4.5 THE FUNDAMENTAL THEOREM OF CALCULUS


 Lucas McGee
 4 years ago
 Views:
Transcription
1 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere inepenently by Newton n Leibniz in the lte 6s, it estblishes the connection between erivtives n integrls, provies wy of esily clculting mny integrls, n ws key step in the evelopment of moern mthemtics to support the rise of science n technology. Clculus is one of the most significnt intellectul structures in the history of humn thought, n the Funmentl Theorem of Clculus is most importnt brick in tht beutiful structure. The previous sections emphsize the mening of the efinite integrl, efine it, n begn to eplore some of its pplictions n properties. In this section, the emphsis is on the Funmentl Theorem of Clculus. You will use this theorem often in lter sections. There re two prts of the Funmentl Theorem. They re similr to results in the lst section but more generl. Prt of the Funmentl Theorem of Clculus sys tht every continuous function hs n ntierivtive n shows how to ifferentite function efine s n integrl. Prt shows how to evlute the efinite integrl of ny function if we know n ntierivtive of tht function. Prt : Antierivtives Every continuous function hs n ntierivtive, even those nonifferentible functions with "corners" such s bsolute vlue. The Funmentl Theorem of Clculus (Prt ) If f is continuous n A() = f(t) t the n ( f(t) t ) = A() = f(). A() is n ntierivtive of f(). Proof: Assume f is continuous function n let A() = f(t) t. By the efinition of erivtive of A, A() = A( + h) # A() lim = lim h" h h" h %' +h & $ f(t)t # $ f(t)t (' By Property 6 of efinite integrls (Section 4.), for h > )' * + ' = lim h" h +h # f(t) t.
2 4.5 The Funmentl Theorem of Clculus Contemporry Clculus +h { min of f on [, +h] }. h f(t) t { m of f on [, +h] }. h. (Fig. ) Diviing ech prt of the inequlity by h, we hve tht +h h f(t) t is between the minimum n the mimum of f on the intervl [, +h]. The function f is continuous (by the hypothesis) n the intervl [,+h] is shrinking (since h pproches ), so lim h" { min of f on [, +h] } = f() n lim { m of f on [, +h] } = f(). Therefore, +h h f(t) t is stuck between two h" { min of f on [, +h] }!! { m of f on [, +h] } f() s h "! h +h Fig. f(t) t s h "!?! f() s h " quntities (Fig. ) which both pproch f(). +h Then h f(t) t must lso pproch f(), n A() = lim h" h +h # f(t) t = f(). Emple : A() = f(t) t for f in Fig.. Evlute A() n A'() for =,, n 4. f(t) t = /, Solution: A() = f(t) t = /, A() = f(t) t =, A() = A(4) = 4 f(t) t = /. Since f is continuous, A '() = f() so A'() = f() =, A'() = f() =, A'() = f() =, A'(4) = f(4) =.
3 4.5 The Funmentl Theorem of Clculus Contemporry Clculus Prctice : A() = f(t) t for f in Fig. 4. Evlute A() n A'() for =,, n 4. Emple : A() = f(t)t for the function f shown in Fig. 5. For which vlue of is A() mimum? For which is the rte of chnge of A mimum? Solution: Since A is ifferentible, the only criticl points re where A'() = or t enpoints. A'() = f() = t =, n A hs mimum t =. Notice tht the vlues of A() increse s goes from to n then the A vlues ecrese. The rte of chnge of A() is A'() = f(), n f() ppers to hve mimum t = so the rte of chnge of A() is mimum when =. Ner =, slight increse in the vlue of yiels the mimum increse in the vlue of A(). Prt : Evluting Definite Integrls If we know n cn evlute some ntierivtive of function, then we cn evlute ny efinite integrl of tht function. The Funmentl Theorem of Clculus (Prt ) If f() is continuous n F() is ny ntierivtive of f ( F '() = f() ), then b f() = F() b = F(b) F(). Proof: If F is n ntierivtive of f, then F() n A() = f(t) t re both ntierivtives of f, F'() = f() = A'(), so F n A iffer by constnt: A() F() = C for ll. At =, we hve C = A() F() = F() = F() so C = F() n the eqution A() F() = C becomes A() F() = F(). Then A() = F() F() for ll so A(b) = F(b) F() n b f() = A(b) = F(b) F(), the formul we wnte.
4 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4 The efinite integrl of continuous function f cn be foun by fining n ntierivtive of f (ny ntierivtive of f will work) n then oing some rithmetic with this ntierivtive. The theorem oes not tell us how to fin n ntierivtive of f, n it oes not tell us how to fin the efinite integrl of iscontinuous function. It is possible to evlute efinite integrls of some iscontinuous functions (Section 4.), but the Funmentl Theorem of Clculus cn not be use to o so. Emple : Evlute ( ). Solution: F() = is n ntierivtive of f() = (check tht D( ) = ), so ( ) = = { } { } = / = /. If friens h picke ifferent ntierivtive of, sy F() = + 4, then their clcultions woul be slightly ifferent but the result woul be the sme: 4 = /. ( ) = + 4 = ( Prctice : Evlute ( ). + 4) ( + 4) = 4/ Emple 4: Evlute.7 INT(). (INT() is the lrgest integer less thn or equl to. