Lessons 2-4: Limits Limit Solving Strtegy for Finl Exm Review Exm 1 Mteril For piecewise functions, you lwys nee to look t the left n right its! If f(x) is not piecewise function, plug c into f(x), i.e., fin f(c). There re three cses tht will occur: Cse 1: f(c) = finite numer, then Cse 2: f(c) = nonzero numer 0 f(x) =? x c f(x) = f(c) x c, then look t left n right its to ecie if the it is,, DNE. Cse 3: f(c) = 0, then mnipulte f(x) to get cse 1 or cse 2. Usully you nee to fctor n 0 simplify f(x), or if you hve squre roots, you nee to rtionlize f(x) y multiplying the numertor n enomintor y the conjugte rememer 2 2 = ( )( + ). One-Sie Limits (nee to consier for Cse 2 ove n for piecewise functions, ut you my e explicitly ske to fin them s well.) Left Limit: f(x) for x < c x c Right Limit: f(x) for x > c x c + f(x) = x c Rememer tht the it f(x) exists if x c A it cn exist even if the function is unefine! If function is unefine, or the vlue of the x c + f(x) function oes not equl the it there, those re issues of continuity. If you re still unsure wht the it is, you cn fin it numericlly or grphiclly. To fin the it numericlly, pick vlues relly close to c. For exmple, if c = 1, pick 0.9999 (for the left it since 0.9999 < 1) n 1.0001 (for the right it since 1.0001 > 1) n plug them into f(x). Fining the it grphiclly will e ifficult unless you lrey hve goo ie how f(x) looks or if you re given the grph of f(x) Prctice Prolems: Exm 1 - # 1, 3, 4 Exm 1 Review - # 1, 2, 4, 6, 8, 9, 12, 13, 14, 17, 18, 20, 30, 31
Lesson 5: Continuity f(x) is continuous t x = c if ALL of the following 3 things re true: f(c) is efine f(x) exists x c f(x) = f(c) x c Types of Discontinuity (which conitions ove hol t the prticulr x-vlue?) Clssifiction Conition Hole Jump Verticl Asymptote Mye Mye No Yes (finite) No (ut oth re finite) Mye (,, or DNE) No No No Hole: Hppen when we cncel out fctor in rtionl function. (Ex. f(x) = (x+1)(x 1) = x + 1 hs hole t x = 1 ecuse we cncele out the x 1 fctor from the enomintor.) They cn lso hppen in piecewise functions when the it exists, ut the function is not efine there or if the function vlue oes not equl the it. x 1 Jump: Hppen only in piecewise functions when the left n right its re finite n o not mtch. Verticl Asymptote: Hppen when we cn t cncel out fctor in the enomintor. (Ex. f(x) = x 1 = 1 hs verticl symptote t x = 1 ecuse we cn t cncel out the x + 1 fctor. (x 1)(x+1) x+1 Notice tht this function lso hs hole t x = 1 ecuse we cn cncel the x 1 fctor.) For piecewise functions, check the x-vlues where the ifferent functions meet to see if the it exists n/or if the it equls the vlue of the function to fin holes or jumps. For non-piecewise functions, fin where the enomintor equls 0. Fctor the numertor n enomintor. If fctor cncels from the enomintor, there will e hole. If fctor oes not cncel from the enomintor, there will e verticl symptote. Prctice Prolems: Exm 1 - # 2, 3 Exm 1 Review - # 2, 3, 8, 9, 17, 19, 32
Lesson 6: Limit Definition of the Derivtive The erivtive of function is the slope of the. tngent line Limit Definition of Derivtive: f (x) = h 0 f(x + h) f(x) h We nee to keep in min cse 3 for fining it ecuse generlly, we en up with 0 when trying to 0 use the it efinition for erivtive. You my nee to expn the numertor or enomintor, rtionlize, fctor, etc. to get the h in the enomintor to cncel with n h in the numertor. Prctice Prolems: Exm 1 - # 5 Exm 1 Review - # 7, 15, 33 Lessons 7-10: Instntneous Rtes of Chnge, Differentition Rules n Trig Functions When you re given function, you tke the erivtive to fin the rte of chnge of the function. Py ttention to units! Rememer if s(t) is position function, s (t) = v(t) the velocity function n v (t) = (t) the ccelertion function Py close ttention to opertions, i.e., ing, sutrcting, multiplying, or iviing. If functions re multiplie (Ex: f(x) = e x sin (x), f(x) = x 2 e x, etc), use the prouct rule to tke the erivtive. If functions re ivie (Ex: f(x) = erivtive. x x 2 +1, f(x) = ex sin(x), etc), use the quotient rule to tke the Prctice Prolems: Exm 1 - # 6, 7, 8, 9, 10, 11, 12 Exm 1 Review - # 5, 10, 11, 16, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40
