Calculus Definitions, Theorems

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1 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. 1. lim b = b. lim = c 3. c c Theorem Properties of Limits lim = c c Let b c be rel umbers, let be positive iteger, let f g be fuctios with the followig limits. lim f() = L c lim g() = K c 1. Sclr Multiple:. Sum or Differece: 3. Prouct: 4. Quotiet: 5. Power: lim [b f() ] = bl c lim [f() ± g() ] = L ± K c lim [f() g() ] = L K c lim f() = L c g() K lim [ f() ] = L c Theorem, Limits of Polyomil Rtiol Fuctios 1. If p is polyomil fuctio c is rel umber, the lim p() = p() c. If r is rtiol fuctio give by r() = p()/q() c is rel umber such tht q(c) 0, the lim r() = r(c) = c p() q(c) \TheoremReview.oc 6/16/01 1

2 Clculus Defiitios, Theorems Theorem: The Limit of Fuctio Ivolvig Ricl Let be positive iteger. The followig limit is vli for ll c if is o, is vli for c > 0 if is eve. lim c = c Theorem: The Limit of Composite Fuctio If f g re fuctios such tht lim g() = L lim f() = f(l), the c L lim f( g() ) = f(l). c Theorem: Limits of Trigoometric Fuctios Let c be rel umber i the omi of the give trigoometric fuctio. 1. lim si = si c. lim cos = cos c 3. c c 4. lim csc = csc c 5. lim sec = sec c 6. c c lim t = t c c lim cot = cot c c Sectio 1.3 Techiques for Evlutig Limits Theorem: Fuctios tht Agree t All But Oe Poit Let c be rel umber let f() = g() for ll c i ope itervl cotiig c. If the limit of g() s pproches c eits, the the limit of f() lso eits lim f() = lim g() c c \TheoremReview.oc 6/16/01

3 A Strtegy for Fiig Limits Clculus Defiitios, Theorems 1. Ler to recogize which limits c be evlute by irect substitutio.. If the limit of f() s pproches c cot be evlute by irect substitutio, try to fi fuctio g tht grees with f for ll other th = c. 3. Apply the Theorem: Fuctios tht Agree t All But Oe Poit to coclue tht lim f() = lim g() = g(c) c c 4. Use grph or tble to reiforce your coclusio. The Squeeze Theorem If h() f() g() for ll i ope itervl cotiig c, ecept possibly t c itself, if lim h() = L = lim g() c c the lim f() eists is equl to L. c Theorem: Two Specil Trigoometric Limits 1. lim si = 1. lim 1 - cos = Sectio 1.4 Cotiuity Oe-Sie Limits Defiitio of Cotiuity Cotiuity t Poit. A fuctio f is cotiuous t c if the followig three coitios re met 1. f(c) is efie.. lim f() eists. c 3. lim f() = f(c). c Cotiuity o Ope Itervl: A fuctio is cotiuous o ope itervl (, b) if it is cotiuous t every poit i the itervl. A fuctio tht is cotiuous o the etire rel lie (, ) is everywhere cotiuous. \TheoremReview.oc 6/16/01 3

4 Clculus Defiitios, Theorems Theorem: Eistece of Limit Let f be fuctio let c L be rel umbers. The limit of f() s pproches c is L if oly if lim f() = L lim f() = L c c + Defiitio of Cotiuity o Close Itervl A fuctio f is cotiuous o the close itervl [, b] if it is cotiuous o the ope itervl (, b) lim f() = lim f() = b b The fuctio f is cotiuous from the right t cotiuous from the left t b. Theorem: Properties of Cotiuity If b is rel umber f g re cotiuous t = c, the the followig fuctios re lso cotiuous t c. 1. Sclr Multiple: b f. Sum Differece: f ± g 3. Prouct: f g 4. Quotiet: f / g, if g(c) 0 Theorem: Cotiuity of Composite Fuctio If g is cotiuous t c f is cotiuous t g(c), the the composite fuctios give by (f o g)() = f(g ()) is cotiuous t c. Itermeite Vlue Theorem (eistece Theorem) A fuctio f is cotiuous o the close itervl [, b] k is y umber betwee f() f(b), the there is t lest oe umber c o [, b] such tht f (c) = k. \TheoremReview.oc 6/16/01 4

5 Clculus Defiitios, Theorems Sectio 1.5 Ifiite Limits Defiitio of Ifiite Limits Let f be fuctio tht is efie t every rel umber i some ope itervl cotiig c (ecept possibly t c itself). The sttemet lim f() = c mes tht for ech M > 0 there eists δ > 0 such tht f() > M wheever 0 < c < δ. Similrly, the sttemet lim f() = c mes tht for ech N < 0 there eists δ > 0 such tht f() < M wheever 0 < c < δ. To efie ifiite limit from the left, replce 0 < c < δ by c δ < < c. To efie ifiite limit from the right, replce 0 < c < δ by c < < c + δ. Defiitio of Verticl Asymptote If f() pproches ifiity (or egtive ifiity) s pproches c from the right or the left, the the lie = c is verticl symptote of the grph of f. Theorem: Verticl Asymptotes A fuctio f is cotiuous o the close itervl cotiig c. If f(c) 0, g(c) = 0, there eists ope itervl cotiig c such tht g() ) 0 for ll c i the itervl, the the grph of the fuctio give by h() = f() g() hs verticl symptote t = c. \TheoremReview.oc 6/16/01 5

