Harold s Calculus Notes Cheat Sheet 26 April 2016

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1 Hrol s Clculus Notes Chet Sheet 26 April 206 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 2 is cotiuous fuctio. f(x) f(c) = (x 2 ) (c 2 ) = x 2 c 2 + = x 2 c 2 = (x + c)(x c) = (x + c) (x c) Sice (x + c) 2c f(x) f(c) 2c (x c) < ε So give ε > 0, we c choose δ = ε > 0 i the 2c Defiitio of Cotiuity. So sustitutig the chose δ for (x c) we get: f(x) f(c) 2c ( ε) = ε 2c Sice oth coitios re met, f(x) is cotiuous. si x lim = x 0 x cos x lim = 0 x 0 x Copyright y Hrol Toomey, WyzAt Tutor

2 Derivtives Defiitio of Derivtive of Fuctio Slope Fuctio Nottio for Derivtives 0. The Chi Rule. The Costt Multiple Rule 2. The Sum Differece Rule 3. The Prouct Rule 4. The Quotiet Rule 5. The Costt Rule 6. The Power Rule 6. The Geerl Power Rule 7. The Power Rule for x 8. Asolute Vlue 9. Nturl Logorithm 0. Nturl Expoetil. Logorithm 2. Expoetil 3. Sie 4. Cosie 5. Tget 6. Cotget 7. Sect (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 [f(g(x))] = f (g(x))g (x) y x = y u u x x [cf(x)] = cf (x) x [f(x) ± g(x)] = f (x) ± g (x) x [fg] = fg + g f x [f g ] = gf fg g 2 x [c] = 0 x [x ] = x x [u ] = u u where u = u(x) x [x] = (thik x = x x 0 = ) x [ x ] = x x x [l x] = x x [e x ] = e x x [log x] = (l ) x x [x ] = (l ) x [si(x)] = cos(x) x [cos(x)] = si(x) x x [t(x)] = sec2 (x) x [cot(x)] = csc2 (x) [sec(x)] = sec(x) t(x) x Copyright y Hrol Toomey, WyzAt Tutor 2

3 Derivtives 8. Cosect 9. Arcsie 20. Arccosie 2. Arctget 22. Arccotget 23. Arcsect 24. Arccosect 25. Hyperolic Sie 26. Hyperolic Cosie 27. Hyperolic Tget 28. Hyperolic Cotget 29. Hyperolic Sect 30. Hyperolic Cosect 3. Hyperolic Arcsie 32. Hyperolic Arccosie 33. Hyperolic Arctget 34. Hyperolic Arccotget 35. Hyperolic Arcsect 36. Hyperolic Arccosect (See Lrso s -pger of commo erivtives) [csc(x)] = csc(x) cot(x) x x [si (x)] = x 2 x [cos (x)] = x 2 x [t (x)] = + x 2 x [cot (x)] = + x 2 x [sec (x)] = x x 2 x [csc (x)] = x x 2 [sih(x)] = cosh(x) x [cosh(x)] = sih(x) x x [th(x)] = sech2 (x) x [coth(x)] = csch2 (x) [sech(x)] = sech(x) th(x) x [csch(x)] = csch(x) coth(x) x x [sih (x)] = x 2 + x [cosh (x)] = x 2 x [th (x)] = x 2 x [coth (x)] = x 2 x [sech (x)] = x x 2 x [csch (x)] = x + x 2 Positio Fuctio s(t) = 2 gt2 + 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) Copyright y Hrol Toomey, WyzAt Tutor 3

4 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) 2. 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)) 2. 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)) 2. 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 2. 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 2. 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 y Hrol Toomey, WyzAt Tutor 4

5 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 = 5 m; y = 20 m; x = 2 m s ; y =? ) 2. Costruct equtio reltig these qutities. (x 2 + y 2 = r 2 ) 3. Differetite oth sies of the equtio. (2xx + 2yy = 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 m 2 s ) Copyright y Hrol Toomey, WyzAt Tutor 5

