2. Limits. Reading: 2.2, 2.3, 4.4 Definitions.

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1 TOPICS COVERED IN MATH 162 This is summry of the topics we cover in 162. You my use it s outline for clss, or s review. Note: the topics re not necessrily liste in the orer presente in clss. 1. Review Reing: Appenix A ( Absolute vlue ), Appenix B, Appenix C ( Circles, prbols ), Section 1.1, Appenix D { x if x 0 Definition. The bsolute vlue of x is efine s x = x if x < 0. Note: the bsolute vlue is efine s function efine piecewise. When working with functions tht inclue bsolute vlues, it is often esier to rewrite them s functions efine piecewise. You nee to be comfortble working with n grphing functions efine piecepise. Definition: The grphs of n equtions in two vribles x,y is the set of points in the x-y plne tht stisfy the eqution. Know how to grph lines, circles, prbols. Complete the squre when necessry. Definition: A function is rule tht ssigns to every element x in set clle the omin unique vlue y in set clle the rnge. Functions cn be represente in wors, by formul, by grph, by tble. Definition: The grph of function f(x) is the grph of the eqution y = f(x). Know how to grph bsic functions, incluing x + b,x n, x, x, qurtics, ny polynomil in fctore form, trnsltions of the bsic functions, functions efine piecewise (eg, Hevisie). Definition: A function f(x) is even if f(x) = f( x). A function f(x) is o if f(x) = f( x). Be ble to check (using this efinition) whether function is even or o. Qurtic mx/min problems: Be ble to fin the bsolute mximum or minimum vlue of qurtic function n where it is ttine. The six bsic trig functions. Know the efinitions, vlues for specil ngles, n grphs of ll six bsic trig functions. Know some simple ientities s in Set 3:Problem 7. Amplitue, perio, frequency, phse shift. The function f(x) = sin(ωx + φ) oscilltes with mplitue. The perio is the length of the intervl in which f unergoes one oscilltion. This is esy to see if φ = 0. In tht cse we know tht the sine unergoes one oscilltion if the rgument ωx [0, 2π], tht is x [0, 2π/ω]. Thus, the perio is 2π/ω. The number ω is clle the frequency (since there re ω oscilltions in the intervl [0, 2π]). The phse shift φ simply mounts to horizontl trnsltion by φ/ω. Reing: 2.2, 2.3, 4.4 Definitions. 2. Limits The limit lim x c f(x) = L. Wht this mens informlly, grphiclly, numericlly (by looking t tbles of vlues). Give grphicl n numericl exmples. One-sie limits lim x c+ f(x) = L r,lim x c f(x) = L l. Wht this mens informlly, grphiclly (n numericlly). Give exmples. Reltion between limit n one-sie limits. 1

2 Infinite limits lim x c f(x) =. This mens tht s x pproches c (from either sie) the function increses without boun. In this cse the limit oes not exist ( is not number). This nottion simply inictes the behviour of f ner c. Limit Theorems. These re use to etermine whether limit exists n to fin them if they o. The following lists the only cses tht I cn think of: Direct substitution. In mny cses the limit of lim x c f(x) cn be foun by simple subsitution (evluting the function t x = c). Limits of frctions, 0/0 cse. Here you nee to o some lgebr to etermine whether limit exists. Limits of frctions, /0 cse, 0. Here the limit oes not exist (numertor stys finite, enomintor vnishes). In this cses there is verticl symptote t x =. When plotting functions with verticl symptotes you must check the one-sie limits to etermine whether the function pproches + or. Functions efine piecewise. To fin the limit t the brekpoints you nee to fin the one-sie limits n check if they exist n gree. Functions incluing bsolute vlues. It is often esiest to rewrite them s function efine piecewise, thereby removing the bsolute vlue. Limits of ifferences,. Here you nee to o some lgebr to etermine whether limit exists. Note: we re skipping the Squeeze Theorem (not in HW). Infinite limits n verticl symptotes. If lim x c + f(x) = or or lim x c f(x) = or then f hs verticl symptote x = c. Only one of the bove is sufficient for verticl symptote. Note: To grph verticl symptotes you must fin the one-sie limits to etermine whether the limiting behviour on ech sie. Limits t infinity n horizontl symptotes. To compute limits s x ± of quotients, ivie by highest power in enomintor first. For rtionl functions, it is sufficient to etermine the leing orer terms in numertor n enomintor. For limits of the form, s before, rewrite using lgebr. If either lim f(x) = L or lim x f(x) = L x then f hs horizontl symptote y = L. (grph of f cn cross symptote repetely) 3. Continuity Reing: 2.5, pge 206, hnout on web Definition. Stte efinition of continuity. Grphicl interprettion. Determine whether given exmples re continuous or not t given points, using efinition. Types of iscontinuities. Removble, jump, infinite iscontinuities. Theorems. Theorems 4,5,7,9 let us quickly etermine lrge clss of continuous functions. One-sie continuity n continuity on close intervl. Define one-sie continuity. The min reson we nee this concept t this point is becuse we nee to know wht it mens for function to be continuous on close intervl. Tht is, continous t every interior point n one-sie continuous t the enpoints. 2