6 Theorem: Properties of Ifiite Limits Clculus Defiitios, Theorems Let c L be rel umbers let f g be fuctios such tht lim f() = lim g() = L c c 1. Sum or Differece: lim [ f() ± g() ] = c. Prouct: lim [ f() g() ] =, L > 0 c lim [ f() g() ] =, L < 0 c 3. Quotiet: lim g() = 0 c f() Similr properties hol for oe-sie limits for fuctios for which the limit of f() s pproches c is. Sectio.1 The Derivtive the Tget Lie Problem Defiitio of Tget Lie with Slope m If f is efie o ope itervl cotiig c, if the limit lim 0 y lim f(c + ) f(c) = = m 0 eists, the the lie pssig through (c, f(c)) with slope m is the tget lie to the grph of f t the poit (c, f(c)). Defiitio of Derivtive of Fuctio The erivtive of f t is give by f () = lim 0 f( + ) f() provie the limit eists. \TheoremReview.oc 6/16/01 6

7 Clculus Defiitios, Theorems Altertive Form of Derivtive f (c) = lim c f() f(c) c This requires tht both the left right limits be the sme. We sy tht f is ifferetible o the close itervl [, b] if it is ifferetible o (, b) if the erivtive from the right t the erivtive from the left t b both eists. Theorem: Differetiblity Implies Cotiuity If f is ifferetible t = c, the f is cotiuous t = c. 1. If fuctio is ifferetible t = c, the it is cotiuous t = c. Thus, ifferetibility implies cotiuity.. It is possible for fuctio to be cotiuous t = c t ot be ifferetible t = c. Thus, cotiuity oes ot imply ifferetibility. (bsolute vlue is emple). Sectio. Bsic Differetitio Rules Rtes of Chge Theorem: The Costt Rule The erivtive of costt fuctio is zero (0). Tht is, if c is rel umber, the [c] = 0 Theorem: The Power Rule If is rtiol umber, the the fuctio f() = is ifferetible [ ] = 1 For f to be ifferetible t = 0, must be umber such tht 1 is efie o itervl cotiig 0. Power Rule whe = 1. [] = 1 \TheoremReview.oc 6/16/01 7

8 Theorem: The Costt Multiple Rule Clculus Defiitios, Theorems If f is ifferetible fuctio c is rel umber, the cf is lso ifferetible [cf()] = c f () Theorem: The Sum Differece Rules The sum (or ifferece) of two ifferetible fuctios is ifferetible is the sum (or ifferece) of their erivtives. [f() + g()] = f () + g () [f() g()] = f () g () Sum Rule Differece Rule Theorem: Derivtives of the Sie Cosie Fuctios [si )] = cos [cos )] = si Positio Fuctio The fuctio s with respect to t is cosiere the positio fuctio reltive to the origi. If, over perio of time t, the object chges its positio by the mout s = s(t + t) s(t) which les to Distce Time s t = Rte, = which is the verge velocity, or Chge i Distce Chge i Time lim s(t + t) s(t) v(t) = = s (t) Velocity t 0 t The spee of object is the bsolute vlue of its velocity. s(t) = ½ gt + v 0 t + s0 Positio Fuctio \TheoremReview.oc 6/16/01 8

9 Clculus Defiitios, Theorems Sectio.3 The Prouct Quotiet Rule Higher-Orer Derivtives Theorem: The Prouct Rule The prouct of two ifferetible fuctios, f g, is itself ifferetible. Moreover, the erivtive of f g is the first fuctio times the erivtive of the seco, plus the seco fuctio times the erivtive of the first. [f() g()] = f() g () + g() f () [f() g() h()] = f () g() h() + f() g () h() + f() g() h () Theorem: The Quotiet Rule The quotiet f/g, of two ifferetible fuctios, f g, is itself ifferetible t ll vlues of for which g() 0. Moreover, the erivtive of f / g is give by the eomitor times the erivtive of the umertor mius the umertor times the erivtive of the eomitor ivie by the squre of the eomitor. f() g() g()f () f()g () = [g()], g() 0 Theorem: Derivtives of Trigoometric Fuctios [t ] = sec [cot ] = -csc [sec ] = sec t [csc ] = -csc cot \TheoremReview.oc 6/16/01 9

10 Sectio.4 The Chi Rule Theorem: The Chi Rule Clculus Defiitios, Theorems If y = f(u) is ifferetible fuctio of u u = g() is ifferetible fuctio of, the y = f(g()) is ifferetible fuctio of y y = u or, equivletly, [f(g())] = f'(g())g'() Theorem: The Geerl Power Rule If y = [u()], where u is ifferetible fuctio of is rtiol umber, the = [u()] 1 or, equivletly, u [u ] = u 1 u' u Theorem: The Derivtice of Absolute Vlue [ u ] = u u u \TheoremReview.oc 6/16/01 10

11 Summry of Differetitio Rules Clculus Defiitios, Theorems Geerl Differetitio Rules Let u v be ifferetible fuctios of. Costt Multiple Rule: Sum or Differece Rule [cu] = cu [u ± v ] = u ± v Prouct Rule: Quotiet Rule: u [uv] = uv + vu = v vu uv v Derivtives of Costt Rule: Simple Power Rule: Algebric Fuctios [c] = 0 [ ] = 1 [] = 1 Derivtives of [si ] = cos [t ] = sec Trigoometric Fuctios [sec ] = sec t [cot ] = csc [cos ] = si [csc ] = csc cot Chi Rule Chi Rule: Geerl Power Rule: [f(u)] = f (u) u [u ] = u 1 u \TheoremReview.oc 6/16/01 11