Exm 2 Mteril Lessons 11-12: The Chin Rule n Derivtive of Nturl Log x [f(g(x))] = f (g(x)) g (x) or x [Out(In)] = Out (In) In The erivtive of the outsie (with the insie plugge into the erivtive) times the erivtive of the insie. Specil Cse: x (ef(x) ) = f (x) e f(x) (Note: the exponent oes NOT chnge.) Prctice Prolems: Exm 2 - # 3, 4, 10 Exm 2 Review - # 6, 9, 10, 11, 12, 13, 18, 19, 29, 33 Lesson 13: Higher Orer Derivtives f (n) (x), n y x n, y(n) ll men to tke the erivtive n times. Prctice Prolems: Exm 2 - # 1, 7 Exm 2 Review - # 1, 2, 22, 24, 25, 30, 38 Lesson 14: Implicit Differentition x [f(y)] = y x f (y) Use when y is not explicitly solve for. For instnce: y 2 = e x, cos(xy) = x, etc. Bsiclly, nytime we touch/chnge y, we nee to multiply the term y y or y, so tke the erivtive of wht you see n multiply y y or y. For exmple, x (ey ) = y x ey x (ln(y)) = y x 1 y x x
x (x y) = 1 y + x y x (prouct rule) x (exy ) = (y + x y ) x exy (chin rule n prouct rule) x (y2 ) = y 2y x y n y will not pper insie trig function, enomintor, or exponent. x Prctice Prolems: Exm 2 - # 6, 11 Exm 2 Review - # 3, 14, 20, 21, 32 Lessons 15-16: Relte Rtes Determine wht formul you nee to use. Plug in ny constnt quntities, i.e. quntities tht o NOT chnge with time. Tke erivtive of oth sies with respect to time. Py ttention to units! For right tringles, we use the Pythgoren Theorem x 2 + y 2 = D 2. For ngles, pick the trig ientity tht hs constnt quntity n the rte of the other quntity given. Prctice Prolems: Exm 2 - # 2, 12 Exm 2 Review - # 4, 8, 15, 16, 17, 23, 28, 36, 37 Lessons 17-18: Reltive Extrem, Criticl Numers, n the First Derivtive Test We cn use the first erivtive to fin out informtion out the function itself. Criticl Numers: where the erivtive of function (f (x)) is = 0 or is unefine. A criticl numer must e in the omin of the function! Reltive Extrem: minimums or mximums of the entire function; function must e efine t the point; cn hve no reltive extrem or cn hve multiple reltive minim or reltive mxim f(x) is incresing when f (x) > 0 f(x) is ecresing when f (x) < 0
Reltive extrem only occur t criticl numers, ut not ll criticl numers hve reltive extrem. o We fin the criticl numers (lwys x-vlues) then use the First or Secon Derivtive Test to check t ech x-coorinte to see if we hve reltive minimum, reltive mximum or neither. First Derivtive Test: Let c e criticl numer for f(x), i.e. f (c) = 0. o If f (x) goes from positive to negtive t x = c, (i.e., f(x) goes from incresing to ecresing), then f(c) is reltive mximum. o If f (x) goes from negtive to positive t x = c, (i.e., f(x) goes from ecresing to incresing), then f(c) is reltive minimum. o If f (x) oes not chnge sign t x = c, f(c) is neither min nor mx. Use the originl function to fin the vlue of the minimum or mximum, i.e. f(c). Nottion: x = c is where the reltive min or mx occurs; f(c) is the vlue of the minimum or mximum. If they sk you to fin the minimum or mximum, they wnt you to fin the vlue of the function t tht point. Prctice Prolems: Exm 2 - # 5, 8, 9 Exm 2 Review - # 5, 7, 26, 27, 31, 34, 35
Exm 3 Mteril Lesson 19: Concvity, Inflection Points, n the Secon Derivtive Test f(x) is concve up when f (x) > 0 f(x) is concve own when f (x) < 0 An inflection point is where f(x) chnges concvity. To fin the y-coorinte, plug the x-vlue into the originl function f(x). Secon Derivtive Test: Let c e criticl numer for f(x) (i.e. f (c) = 0). o If f (c) > 0, f(x) is concve up t x = c, so f(c) is reltive minimum. o If f (c) < 0, f(x) is concve own t x = c, so f(c) is reltive mximum. o If f (c) = 0, the test is inconclusive, so use the first erivtive test to clssify the criticl numer. Prctice Prolems: Exm 3 - # 1 Exm 3 Review - # 1, 2, 11, 16, 23, 24, 33, 38, 39 Lesson 20: Asolute Extrem on n Intervl How to fin the solute extrem of f(x) on close intervl [, ]: Fin ll criticl numers of f(x) (where f (x) = 0). Evlute f(x) t ll criticl numers in [, ], n t x = n x =. Compre the f(x) vlues n ientify the solute min n solute mx. Prctice Prolems: Exm 3 - # 2 Exm 3 Review - # 12, 20, 36 Lesson 21: Grphicl Interprettions of Derivtives When given grph of f (x), we cn fin where f(x) is/hs the following: 1. criticl numers where f (x) = 0 or oes not exist, i.e. where the grph of f (x) touches the x-xis. 2. incresing - where f (x) > 0, i.e where the grph of f (x) is ove the x-xis. 3. ecresing - where f (x) < 0, i.e where the grph of f (x) is elow the x-xis. 4. reltive mximum - where the grph of f (x) goes from positive to negtive.
