7! 9! = x + 3 x3 + 5 x x x9 + x! + 5 x4 x6 x8 x ,5 4! 6! 8! 0! + π i x + x x x π x3 x 3 3x x x x3 3 + x5 5 x7 7 + x9 9 x + x3 3! + x5 5! + x7 7! + x9 9! + + x! + x4 4! + x6 6! + x8 8! + x x3 x5 x7 x ! 5! 7! 9! x 3 x3 + 5 x x x9 x x x x i x3 i 3x5 i 3 5x7 i x x + x3 3 + x5 5 + x7 7 + x9 9 + Copyright 05 y Hrol Toomey, WyzAt Tutor
Calculus Summary Sheet
Clculus Summry Sheet Limits Trigoometric Limits: siθ lim θ 0 θ = 1, lim 1 cosθ = 0 θ 0 θ Squeeze Theorem: If f(x) g(x) h(x) if lim f(x) = lim h(x) = L, the lim g(x) = L x x x Ietermite Forms: 0 0,,, 0,
More informationExponential and Logarithmic Functions (4.1, 4.2, 4.4, 4.6)
WQ017 MAT16B Lecture : Mrch 8, 017 Aoucemets W -4p Wellm 115-4p Wellm 115 Q4 ue F T 3/1 10:30-1:30 FINAL Expoetil Logrithmic Fuctios (4.1, 4., 4.4, 4.6) Properties of Expoets Let b be positive rel umbers.
More informationMTH213 Calculus. Trigonometry: Unit Circle ( ) ( ) ( )
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
More informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationCalculus Definitions, Theorems
Sectio 1.1 A Itrouctio To Limits Clculus Defiitios, Theorems f(c + ) f(c) m sec = = c + - c f(c + ) f(c) Sectio 1. Properties of Limits Theorem Some Bsic Limits Let b c be rel umbers let be positive iteger.
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationThings I Should Know In Calculus Class
Thigs I Should Kow I Clculus Clss Qudrtic Formul = 4 ± c Pythgore Idetities si cos t sec cot csc + = + = + = Agle sum d differece formuls ( ) ( ) si ± y = si cos y± cos si y cos ± y = cos cos ym si si
More informationReview Handout For Math 2280
Review Hout For Mth 80 si(α ± β siαcos β ± cos α si β si θ 1 [1 cos(θ] cos si α cos β 1 [si(α + β + si(α β] si cos α cos β 1 [cos(α + β + cos(α β] si x +siy si ( ( x+y cos x y Trigoometric Ietites cos(α
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Nme : Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits c e prite give to the istructor or emile to the istructor t jmes@richl.eu.
More informationDifferentiation Formulas
AP CALCULUS BC Fil Notes Trigoometric Formuls si θ + cos θ = + t θ = sec θ 3 + cot θ = csc θ 4 si( θ ) = siθ 5 cos( θ ) = cosθ 6 t( θ ) = tθ 7 si( A + B) = si Acos B + si B cos A 8 si( A B) = si Acos B
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
More informationAP Calculus AB AP Review
AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationMAS221 Analysis, Semester 2 Exercises
MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)
More informationBC Calculus Path to a Five Problems
BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationTest Info. Test may change slightly.
9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More information10.5 Test Info. Test may change slightly.
0.5 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum)
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationCourse 121, , Test III (JF Hilary Term)
Course 2, 989 9, Test III (JF Hilry Term) Fridy 2d Februry 99, 3. 4.3pm Aswer y THREE questios. Let f: R R d g: R R be differetible fuctios o R. Stte the Product Rule d the Quotiet Rule for differetitig
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More information1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationMath 104: Final exam solutions
Mth 14: Fil exm solutios 1. Suppose tht (s ) is icresig sequece with coverget subsequece. Prove tht (s ) is coverget sequece. Aswer: Let the coverget subsequece be (s k ) tht coverges to limit s. The there
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More information4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why?
AB CALCULUS: 5.3 Positio vs Distce Velocity vs. Speed Accelertio All the questios which follow refer to the grph t the right.. Whe is the prticle movig t costt speed?. Whe is the prticle movig to the right?
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationCalculus BC Bible. (3rd most important book in the world) (To be used in conjunction with the Calculus AB Bible)
PG. Clculus BC Bile (3rd most importt ook i the world) (To e used i cojuctio with the Clculus AB Bile) Topic Rectgulr d Prmetric Equtios (Arclegth, Speed) 3 4 Polr (Polr Are) 4 5 Itegrtio y Prts 5 Trigoometric
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More informationThe Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
More informationCalculus BC Bible. (3rd most important book in the world) (To be used in conjunction with the Calculus AB Bible)
PG. Clculus BC Bile (rd most importt ook i the world) (To e used i cojuctio with the Clculus AB Bile) Topic Rectgulr d Prmetric Equtios (Arclegth, Speed) 4 Polr (Polr Are) 4 5 Itegrtio y Prts 5 Trigoometric
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More information( ) = A n + B ( ) + Bn
MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = + 3 + This is coverget geometric series where r = d
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationNational Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationMathematical Notation Math Calculus for Business and Social Science
Mthemticl Nottio Mth 190 - Clculus for Busiess d Socil Sciece Use Word or WordPerfect to recrete the followig documets. Ech rticle is worth 10 poits d should e emiled to the istructor t jmes@richld.edu.