12 Sectio.5 Implicit Differetitio Guielies for Implicit Differetitio Clculus Defiitios, Theorems 1. Differetite both sies of the equtio with respect to.. Collect ll terms ivolvig y/ o the left sie of the equtio move ll other terms to the right sie of the equtio. 3. Fctor y/ out of the left sie of the equtio 4. Solve for y/ by iviig both sies of the equtio by the left-h fctor tht oes ot coti y/. Sectio.6 Relte Rtes Guielies for Solvig Relte-Rte Problems 1. Ietify ll give qutities qutities to be etermie. Mke sketch lbel the qutities.. Write equtio ivolvig the vribles whose rtes of chge re give or they re to be etermie. 3. Usig the Chi Rule, implicitly ifferetite both sies of the equtio with respect to time t. 4. After completig Step 3, substitute ito the resultig equtio ll kow vlues for the vribles their rtes of chge. The solve for the require rte of chge. \TheoremReview.oc 6/16/01 1

13 Sectio 3.1 Etrem o Itervl Clculus Defiitios, Theorems Defiitio of Etrem ( eistece efiitio) Let f be efie o itervl I cotiig c, 1. f(c) is the miimum of f o I if f(c) f() for ll i the itervl.. f(c) is the mimum of f o I if f(c) f() for ll i the itervl. The miimum mimum of fuctio o itervl re the etreme vlues, or etrem, of the fuctio o the itervl. The miimum mimum of fuctio o itervl re lso cll the bsolute miimum bsolute mimum o the itervl, respectively. The Etreme Vlue Theorem ( eistece theorem) If f is cotiuous o close itervl [, b], the f hs both miimum mimum o tht itervl. Defiitio of Reltive Etrem 1. If there is ope itervl cotiig c o which f(c) is mimum, the f(c) is clle reltive mimum of f.. If there is ope itervl cotiig c o which f(c) is miimum, the f(c) is clle reltive miimum of f. The plurl of reltive mimum is reltive mim, the plurl of reltive miimum is reltive miim. Defiitio of Criticl Number Let f be efie t c. If f (c) = 0 or f is uefie t c, the c is criticl umber of f. Reltive Etrem Occur oly t Criticl Numbers If f hs reltive miimum or reltive mimum t = c, the c is criticl umber of f. \TheoremReview.oc 6/16/01 13

14 Clculus Defiitios, Theorems Guielies for Fiig Etrem o Close Itervl To fi the etrem of cotiuous fuctio f o close itervl [, b], use the followig steps. 1. Fi the criticl umbers of f i [, b].. Evlute f t ech criticl umber i (, b). 3. Evlute f t ech epoit of [, b]. 4. The lest of these vlues is the miimum. The gretest is the mimum. Sectio 3. Rolle s Theorem the Me Vlue Theorem Theorem: Rolle s Theorem (eistece theorem) Let f be cotiuous o the close itervl [, b] ifferetible o the ope itervl (, b). If f()= f(b) the there is t lest oe umber c i (, b) such tht f (c) = 0. (mes sice f goes from positive to egtive slope (or vice vers) the there is poit, c, betwee (, b) where the slope is equl to zero) Theorem: The Me Vlue Theorem If f is cotiuous o the close itervl [, b] ifferetible o the ope itervl (, b), the there eists umber c i (, b) such tht f (c)= f(b) f() b (mes there is istteous slope tht is equl to the verge slope). \TheoremReview.oc 6/16/01 14

15 Clculus Defiitios, Theorems Sectio 3.3 Icresig Decresig Fuctios th First Derivtive Test Defiitio: Icresig Decresig Fuctios A fuctio f is icresig o itervl if for y two umbers 1 i the itervl, 1 < implies f( 1 ) < f( ). A fuctio f is ecresig o itervl if for y two umbers 1 i the itervl, 1 < implies f( 1 ) > f( ). Theorem: Test for Icresig or Decresig Fuctios Let f be fuctio tht is cotiuous o the close itervl [, b] ifferetible o the ope itervl (, b). 1. If f () > 0 for ll i (, b), the f is icresig o [, b].. If f () < 0 for ll i (, b), the f is ecresig o [, b]. 3. If f () = 0 for ll i (, b), the f is costt o [, b]. Theorem: The First Derivtive Test Let c be criticl umber of fuctio f tht is cotiuous o ope itervl I cotiig c. If f is ifferetible o the itervl, ecept possibly t c, the f (c) c be clssifie s follows. 1. If f () chges from egtive to positive t c, the f(c) is reltive miimum of f.. If f () chges from positive to egtive t c, the f(c) is reltive mimum of f. grphs \TheoremReview.oc 6/16/01 15