5. reltive minimum - where the grph of f (x) goes from negtive to positive. 6. concve up - where f (x) > 0, i.e. where f (x) is incresing 7. concve own - where f (x) < 0, i.e. where f (x) is ecresing 8. inflection point - f (x) chnges sign, i.e. f (x) hs horizontl tngent line n switches from incresing to ecresing or ecresing to incresing Prctice Prolems: Exm 3 - # 5 Exm 3 Review - # 4, 40 Lesson 22: Limits t Infinity When evluting it with x or x, tke the highest x power in the numertor n enomintor with their coefficients, cncel, n then evlute. Ex. Ex. 1 x 2 3x 3 x x 3 x + 1 = 3x 3 x x 3 = 3 = 3 x x x 2 7 23x + 1 = x x 2 23x = x x 23 = Ex. x x 2 + x 1 7x 3 + x 2 = x x 2 7x 3 = x 1 7x = 0 Asymptotes (for f(x)) o Verticl: Simplify f(x) n set the enomintor equl to 0 n solve for x. (x = #) o Horizontl: Fin f(x) n f(x). If one of the its equls finite L, y = L is the HA. If x x the its re infinite ( or ), there is no HA. o Slnt: Occur when the egree of numertor is exctly one more thn egree of the enomintor. Fin y polynomil ivision. (Of the form y = mx +.) Prctice Prolems: Exm 3 - # 3, 4, 8, Exm 3 Review - # 5, 6, 13, 14, 26, 27, 35, 37
Lesson 23: Curve Sketching To sketch curve, fin the following: 1. x-intercepts (when y = 0) 2. y-intercept (when x = 0) 3. Incresing/Decresing Intervls (use first erivtive) 4. Concve Up/Concve Down Intervls (use secon erivtive) 5. Inflection Points (use secon erivtive) 6. Verticl Asymptotes 7. Horizontl Asymptotes 8. Slnt Asymptotes When mking numer lines for the first n secon erivtives, e sure to mrk where the function is unefine, check the sign (positive or negtive) on oth sies of the iscontinuity. None of the intervls shoul inclue the x-vlues where f(x) is unefine! Prctice Prolems: Exm 3 - # 8 Exm 3 Review - # 3, 25 Lessons 24-26: Optimiztion Drw picture! 1. Ientify the quntity to e optimize (mximize or minimize). 2. Ientify the constrint(s). 3. Solve the constrint for one of the vriles. Sustitute this into the eqution from Step 1. 4. Tke the erivtive with respect to the vrile. 5. Set the erivtive equl to zero n solve for the vrile. 6. Fin the esire quntity. Note: In most cses, you only get one plusile solution. However, you cn use the 1 st DT or 2 n DT to check if your nswer in step 5 is the criticl numer where the minimum or mximum occurs. Prctice Prolems: Exm 3 - # 10, 11, 12 Exm 3 Review - # 7, 8, 9, 10, 15, 19, 21, 22, 28, 29, 30, 44, 45
Lessons 27-28: Antierivtives n Inefinite Integrtion Rewrite the function, if necessry, to utilize the tle of integrtion. We hve NOT lerne methos to uno the chin, prouct, or quotient rules. Once you hve integrte, you cn tke the erivtive to oule check your integrtion. When given n initil vlue or initil vlues, you cn use them to fin the prticulr solution (i.e. you cn solve for C). Prctice Prolems: Exm 3 - # 6, 7, 9 Exm 3 Review - # 17, 18, 31, 32, 34, 41, 42, 43
Lesson 29: Are n Riemnn Sums Summtions: 5 Post Exm 3 Mteril Ex. (i + i 2 ) = (2 + 2 2 ) + (3 + 3 2 ) + (4 + 4 2 ) + (5 + 5 2 ) = (6) + (12) + (20) + (30) = 68 i=2 Estimte the re uner the curve f(x) with n rectngles on the intervl [, ]. Left Riemnn Sum: Right Riemnn Sum: where x = n x i = + i Δx, n 1 f(x i ) Δx i=0 n f(x i ) Δx i=1 i = 0, 1, 2, 3,, n After Exm 3 Review - # 4, 13, 24, 30, 34, 35, 45, 46, 49 Lesson 30: Definite Integrls f(x) x is the re uner the curve f(x) (etween f(x) n the x-xis) on the intervl [, ]. Py ttention to coefficients in given integrls!! Properties of Definite Integrls: o f(x)x o f(x)x o k f(x)x = 0 o (f(x) ± g(x)) x c o f(x)x = f(x)x = k f(x)x = f(x)x = f(x)x c + f(x)x ± g(x)x After Exm 3 Review - # 5, 23, 47, 48
Lessons 31-32: The Funmentl Theorem of Clculus If f (x)x = f(x), then f (x)x = f() f(). When we re given n initil vlue, we cn fin the prticulr solution for inefinite integrtion (lessons 27-28). When we o NOT hve n initil vlue, we cn fin the chnge in the function etween two points n y using the funmentl theorem of clculus. After Exm 3 Review - # 3, 6, 7, 8, 9, 14, 15, 16, 21, 22, 25, 26, 27, 36, 37, 38, 39, 40, 41, 44 Lesson 33: Numericl Integrtion Approximte the re uner f(x) on [, ] with n trpezois: T n = 1 2 Δx [f(x 0) + 2f(x 1 ) + 2f(x 2 ) + + 2f(x n 1 ) + f(x n )] where x = n x i = + i Δx, i = 0, 1, 2, 3,, n After Exm 3 Review - # 10, 28, 42, 49 Lessons 34/36: Exponentil Growth/Decy If the rte of chnge of y is proportionl to y, i.e. y = ky where k is the growth/ecy rte, then C = y(0) t y(t) = Ce kt If k is negtive, we hve exponentil ecy. If k is positive, we hve exponentil growth. For svings ccounts with interest compoune continuously, we hve A(t) = Pe rt, where r is the interest rte s eciml, P is the principle (originl mount investe), n t is time in yers. The hlf-life t 1/2 for exponentil/rioctive ecy is the time it tkes for hlf of n mount to ecy. k = ln (1 2 ) t 1/2 After Exm 3 Review - # 1, 2, 11, 12, 17, 18, 19, 20, 29, 31, 32, 33, 43
Tle of Derivtives x (c) = 0 Tle of Integrtion 0 x = C x (xn ) = n x n 1, n 0 x n x = 1 n + 1 xn+1 + C, n 1 k x = kx + C x (ex ) = e x e x x = e x + C x (ln(x)) = 1 x, x > 0 1 x = ln x + C x (sin(x)) = cos (x) x (cos(x)) = sin (x) x x (tn(x)) = sec2 (x) (sec(x)) = sec(x) tn (x) x x (cot(x)) = csc2 (x) (csc(x)) = csc(x) cot (x) x (c f(x)) = c f (x) x x (f(x) ± g(x)) = f (x) ± g (x) x (f(x) g(x)) = f (x)g(x) + f(x)g (x) x ( x (f(x) g(x) ) = f (x)g(x) f(x)g (x) (g(x)) 2 OR Top Bottom ) = Top Bottom Top Bottom (Bottom) 2 [f(g(x))] = f (g(x)) g (x) x OR [Out(In)] = Out (In) In x cos(x) x = sin(x) + C sin(x) x = cos(x) + C sec 2 (x) x = tn(x) + C sec(x) tn(x) x = sec(x) + C csc 2 (x) x = cot(x) + C csc(x) cot(x) x = csc(x) + C cf (x) x = c f (x) x (f (x) ± g (x)) x = f (x) x ± g (x) x We on t hve wy to uno proucts or quotients except for the specific trig functions ove. We lso cnnot uno the chin rule yet. To integrte things tht look like proucts, quotients, or chins, we nee to rewrite them! x [ef(x) ] = f (x) e f(x)