3 Intermeite n extreme vlue theorems. Stte them. Simple exmples. Fining where f(x) 0. A function cn only chnge sign t points t which either f(x) = 0 or f is iscontinuous. Use these points to fin intervls where function is positive or negtive. See hnout on web. 4. Rtes of chnge Reing: 2.1, 3.1 Slope of grph y = f(x). (1) Slope of secnt line through two points (x 1,f(x 1 ) n (x 2,f(x 2 ): lso referre to s the verge rte of chnge of f on [x 1,x 2 ] (2) Slope of tngent line t the point (c,f(c)): f(x 2 ) f(x 1 ) x 2 x 1 f(x) f(c) f(c + x) f(c) lim = lim x c x c x 0 x lso referre to s the instntneous rte of chnge of f t x = c, provie the limit exists. (Why re these two limits the sme?) Position n velocity. Suppose position p(t) of n object t time t is given. Then p(t 2 ) p(t 1 ) (1) Averge velocity on the intervl [t 1,t 2 ]: t 2 t 1 (2) Instntneous velocity v(t) t fixe time t = c is limit: provie the limit exists. p(t) p(c) p(c + t) p(c) v(c) = lim = lim t c t c t 0 t Rtes of chnge. Suppose function f(t) is given. f(t 2 ) f(t 1 ) (1) Averge rte of chnge of f(t) on [t 1,t 2 ]: t 2 t 1 (2) Instntneous rte of chnge of f(t) t t = c: provie the limit exists. f(t) f(c) f(c + t) f(c) lim = lim t c t c t 0 t Skills you nee: Compute verge n instntneous rtes of chnges, velocities, slopes. Stte uner wht conition the tngent line to grph y = f(x) t the point (c,f(c)) exists. Fin the eqution of tngent n norml lines. Give exmples when the tngent line oes not exist. Unerstn the mening of rte of chnge (either verge or instntneous) n its units, if the vribles re imensionl. 3

4 Reing: 3.1, Definition of the erivtive Definition of erivtive s limit. The erivtive of f t c is the instntneous rte of chnge of f t c n is enote by f (c). Memorize the efinition of the erivtive s limit: f f(c + h) f(c) f(x) f(c) (c) = lim = lim h 0 h x c x c provie the limit exists. (Why re these two limits the sme?) Interpret the erivtive. Interpret erivtive s slope of grph, velocity, or other instntneous rte of chnge, n etermine its units (if f n x hve units). Use the efinition to compute erivtives. Grphs of functions n their erivtives. The erivtive f (c) is the slope of the grph of f(x) t x = c. Use this fct to euce the grph of f from the grph of f. Differentibility n continuity. Define f is ifferentible t x = c. Be ble to show when function is not ifferentible using efinition. Differentibilty implies continuity but continuity oes not imply ifferentibility ( Differentibility is stronger thn continuity ). Give exmples. Other nottions for the erivtive of y = f(x): f (x) = f x = y x = x [f]. Derivtives of expressions. Exmple: x [x3 3x 2 ] Reing: 3.2, pges 129 bottom Higher erivtives Nottion. For exmple f (x) = 3 x 3 [f(x)] = 3 f x 3 (x) = y (x) = f (3) (x) Interpreting higher erivtives: If s(t) be position of prticle on line. Then s (t) = v(t) instntneous velocity t time t, n s (t) = (t) = instntneous ccelertion. Interpreting higher erivtives: Time permitting, mention the grphicl interprettion of f (x) in terms of concvity of the grph of f. (however, no homework on this n we will get bck to this lter) Reing: 3.3, 3.4 Derivtive of constnt: [c] = 0. x Power rule: x [xn ] = nx n 1, ny integer n, 7. Rules for ifferentition The erivtive of sum is the sum of the erivtives: x [xp ] = px p 1, ny rel number p f [f + g] = x x + g x or (f + g) (x) = f (x) + g (x) The erivtive of constnt times f, is the constnt times the erivtive of f: f [cf] = c x x Derivtives of polynomils (use ll bove) or 4 (cf) (x) = cf (x)

