Harold s Calculus Notes Cheat Sheet 15 December 2015

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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

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,