16 Clculus Defiitios, Theorems Sectio 3.4 Cocvity the Seco Derivtive Test Defiitio of Cocvity Let f be ifferetible o ope itervl, I. The grph of f is Cocve upwr o I if f is icresig o the itervl Cocve owwr if f is ecresig o the itervl. Theorem: Test for Cocvity Let f be fuctio whose seco erivtive eists o ope itervl I. 1. If f () > 0 for ll i I, the the grph of f is cocve upwr. 3. If f () < 0 for ll i I, the the grph of f is cocve owwr Theorem: Poit of Iflectio If (c, f (c)) is poit of iflectio of the grph of f, the either f (c) = 0 of f is uefie t = c. Theorem: Seco Derivtive Test Let f be fuctio such tht f (c) = 0 the seco erivtive of f eists o itervl cotiig c. 1. If f (c) > 0, the f (c) is reltive miimum. If f (c) < 0, the f (c) is reltive mimum If f (c) = 0, the the test fils. Sectio 3.5 Limits t Ifiity Defiitio of Limits t Ifiity (let L be rel umber) 1. The sttemet limit f (), s, = L mes tht for ech ε > 0 there eists M > 0 such tht f () L < ε wheever > M. The sttemet limit f (), s, = L mes tht for ech ε > 0 there eists N <0 such tht f () L < ε wheever < N \TheoremReview.oc 6/16/01 16

17 Clculus Defiitios, Theorems Defiitio of Horizotl Asymptote The lie y = L is horizotl symptote of the grph f if Limit f(), s, = L or Limit f(), s, = L Theorem: Limits t Ifiity If r is positive rtiol umber c is y rel umber the lim c r = 0 lim Furthermore if r is efie whe < 0 the c r = 0 A ietermite form is / or 0 / 0 for tkig limits, wheres c / 0 is uefie. Whe elig with ietermite limit, s geerl rule ivie by the highest power i the eomitor whe rewritig the limit. Sectio 3.6 A summry of Curve Sketchig Guielies for Alyzig the Grph of Fuctio 1. Determie the omi rge of the fuctio.. Determie the itercepts symptotes of the grph. 3. Locte the -vlues where f '() f ''() re either zero or uefie. Use the results to etermie reltive etrem poits of iflectio. \TheoremReview.oc 6/16/01 17

18 Sectio 3.7 Optimiztio Problems Clculus Defiitios, Theorems Problem-Solvig Strtegy for Applie Miimum Mimum Problems 1. Assig symbols to ll give qutities qutities to be etermie. Whe fesible, mke sketch.. Write primry equtio for the qutity tht is to be mimize (or miimize). 3. Reuce the primry equtio to oe hvig sigle iepeet vrible. This my ivolve the use of secory equtios reltig the iepeet vribles of the primry equtio. 4. Determie the omi of the primry equtio. Tht is, etermie the vlues for which the stte problem mkes sese. 5. Determie the esire mimum or miimum vlue by the clculus techiques. Sectio 3.8 Newto s Metho Newto s Metho for Approimtig the Zeros of fuctio. Let f (c) = 0, where f is ifferetible o ope itervl cotiig c, The, to pproimte c, use the followig steps. 1. Mke iitil estimte 1 tht is close to c. (A grph is helpful.). Determie ew pproimtio +1 = f ( ) f ( ) 3. If +1 is less th the esire ccurcy, let +1 serve s the fil pproimtio, otherwise, retur to Step clculte ew pproimtio. Ech successive pplictio of this proceure is clle itertio. \TheoremReview.oc 6/16/01 18

19 Sectio 3.9 Differetils Clculus Defiitios, Theorems Defiitio: Differetils Let y = f () represet fuctio tht is ifferetible i ope itervl cotiig. The ifferetil of (eote by ) is y ozero rel umber. The ifferetil of y (eote by y) is y = f () Differetil Formuls Let u v be ifferetible fuctios of. Costt Multiple Sum or Differece Prouct Quotiet [cu]= c u [u + v]= u + v [uv]= v u + u v [ u / v ]= v u u v v f( + ) f() + y = f() + f () The key to usig this formul is to choose vlue for tht mkes the clcultios esy. \TheoremReview.oc 6/16/01 19

20 Clculus Defiitios, Theorems Sectio 3.10 Busiess Ecoomics Applictios Summry of Busiess Terms Formuls Bsic Terms is the umber of uits prouce (or sol) p is the price per uit R is the totl reveue from sellig uits C is the totl cost of proucig uits Bsic Formuls R = p C is the verge cost per uit C = P is the totl profit from sellig uits The brek-eve poit is the umber of uits for which R = C. C P = R C Mrgils R C P = (mrgil reveue) ~ (Etr reveue for sellig oe itiol uit) = (mrgil cost) ~ (Etr cost for proucig oe itiol uit) = (mrgil profit) ~ (Etr profit for sellig oe itiol uit) Dem Fuctio: the umber of uits tht cosumers re willig to purchse t give price p per uit. Sectio 4.1 Atierivtives Iefiite Itegrtio Defiitio of Atierivtives A fuctio F is tietivtive of f o itervl I if F () = f() for ll i I. \TheoremReview.oc 6/16/01 0

21 Clculus Defiitios, Theorems Theorem: Represettio of Atierivtives If F is tierivtive of f o itervl I, the G is tierivtive of f o the itervl I if oly if G is of the form G() = F() + C for ll i I where C is costt. Bsic Itegrtio Rules Differetitio Formuls / [C] = 0 / [k] = k / [k f()] = k f () / [ f() ± g()] = f () ± g () Itegrtio Formuls 0 = C k = k + C k f() = k f() [f() ± g()] = f() ± g() / [ ] = 1 [ ] = + 1 / C, 1 / [si ] = cos / [cos ] = si / [t ] = sec / [sec ] = sec t / [cot ] = csc / [csc ] = csc cot cos = si + C si = cos + C sec = t + C sec t = sec + C csc = cot + C csc cot = csc + C \TheoremReview.oc 6/16/01 1