5 The prouct rule: The quotient rule: [ ] f f g = x x g + f g x f ] = x[ g f x f g x g g 2 or or ( f g (fg) (x) = f (x)g(x) + f(x)g (x) ) (x) = g(x)f (x) f(x)g (x) [g(x)] 2 Derivtives of sin(x) n cos(x): Derivtives of exponentil Reing: 3.5 x [ex ] = e x [sin(x)] = cos(x), x [cos(x)] = sin(x). x (next week, liste here for completeness) 8. Chin Rule Chin rule (the erivtive of composition): x [f(g(x))] = f ( g(x) ) g (x) In Leibnitz nottion, if y = f(u) n u = g(x) then y x = y u u u=g(x) x where the br enotes tht y/u is evlute t u = g(x). Note tht while you cnnot cncel ifferentils u (they re not vribles or functions), the fct tht it looks like you re cncelling u mkes the rule esy to remember. Its ese of use is wht me Leibnitz s nottion so useful. Repete ppliction of the chin rule (simply be systemtic). Exmple: [ (sin(5x)) 3 ] = 3sin 2 (5x) x x [sin(5x)] = 3sin2 (5x)cos(5x) x [5x] = 3sin2 (5x)cos(5x) 5 Reing: Implicit ifferentition. Simple relte rtes. If y = f(x) is given, we sy tht y is given explicitely in terms of x. If y is not given explicitely but is efine through n eqution, such s x 2 + y 2 = 1, then we sy y is given implicitely by the eqution. In tht cse we cn still fin y/x by ifferentiting both sies of the eqution with respect to x, n remembering to use the chin rule, prouct rule, etc, correctly. In this exmple: 2x + 2y y x = 0 Relte rtes of chnge. If x = x(t), y = y(t) n z = z(t) re ll epenent vribles on time, n you re given (or know) reltion between x, y, n z, for exmple x = y z 2z 2 then you cn fin reltion between the erivtives x/t, y/t n z/t using implicit ifferentition with respect to t. In the exmple x t = y y z + t 2 z z t 5zz t 5

6 NOTE: The problems on relte rtes in the homework re much simpler thn ny of the exmples worke out in 3.8, which is why no reing in 3.8 is ssigne. Instructors, plese o exmples s bove in clss. This is sufficient to prepre for homework. 10. Exponentil functions Reing: 7.2 Algebric rules for exponents. (b) x = x b x, ( ) x x = b b x, 0 = 1, xy = ( x ) y, x = 1 x Grphs of x. For vrious > 1, = 1 n < 1. Limits of exponentil functions. e h 1 Definition of e: e is the number such tht lim h 0 h Derivtive of the exponentil x [ex ] = e x. Reing: Inverse functions Definition: function is invertible if there exists nother function g such tht (f g)(x) = x n (g f)(x) = x for ll x in the omins of g n f respectively. In tht cse we enote g by f 1, n f(f 1 (x)) = x, f 1 (f(x)) = x Uner wht conitions is f invertible? Grphs of f n f 1 Derivtives of f n f 1 : use implicit ifferentition on the inverse reltion f(f 1 (x)) = x to obtin tht x [f 1 1 (x)] = f (f 1 (x)) (You shoul be ble to go through this erivtion!) Wht oes this result men grphiclly? = 1, 12. The Nturl Logrithm, n erivtives of lnx, log x, x Reing: 7.3, 7.4 (Exmples 1-5, 12. Tht is, skip ll exmples involving integrls. Also logrithmic ifferentition not neee for homework.) Review of lgebr of logrithms n exponentils. Review of grphs of logrithms, incluing limits. Obtin erivtive of ln x using inverse reltions: Obtin erivtive of log x using reltion: log x = ln x ln x [log x] = x x [ln x] = 1 x [ ] ln x = ln 1 (ln )x Obtin erivtive of x using lgebr, the erivtive of e x, n the chin rule: x [x ] = x [e(ln )x ] = (ln )e (ln )x = (ln ) x Derivtives of functions with vribles in both the bse n the exponent, such s f(x) = x x. You cn o these problems by rewriting bse in terms of logrithm, in the exmple given 6