More informationis continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a
. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
More informationArea, Volume, Rotations, Newton s Method
Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y)
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More informationBasic Limit Theorems
Bsic Limit Theorems The very "cle" proof of L9 usig L8 ws provided to me by Joh Gci d it ws this result which ispired me to write up these otes. Absolute Vlue Properties: For rel umbers x, d y x x if x
More informationThe Basic Properties of the Integral
The Bsic Properties of the Itegrl Whe we compute the derivtive of complicted fuctio, like x + six, we usully use differetitio rules, like d [f(x)+g(x)] d f(x)+ d g(x), to reduce the computtio dx dx dx
More informationF x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2
0 微甲 07- 班期中考解答和評分標準 5%) Fid the poits o the surfce xy z = tht re closest to the origi d lso the shortest distce betwee the surfce d the origi Solutio Cosider the Lgrge fuctio F x, y, z, λ) = x + y + z
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationAP Calculus Formulas Matawan Regional High School Calculus BC only material has a box around it.
AP Clcls Formls Mtw Regiol High School Clcls BC oly mteril hs bo ro it.. floor fctio (ef) Gretest iteger tht is less th or eql to.. (grph) 3. ceilig fctio (ef) Lest iteger tht is greter th or eql to. 4.
More informationCITY UNIVERSITY LONDON
CITY UNIVERSITY LONDON Eg (Hos) Degree i Civil Egieerig Eg (Hos) Degree i Civil Egieerig with Surveyig Eg (Hos) Degree i Civil Egieerig with Architecture PART EXAMINATION SOLUTIONS ENGINEERING MATHEMATICS
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationFor students entering Honors Precalculus Summer Packet
Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success
More informationSharjah Institute of Technology
For commets, correctios, etc Plese cotct Ahf Abbs: hf@mthrds.com Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm:
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationBRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I
EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More informationAP Calculus BC Formulas, Definitions, Concepts & Theorems to Know
P Clls BC Formls, Deiitios, Coepts & Theorems to Kow Deiitio o e : solte Vle: i 0 i 0 e lim Deiitio o Derivtive: h lim h0 h ltertive orm o De o Derivtive: lim Deiitio o Cotiity: is otios t i oly i lim
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More information8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
More information5.1 - Areas and Distances
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 63u Office Hours: R 9:5 - :5m The midterm will cover the sectios for which you hve received homework d feedbck Sectios.9-6.5 i your book.
More information0 dx C. k dx kx C. e dx e C. u C. sec u. sec. u u 1. Ch 05 Summary Sheet Basic Integration Rules = Antiderivatives Pattern Recognition Related Rules:
Ch 05 Smmry Sheet Bic Itegrtio Rle = Atierivtive Ptter Recogitio Relte Rle: ( ) kf ( ) kf ( ) k f ( ) Cott oly! ( ) g ( ) f ( ) g ( ) f ( ) g( ) f ( ) g( ) Bic Atierivtive: C 0 0 C k k k k C 1 1 1 Mlt.
More information{ } { S n } is monotonically decreasing if Sn
Sequece A sequece is fuctio whose domi of defiitio is the set of turl umers. Or it c lso e defied s ordered set. Nottio: A ifiite sequece is deoted s { } S or { S : N } or { S, S, S,...} or simply s {
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More informationName: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More informationf(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that
Uiversity of Illiois t Ur-Chmpig Fll 6 Mth 444 Group E3 Itegrtio : correctio of the exercises.. ( Assume tht f : [, ] R is cotiuous fuctio such tht f(x for ll x (,, d f(tdt =. Show tht f(x = for ll x [,
More informationCalculus. dx dx + v du. v du dx - u dv
Clculus I. Derivtives dy f (x + Dx)- f (x) = f '(x) = Dx Æ0 Dx The Product Rule: d dv (uv) = u + v du The Quotiet Rule: d Ê u ˆ v du Á = - u dv Ë v v Positio, Velocity, d Accelertio v = ds = dv dt dt Chi
More informationThe Definite Riemann Integral
These otes closely follow the presettio of the mteril give i Jmes Stewrt s textook Clculus, Cocepts d Cotexts (d editio). These otes re iteded primrily for i-clss presettio d should ot e regrded s sustitute
More information82A Engineering Mathematics
Clss Notes 9: Power Series /) 8A Egieerig Mthetics Secod Order Differetil Equtios Series Solutio Solutio Ato Differetil Equtio =, Hoogeeous =gt), No-hoogeeous Solutio: = c + p Hoogeeous No-hoogeeous Fudetl
More informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationAP Calculus Notes: Unit 6 Definite Integrals. Syllabus Objective: 3.4 The student will approximate a definite integral using rectangles.
AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.4 The studet will pproimte defiite itegrl usig rectgles. Recll: If cr is trvelig t costt rte (cruise cotrol), the its distce trveled is equl to rte
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationSecond Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem
Secod Me Vlue Theorem for Itegrls By Ng Tze Beg This rticle is out the Secod Me Vlue Theorem for Itegrls. This theorem, first proved y Hoso i its most geerlity d with extesio y ixo, is very useful d lmost
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationAlgebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by
Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric
More informationDefinition Integral. over[ ab, ] the sum of the form. 2. Definite Integral
Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More informationLecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,
More information9.5. Alternating series. Absolute convergence and conditional convergence
Chpter 9: Ifiite Series I this Chpter we will be studyig ifiite series, which is just other me for ifiite sums. You hve studied some of these i the pst whe you looked t ifiite geometric sums of the form:
More information