22 Clculus Defiitios, Theorems Sectio 4. Are Sigm Nottio The sum of terms 1,, 3,, is writte s Σ i = where i is the ie of summtio, i, is the i th term of the sum, the upper lower bous of summtio 1. Theorem : Summtio Formuls 1. Σ c = c. Σ i = i=1 i=1 ( + 1) 3. i ( + 1) ( + 1) Σ = 4. Σ i 3 = 6 i=1 i=1 ( + 1) 4 Upper Lower Sums Defie iscribe rectgle lyig isie the i th subregio of grph prtitio the circumscribe rectgle eteig outsie the i th subregio of grph prtitio. The height of the i th iscribe rectgle is f(m i ) height of the circumscribe rectgle is f(m i ). So, for ech i, the re of the iscribe rectgle is less th or equl to the re of the circumscribe rectgle. If the rectgles re to be lower sum, the ll the rectgles forme MUST be betwee the -is the grph. If the rectgles re to be upper sum, the the rectgles forme must hve the grph betwee the top of the rectgle the -is. Lower sum = s() = Σ i=1 f(m i ) Upper sum = S() = Σ i=1 f(m i ) Left Epoits m i = 0 + (i 1) ( / ) = (i 1) / Right Epoits M i = 0 + i( / ) = i / \TheoremReview.oc 6/16/01

23 Clculus Defiitios, Theorems Theorem: Limits of the Upper Lower Sums Let f be cotiuous oegtive o the itervl [, b]. The limits s of both the lower upper sums eist re equl to ech other. Tht is. lim s() = lim Σ i=1 f(m i ) = lim Σ f(m i ) i=1 = lim S() where = (b ) / f(m i ) f(m i ) re the miimum mimum vlue of f o the subitervl. Defiitio of the Are of Regio i Ple Let f be cotiuous oegtive o the itervl [, b]. the re of the regio boue by the grph of f, the -is, the verticl lies = = b is Are = lim f(c i ), i 1 c i i where = (b ) /. The followig is from the Note tht the Riem sum whe ech i is the right-h epoit of the subitervl [ i-1, i ] is whe ech i is the left-h epoit of the subitervl [ i-1, i ] is whe ech i is the left-h mipoit of the subitervl [ i-1, i ] is. \TheoremReview.oc 6/16/01 3

24 Clculus Defiitios, Theorems Sectio 4.3 Riem Sums Defiite Itegrls The lrgest subitervl of is clle the orm of the prtitio is eote by the symbol. If every subitervl is of equl with, the the prtitio is clle regulr the orm is b = = is the prtitio use to ivie up the itervl [, b]. To efie the efiite itegrl, cosier the followig limit. lim 0 Σ i = 1 f(c i ) i = L To sy tht this limit eist mes tht for ε > 0 there eists δ > 0 such tht for every prtitio with < δ it follows tht L Σ i = 1 f(c i ) i < ε (This must be true for y choice of c i, i the i th subitervl). Defiitio of Riem Sum If f is efie o the close itervl [, b] the limit lim 0 Σ i = 1 f(c i ) i eists ( s escribe bove), the f is itegrble o [, b] the limit is eote by lim Σ f(c i ) i = f() 0 b i = 1 The limit is clle the efiite itegrl of f from to b. The umber is the lower limit of itegrtio, the umber b is the upper limit of itegrtio. The with of the lrgest subitervl of prtitio is clle the orm of the prtitio is eote by. If ll the prtitios re equl the the prtitio is clle regulr. \TheoremReview.oc 6/16/01 4

25 Clculus Defiitios, Theorems Defiite Itegrls To efie the efiite itegrl, cosier the followig limit lim 0 Σ i = 1 f(c i ) i = L To sy tht this limit eists mes tht for ε > 0 there eists δ > 0 such tht for every prtitio with < δ it follows tht Σ L f(c i ) i < ε i = 1 (This must be true for y choice of c i i the i th subitervl of.) Defiitio of Defiite Itegrl If f is efie o the close itervl [, b] the limit is lim 0 Σ i = 1 f(c i ) i eists (s escribe by the epsilo-elt efiitio), the f is itegrble o [, b] the limit is eote by lim Σ f(c i ) i = b f() i = 1 0 The limit is clle the efiite itegrl of f from to b. The umber is the lower limit of itegrtio, the umber b is the upper limit of itegrtio. Theorem: Cotiuity Implies Itegrbility If fuctio f is cotiuous o the close itervl [, b], the f is itegrble o [, b]. \TheoremReview.oc 6/16/01 5

26 Clculus Defiitios, Theorems Theorem: The efiite Itegrl s the Are of Regio If f is cotiuous oegtive o the close itervl [, b], the the re of the regio boue by the grph of f, the -is, the verticl lies = = b is give by Are = b f() Defiitio of Two Specil Defiite Itegrls 1. If f is efie t =, the we efie f() = 0.. If f is itegrble o [, b], the we efie f() = f(). b b Theorem: Aitive Itervl Property If f is itegrble o the three close itervls etermie by, b, c, the b f() = c f() = b f() c Theorem: Properties of Defiite Itegrls If f g re itegrble o [, b] k is costt, the the fuctios of kf f ± g re itegrble o [, b], 1. b k f() = k b f(). b [f() ± g()] = b f() ± b f() \TheoremReview.oc 6/16/01 6