7 here use x = e ln x to write f(x) = e (ln x)x n then ifferentite this exponentil. (No nee for logrithmic ifferentition here.) Note: such expressions only mke sense if the bse is positive. Reing ssignment: 7.8 (pges ) 13. L Hôpitl s rule. Reltive rtes of growth. Improper forms: give exmples n explin why they re improper. Here, we won t emphsize hnling ll possible ineterminte forms, but unerstning wht n improper form is n how to recognize them. We ll see how to hnle some of some common ones. Derive L Hôpitl s rule for 0/0 cse. Stte tht it lso pplies for / cse. Know to use L Hôpitl s only when pplicble. Even if pplicble, lgebr to simplify the function cn be better lterntive. Show exmples on how to hnle ineterminte forms 0, n 1,0 0,. Reltive rtes of growth. Assume f(x) n g(x) be positive for x sufficiently lrge. We sy f(x) (i) f(x) grows fster thn g(x) s x if lim x g(x) =. (ii) f(x) grows slower thn g(x) s x if lim x f(x) g(x) = 0. (iii) f(x) n g(x) grow t the sme rte s x if lim x finite number. f(x) g(x) L Hôpitl s rule my pply to etermine the limits tht pper in some cses. We cn estblish tht = L 0, where L is some ln x x n x m x b x x x, where 0 < n < m,1 < < b where f g mens tht f grows slower thn g s x. Reing ssignment: Mxim n minim of functions Define bsolute mximum (minimum) vlue of function: lrgest or smllest function vlue (not the point where it is ttine!). Define criticl numbers: points in the omin where f = 0 or unefine. Fining the bsolute mximum/minimum of function tht is continuous on close intervl: Fin ll criticl numbers n compre function vlues t criticl numbers n t enpoints. (Abs mx/min exist by Extreme vlue theorem.) Reing ssignment: Intervls of incresing/ecresing, concve up/own Fin intervls where f is incresing or ecresing. Use signchrt for f (s in hnout Fining where function is positive/negtive ). Use intervls of incresing/ecresing to fin locl mximum or minimum vlues of functions. (This is clle the First Derivtive Test.) Fin intervls where f is concve up or own. Use signchrt for f. Fin inflection points. 7

8 Use intervls of incresing/ecresing, concve up/own to grph functions. Use concvity to etermine whether function hs locl mx or min t given criticl number. (This is clle the Secon Derivtive Test.) Note: you cnnot use the secon erivtive test to euce bsolute mximum/minimum vlues of function! 16. Using clculus to grph/fin bs mx/min Reing ssignment: 4.5 (skip slnt symptotes), 4.6 (exmples 1,2,3) Use intervls of incresing/ecresing, intervls of concve up/own, symptotes, intercepts n symmetries to grph functions. This is the topic of 4.5, bsiclly summry of wht we lrey know. Exmples 1,2,3 in 4.6 illustrte how clculus cn help refine grph obtine using grphing clcultor or Mtlb. Fin bsolute mxim n minim of functions on ny omins using enough informtion (such s first erivtive test n behviour t enpoints n ner iscontinuities) Reing ssignment: Using clculus to solve optimiztion problems These re wor problem in which you nee to ientify the function tht is to be mximize or minimize n its omin, n then use clculus to fin the bsolute mx or min of tht function on the prticulr omin. Plese follow the outline: Step 1: Drw picture. Lbel ll vribles, n ll constnts. Step 2: Wht is given? Wht o you wnt to mximize or minimize? Step 3: Fin formul for the function you wnt to mx/min. If more tht one vrible is involve, use wht is known to reuce the formul to function of one vrible. Stte the omin of the function. Step 4: Use ny metho you wish to fin the bsolute mx or min. If function is continuous on close intervl, only nee vlues t criticl points n enpoints. Otherwise, use (i) first erivtive test, (ii) secon erivtive test plus some rgument (remember tht 2n erivtive gives only locl informtion, not globl), or (iii) simply grph if f is prticulrly simple, such s qurtic. In ny cse, nswer hs to be clerly justifie, n ccompnie by grph. Step 5: Answer the question with sentence (in correct English). 8