6 Summtio Formuls Sum of Powers Misc. Summtio Formuls c = c i = i 2 = ( + ) 2 i 3 = ( i) i 4 i 5 i 6 i 7 = ( + )(2 + ) 6 2 = 2 ( + ) 2 4 = = = ( + )(2 + )( ) 30 = 2 ( + ) 2 ( ) 2 = = = ( + )(2 + )( ) 42 = 2 ( + ) 2 ( ) 24 S k () = i k ( + )k+ = k + k + (k + r ) S r() i(i + ) = i 2 + i = i(i + ) = + = i(i + )(i + 2) ( + 3) 4( + )( + 2) k r=0 ( + )( + 2) 3 Copyright y Hrol Toomey, WyzAt Tutor 6

7 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 2 < < 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 2) + f(x 3) + + f(x )] where x = x i = (x 2 i + x i ) = mipoit of [x i, x i ] Error Bous: E M K( ) P (x) = f(x) x x 2 [f(x 0) + 2f(x ) + 2f(x 3 ) + + 2f(x ) + f(x )] where x = x i = + i x Error Bous: E T K( )3 2 2 P 2 (x) = f(x)x x 3 [f(x 0) + 4f(x ) + 2f(x 2 ) + 4f(x 3 ) + + 2f(x 2 ) + 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^2,x,0,) x 2 x = 0 3 [MENU] [4] Clculus [3] Itegrl [TAB] [TAB] [X] [^] [2] [TAB] [TAB] [X] [ENTER] Copyright y Hrol Toomey, WyzAt Tutor 7

8 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 Me Vlue Theorem for Itegrls The Averge Vlue for Fuctio (See Hrol s Fumetl Theorem of Clculus Chet Sheet) 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 y Hrol Toomey, WyzAt Tutor 8

9 Itegrtio Methos. Memorize See Lrso s -pger of commo itegrls 2. 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 2 x 2 Pick u usig the LIATED Rule: L Logrithmic : l x, log x, etc. I Iverse Trig.: t x, sec x, etc. A Algeric: x 2, 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 2: If egree of P(x) < Q(x) the o prtil frctio expsio 2 x 2 x Sustututio: x = si θ Ietity: si 2 θ = cos 2 θ 5. Trig Sustitutio for x 2 2 x 2 2 x Sustututio: x = sec θ Ietity: sec 2 θ = t 2 θ x x 5c. Trig Sustitutio for x Sustututio: x = t θ Ietity: t 2 θ + = sec 2 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 y Hrol Toomey, WyzAt Tutor 9

10 Coitio Prtil Frctios (See Hrol s Prtil Frctios Chet Sheet) f(x) = P(x) Q(x) where P(x) Q(x) re polyomils egree of P(x) < Q(x) If egree of P(x) Q(x) the o log ivisio first P(x) (x + )(cx + ) Exmple Expsio 2 (ex 2 + fx + g) A = (x + ) + B (cx + ) + C (cx + ) 2 + Dx + E (ex 2 + fx + g) Typicl Solutio x = l x + + C x + Sequece Sequeces & Series Geometric Series (See Hrol s Series Chet Sheet) lim = L (Limit) Exmple: (, +, +2, ) S = lim ( r ) r = r oly if r < where r is the rius of covergece ( r, r) is the itervl of covergece Covergece Tests Series Covergece Tests (See Hrol s Series Covergece Tests Chet Sheet). Divergece or th Term 6. Rtio 2. Geometric Series 7. Root 3. p-series 8. Direct Compriso 4. Altertig Series 9. Limit Compriso 5. Itegrl 0. Telescopig Tylor Series Tylor Series (See Hrol s Tylor Series Chet Sheet) + = f() (c)! =0 f(x) = P (x) + R (x) (x c) + f(+) (x ) ( + )! (x c) + where x x c (worst cse scerio x ) lim x + R (x) = 0 Copyright y Hrol Toomey, WyzAt Tutor 0