27 Clculus Defiitios, Theorems Theorem: Preservtio of Iequlity 1. If f is itegrble oegtive o the close itervl [, b], the 0 < b f(). If f g re o the close itervl [, b] f() g() for every i [, b], the b f() b g() Sectio 4.4 The Fumetl Theorem of Clculus Theorem: The Fumetl Theorem of Clculus If fuctio f is cotiuous o the close itervl [, b] F is tierivtive of f o the itervl [, b], the b f() = F(b) F() Theorem: Me vlue Theorem for Itegrls If fuctio f is cotiuous o the close itervl [, b], the there eist umber c i the close itervl {, b] such tht b f() = f(c) (b ) Defiitio of the Averge vlue of Fuctio o Itervl If f is itegrble o the close itervl [, b], the the verge vlue of f o the itervl is 1 b b f() \TheoremReview.oc 6/16/01 7

28 Clculus Defiitios, Theorems Theorem: The Seco Fumetl Theorem of Clculus If fuctio f is cotiuous o the opee itervl I cotiig, the, for every i the itervl, f ( t) t = f ( ) Sectio 4.5 Itegrtio by Substitutio Theorem: Atiifferetitio of Composite Fuctio Let g be fuctio whose rge is itervl I, let f be fuctio tht is cotiuous o I. If g is ifferetible o its omi F is tierivtive of f o I, the f ( g( )) g '( ) = F( g( )) + C If u = g(), the u = g () f ( u) u = F( u) + C Costt Multiple Rule kf ( ) = k f ( ) Chge of Vribles Rule f ( g( )) g '( ) = f ( u) u = F( u) + C \TheoremReview.oc 6/16/01 8

29 Clculus Defiitios, Theorems Theorem: The Geerl Power Rule for Itegrtio If g is ifferetible fuctio of, the [ ] [ g( ) ] + 1 g( ) g '( ) = + C, Equivletly, if u = g(), the + 1 u u u = + C, Sectio 4.6 Numericl Itegrtio Some elemetry fuctios o ot hve tierivtives tht re themselves elemetry fuctios. So whe we ee to evlute oe of these elemetry fuctios, the the Fumetl Theorem of Clculus cot be pplie. I these cses, we hve to resort to pproimtio techiques to fi the swer. We will tlk bout work with two pproimtios, the Trpezoi Rule, Simpso s Rule. However, you will oly be resposible for Trpezoi Rule o the AP test. Theorem: Trpezoi Rule b Let f be cotiuous o [, b]. The Trpezoi Rule for pproimtig f ( ) is give by b b f ( ) f ( 0 ) + f ( 1 ) + + f ( 1) f ( ) + [ ] b b f ( ) f ( ) + f ( 1 ) + + f ( 1) f ( b) + or [ ] Essetilly, the Trpezoi Rule pproimtes the re with first egree equtio for ech prtitio of the re beeth the grph. Simpso s Rule uses seco egree equtio to pproimte the re for ech prtitio is thus more ccurte th the Trpezoi Rule. The followig theorem is use to evelop Simpso s Rule the itegrls seco egree or less. \TheoremReview.oc 6/16/01 9

30 Theorem: Itegrl of p() = A + B + C Clculus Defiitios, Theorems If p( ) = A + B + C, the b b + b p( ) = p( ) + 4 p + p( b) 6 Theorem: Simpso s Rule ( is eve) b Let f be cotiuous o [, b]. Simpso s Rule for pproimtig f ( ) b b f ( ) f ( 0 ) + 4 f ( 1 ) + f ( ) + 4 f ( 3) f ( 1) f ( ) + 3 [ ] The coefficiet ptter is Or you c look t the 4 coefficiets beig ssocite with the o subscript of. is Theorem: Errors i the Trpezoi Rule Simpso s Rule If f hs cotiuous seco erivtive, b f ( ) by the Trpezoi Rule is f (), o [, b], the the error E i pproimtig 3 ( b ) () E m f ( ), 1 b Trpezoi Rule Moreover, if f hs cotiuous fourth erivtive, b pproimtig f ( ) by Simpso s Rule is f (4), o [, b], the the error E i 5 ( b ) (4) E m f ( ), b Simpso s Rule \TheoremReview.oc 6/16/01 30

31 Clculus Defiitios, Theorems () The term m f ( ) mes to tke the seco erivtive use it mimum bsolute vlue withi the itervl [, b]. For Simpso s Rule you hve to fi the mimum bsolute vlue of the fourth erivtive withi the itervl [, b]. Theorem: Chge of Vribles for Defiite Itegrls If the fuctio u = g() hs cotiuous erivtive o the close itervl [, b] f is cotiuous o the rge of g, the g( b) f ( g ( )) g '( ) = f ( u ) u g( ) NOTE: A Eve Fuctio is fuctio such tht f( ) = f(), which mes it is symmetric bout the y-is. A O Fuctio is fuctio such tht f( ) = f(), which mes it is symmetric bout the origi. Theorem: Itegrtio of Eve O Fuctios 1. If f is eve fuctio, the. If f is o fuctio, the. f ( ) = f ( ) 0 f ( ) = 0. Sectio 5.1 The Nturl Logrithmic Fuctio Defiitio of the Nturl Logrithmic Fuctio The turl logrithmic fuctio is efie by 1 l = t, > 0. 1 t The omi of the turl logrithmic fuctio is the set of ll positive rel umbers. \TheoremReview.oc 6/16/01 31