9 Reing ssignment: Antierivtives, Differentil equtions. Definition: n ntierivtive of f(x) is function F(x) such tht F/x = f(x) Definition: the most generl ntierivtive of f(x) is the set of functions F(x) + C where F/x = f(x). It is cler grphiclly n using rules for ifferentition tht ing constnt to F oes not chnge its erivtive. Exmples: The most generl ntierivtive of x n is xn+1 + C, for n 1 n + 1 The most generl ntierivtive of 1 is ln(x) + C, if x > 0. Wht if x < 0? x Alwys check your results by confirming tht the erivtive of the ntierivtive is wht you strte with! Nottion: the most generl ntierivtive of f(x) is lso enote by the inefinite integrl f(x)x. Thus f(x)x = F(x) + C where F is n ntierivtive of f. Plese remember tht the most generl ntierivtive f(x)x enotes fmily of functions! This nottion is not use in the book until 5.4, but you my see it in other courses sooner (such s physics n engineering courses), which is why we mention it here. A tble of some functions n ntierivtives is given below. The top entry sttes tht the right column is the ntierivtive of the left column. The bottom entry emphsizes tht the left column is the erivtive of the right column. f(x) x n, n 1 1 x sin(x) cos(x) e x f(x)x x n+1 n C ln x + C cos(x) + C sin(x) + C e x + C F (x) F(x) Applictions to Velocity n ccelertion. Since velocity is the erivtive of position, n ccelertion is the erivtive of velocity, it follows tht position is n ntierivtive of velocity velocity is n ntierivtive of ccelertion Given velocity, in orer to etermine the position we nee n initil position p(0) = p 0 tht etermines the constnt C. Similrly, given n ccelertion, in orer to etermine the velocity we nee n initil velocity v(0) = v 0. In mny of the problems below we nee tht ccelertion on erth of flling boy is 9.8 m/s 2, which equls 32 ft/s 2. 9

10 19. The re problem. Reing ssignment: 5.1, pges Sigm nottion. Definition: N j=1 j = n. Evlute for exmples. Explin the first two of the following formuls, know where to fin thir one when neee (if you nee thir one in n exm it will be given to you) 1 = N, j=1 j = j=1 N(N + 1) 2, j 2 = j=1 N(N + 1)(2N + 1) 6 Approximting res uner grphs y = f(x), f 0. Approximte the re between x =, x = b, y = f(x) n y = 0 by finite sum corresponing to the re of N rectngles of equl with, Are [ f(x 1 ) + f(x 2 ) + f(x 3 ) f(x N )] x = k=1 (1) f(x k ) x (2) Wht is x j? Be ble to etermine the vlues of x j corresponing to the left enpoint, the right enpoint n the mipoint rule. Wht is x? Be ble to write the pproximting sum for fixe vlue of N (such s N = 4 or N = 10) n for rbitrry N. Evlute the pproximtions. Evlute the pproximtion (2) either (i) irectly using smll vlues of N, n using mipoint, left or right enpoint for x j (ii) using MATLAB or progrm in your grphing clcultor for lrger N (optionl) (iii) using the formuls (1) bove, when possible (selecte cses only!) Exct re is obtine in limit. Are = lim N k=1 f(x k ) x (3) Sometimes you cn evlute this limit nlyticlly using formuls (1) bove. Alterntive: pproximte this limit numericlly, using (2) with incresing vlues of N. 20. The istnce problem. Reing ssignment: 5.1, Fining istnce trvelle from velocity t. Given velocity v(t k ) t some times t k in intervls [t k 1,t k ] of length t k, cn pproximte istnce trvelle on tht time intervl by v(t k ) t k, n totl istnce trvelle in the intervl [t 0,t N ] by Distnce v(t 1 ) t 1 + v(t 2 ) t 2 + f(x 3 ) t v(t N ) t N = Obtin exct istnce trvelle by tking limit Distnce = lim N k=1 v(t k ) t k v(t k ) t k Thus the problem of fining istnce trvelle from velocity t is the Thus the problem of fining istnce trvelle from velocity t is the sme kin of problem s the one of fining res uner grphs using function vlues. k=1 10