32 Clculus Defiitios, Theorems Theorem: Properties of Nturl Logrithmic Fuctio The turl logrithmic fuctio hs the followig properties. 1. The omi is (0, ) the rge is (, ).. The fuctio is cotiuous, icresig, oe-to-oe. 3. The grph is cocve ow. Theorem: Logrithmic Properties If b re positive umbers is rtiol, the the followig properties re true. 1. l(1) = 0 l b = l + l b. ( ) 3. l = l l b b 4. l( ) = l Defiitio of e The letter e eotes the positive rel umber such tht e 1 l e = t = 1 1 t NOTE: e Theorem: Derivtive of the Nturl Logrithmic Fuctio Let u be ifferetible fuctio of 1 1. [ l ] =, > 0. [ u] 1 u u ' l = =, u > 0 u u \TheoremReview.oc 6/16/01 3

33 Clculus Defiitios, Theorems Theorem: Derivtive Ivolvig Absolute Vlue If u is ifferetible fuctio of such tht u 0, the u ' l =. u [ u ] Sectio 5. The Nturl Logrithmic Fuctio Theorem: Log Rule for Itegrtio Let u be ifferetible fuctio of = l + C. 1 u u = l u + C Altertive form of Log Rule u ' u = l u + C u Itegrls of the Si Bsic Trigoometric Fuctios si = cos + C cos u u = si u + C t = l cos + C cot u u = l si u + C sec = l sec + t + C csc u u = l csc u + cot u + C \TheoremReview.oc 6/16/01 33

34 Clculus Defiitios, Theorems Sectio 5.3 The Nturl Logrithmic Fuctio Defiitio of Iverse Fuctio A fuctio g is the iverse fuctio of the fuctio f if f ( g( )) g( f ( )) = for ech i the omi of g = for ech i the omi of f. The fuctio g is eote by f 1 (re f iverse ). NOTE: See sectio 3.3 for the meig of strictly mootoic. Theorem: Reflective Property of Iverse Fuctios The grph of f cotis the poit (, b) if oly if the grph of f 1 cotis the poit (b, ). Theorem: The Eistece of Iverse Fuctio 1. A fuctio hs iverse fuctio if oly if it is oe-to-oe.. If f is strictly mootoic o its etire omi, the is oe-to-oe therefore hs iverse fuctio. Theorem: Cotiuity Differetibility of Iverse Fuctios Let f be fuctio whose omi is the itervl I. If f hs iverse fuctio, the the followig sttemets re true. 1. If f is cotiuous o its omi, the f 1 is cotiuous o its omi.. If f is icresig o its omi, the f 1 is icresig o its omi. 3. If f is ecresig o its omi, the f 1 is ecresig o its omi. 4. If f is ifferetible t c f (c) 0, the f 1 is ifferetible t f(c). \TheoremReview.oc 6/16/01 34

35 Clculus Defiitios, Theorems Theorem: The Derivtive of Iverse Fuctio Let f be fuctio tht is ifferetible i itervl I. If f hs iverse fuctio g, the g is ifferetible t y for which f (g()) 0. Moreover, 1 g '( ) =, f f '( g( )) '( g ( )) 0. Sectio 5.4 Epoetil Fuctios: Differetitio Itegrtio Defiitio of the Nturl Epoetil Fuctio The iverse fuctio of the turl logrithmic fuctio f() = l is clle the turl epoetil fuctio is eote by Tht is, f 1 ( ) = e y = e if oly if = l y. Iverse reltioship l l( e ) = e = Theorem: Opertios with Epoetil Fuctios Let b be y rel umbers. 1. b b e e e + =. e e b = e b \TheoremReview.oc 6/16/01 35

36 Clculus Defiitios, Theorems Properties of the Nturl Epoetil Fuctio 1. The omi of f() = e is (, ), the rge is (0, ).. The fuctio f() = e is cotiuous, icresig, oe-to-oe o its etire omi. 3. The grph of f() = e is cocve upwr o its etire omi. 4. lim e = 0 lim e = Theorem: Derivtive of the Nturl Epoetil Fuctio Let u be ifferetible fuctio of. 1. e e =. u e = e u u Theorem: Itegrtio Rules for Epoetil Fuctios Let u be ifferetible fuctio of. 1. e e = + C. e = e + C Sectio 5.5 Bses Other Th e Applictios Defiitio of Epoetil Fuctio to Bse If is positive rel umber ( 1) is y rel umber, the the epoetil fuctio to the bse is eote by is efie by = e (l ), If = 1, the y = 1 = 1 is costt fuctio. \TheoremReview.oc 6/16/01 36

37 Clculus Defiitios, Theorems Theorem: Derivtives for Bses Other Th e Let be positive rel umber ( 1) let u be ifferetible fuctio of. 1. (l ) =. u = (l ) u u logc 1 = = log c (l ) 3. [ log ] 4. [ log ] 1 u = (l ) u Itegrl of Bse Other Th e 1 = + C. The sme thig woul hppe if you use the Chge of l Bse Formul the itegrte. The erivtive power rule ws itrouce i chpter, but it ws limite to rtiol umbers (of the form m /. The followig theorem etes tht rule to cover y rel vlue of the epoet. Theorem: The Power Rule for Rel Epoets Let be y rel umber let u be ifferetible fuctio of. 1. = 1. u = u 1 u Theorem: A Limit Ivolvig e lim 1+ = lim = e t r We showe tht A = P 1 +, where P is the pricipl, r is the rte i eciml form, is the umber of py perios per yer, t is the umber of yers. A is the mout ccumulte over t r t yers. We c erive the cotiuous compouig equtio s A = Pe. \TheoremReview.oc 6/16/01 37

38 Clculus Defiitios, Theorems Sectio 5.6 Iverse Trigoometric Fuctios: Differetitio Noe of the si bsic trigoometric fuctios hve iverse fuctios. Like the squre root fuctio, we hve to limit the omi of the si bsic fuctios the we c erive iverse for ech of the bsic trigoometric fuctios. The sie, cosect, tget fuctios π π re limite to the itervl,, the cosie, sect fuctios re limite to [ ] 0, π. These restrictios c epli why cre must be use whe tkig the iverse of y trig fuctio tht is outsie the omi bove. For emple: if you took the iverse sie of / the clcultor woul give you swer of π / 4, which is 315 o ot the itee 5 o. This mes tht cotet of the iverse trig fuctio must be cosiere sice oe iverse hs two possible swers. Properties of Iverse Trigoometric Fuctios π 1 y rcsi() si() rccos() 3 1 y π π π 1 3 cos() Domi: [-1, 1] Rge: [ π /, π / ] Domi: [-1, 1] Rge: [0, π] Domi Rge of the Arc Fuctios \TheoremReview.oc 6/16/01 38

39 Clculus Defiitios, Theorems sec() rcsec() π y π rccsc() π y csc() π Domi: (, 1] [1, ) Domi: (, 1] [1, ) Rge: [0, π / ] ( π /, π] Rge: [ π /, 0) (0, π / ] Domi Rge of the Arc Fuctios 4 3 y t() 4 3 y cot() π π 1 rct() π 1 rccot() π Domi: (, ) Rge: [ π /, π / ] Domi: (, ) Rge: [0, π] Domi Rge of the Arc Fuctios \TheoremReview.oc 6/16/01 39

40 Clculus Defiitios, Theorems Theorem: Derivtives of Iverse Trigoometric Fuctios Let u be ifferetible fuctio of. rcsi u = u 1. [ ] 1 u rccos u = u. [ ] 1 u 3. [ rct u] = 4. + [ rc cot u] 1 u u u = 1 + u u u u 1 5. [ rc secu] = u u u 1 6. [ rc cscu] = Sectio 5.7 Iverse Trigoometric Fuctios: Itegrtio Theorem: Itegrls Ivolvig Iverse Trigoometric Fuctios Let u be ifferetible fuctio of, let > u u = rcsi + C. u u + u 1 u = rct + C 3. u u 1 u = rcsec + C Sectio 6.1 Slope Fiels Euler s Metho Sectio 6. Differetil Equtios: Growth Decy Theorem: Epoetil Growth Decy Moel If y is ifferetible fuctio of t such tht y > 0 y = ky, for some costt k, the y = Ce kt. C is the iitil vlue of y, k is the proportiolity costt. Epoetil growth occurs whe k > 0, epoetil ecy occurs whe k < 0. \TheoremReview.oc 6/16/01 40

41 Newto s Lw of Coolig Clculus Defiitios, Theorems Sttes tht the rte of chge i the temperture of object is proportiol to the ifferece betwee the object s temperture the temperture of the surrouig meium. y = k( y T m ), where y is the rte of chge of the objects temperture, k is the proportiolity costt, y is the object s curret temperture, T m is the temperture of the surrouig meium. Sectio 7.1 Are of Regio Betwee Two Curves Are of Regio Betwee Two Curves If f g re cotiuous o[, b] g() f() for ll i [, b], the the re of the regio boue by the grphs of f g the verticl lies = = b is b [ ] A = f ( ) g( ). Sectio 7. Volume: The Disk Metho Sectio 7.5 Work Defiitio of Work Doe by Costt Force If object is move istce i the irectio of pplie costt force F, the the work W oe by the force is efie s W = FD. Hooke s Lw The force F require to compress or stretch sprig (withi its elstic limits) is proportiol to the istce tht the sprig is compresse or stretche from its origil legth. Tht is F = k, where the costt of proportiolity k (the sprig costt) epes o the specific ture of the sprig. \TheoremReview.oc 6/16/01 41

42 Clculus Defiitios, Theorems Defiitio of Work Doe by Vrible Force If object is move log stright lie by cotiuously vryig force F(), the the work oe by the force s the object is move from = to = b is W = lim W = 0 i = 1 b F( ) i Newto s Lw of Uiversl Grvittio The force F of ttrctio betwee two prticles of msses m 1 m is proportiol to the prouct of the msses iversely proportiol to the squre of the istce betwee the two prticles. Tht is Coulomb s Lw m1m F = k, where the costt of proportiolity k epes o the uits use for the msses the istce. If grms cetimeters re use, the 8 cm k = g s 3 The force F betwee two chrges of msses q 1 q i vcuum is proportiol to the prouct of the chrges iversely proportiol to the squre of the istce betwee the two chrges. Tht is q1q F = k, where the costt of proportiolity k = 1 whe q is i electrosttic uits is i cetimeters. \TheoremReview.oc 6/16/01 4