11 21. The efinite integrl: efinition, properties. Reing ssignment: 5.2, 5.3, 5.4 pges Definition of the efinite integrl whether or not f 0: b f(x)x = lim N k=1 f(x k ) x (4) provie the limit exists, for ny points x k in the kth subintervl. If it oes, the function f is si to be integrble on [, b]. The efinite integrl is lso clle the Riemnn integrl n the sum N k=1 f(x k ) x is clle the Riemnn sum. Theorem: If f is continuous then f is integrble. If f 0 on [,b], then the efinite integrl equls the re uner grph. If f 0 on [,b], then the efinite integrl equls the negtive re uner grph. In some cses you cn evlute efinite integrls by interpreting them s res. Properties: (i) b f(x) + g(x)x = b f(x)x + b g(x)x (ii) b cf(x)x = c b f(x)x (iii) c f(x)x + b c f(x)x = b (iv) f(x)x = 0 (v) b f(x)x = b f(x)x (vi) If f M on [,b], then b f(x)x f(x)x M(b ) Funmentl Theorem of Clculus, Prt I. If f is continuous on [,b] then the erivtive of the function x [ g(x) = f(t)t is g x ] (x) = f(x). In other wors: f(t)t = f(x) x Be ble to use this Prt to ifferentite integrls with vrible bouns. Unerstn the bsic ie of the proof of this theorem, on pges Be ble to ifferentite integrls with vrible bouns, using the Funmentl Theorem n the Chin Rule, when necessry. The Funmentl Theorem of Clculus, Prt II. If f is continuous on [,b], then b f(t)t = F(b) F() where F(x) is ny ntierivtive of f(x), tht is F (x) = f(x). Be ble to use this Prt to evlute efinite integrls. Nottion: b f(t)t = [ F(x) ] b = F(b) F() where F is n ntierivtive. Alterntive sttement of the funmentl theorem, Prt II, b F (t)t = F(b) F() which is wht the book clls the net chnge theorem. Note: The Funmentl Theorem, Prt I, sttes tht x f(t)t is n ntierivtive of f. Note: 1 x x = ln x + C, if x 0. Check this! 11

12 Reing ssignment: 5.4, pges The inefinite integrl A convenient nottion for ntierivtives f(x)x = F(x) + C where F is n ntierivtive. Note: the inefinite integrls is fmily of functions. The efinite integrl, on the other hn, is number tht epens on the bouns. Do not confuse these two. 23. The verge of function Reing ssignment: 6.5 (skip Men Vlue Theorem for Integrls ) Definition. We efine the verge f v of f on [,b] s the verge of finitely mny function vlues f(x j ) smple t uniformly istribute points x j [,b],j = 0,...,n, in the limit s n. This limit equls f v = 1 b f(x)x b This give new wy to view n integrl. Here re four wys to think of the efinite integrl b f(x)x, which summrize wht we hve lernt: (i) The limit of sum of the form below, or, which (sometimes more useful wy of thinking, for me) pproximtely finite sum of the form b f(x)x = lim n j=1 where {x j } n j=0 is uniform prtition of [,b]. n f(x j ) x n f(x j ) x (ii) An re or ifference of res: b f(x)x = the re uner f if f 0, the negtive of the re if f 0, the ifference of res if f chnges sign. (iii) The ifference of the ntierivtive b F (x)x = F(b) F() (iv) The verge vlue f v of function times intervl-length, b f(x)x = f v(b ) A nice ppliction: Smoothing. Integrtion verges t n is therefore smoothing. We cn think of the integrl A f (x) = 1 x+h f(t)t 2h x h s n opertion tht verges the vlues of f over the 0.4 intervl [x h,x + h]. The figure to the right shows 0.3 the function f(x) = x (she) n its smoothe 0.2 versions A f (x) for h = 1/2 (blue) n h = 1/10 (re). 0.1 Cn you figure out formul for A f (x)? y j= x f(x)= x A f (x), h=1/2 A f (x), h=1/10 12

B Veitch. Calculus I Study Guide

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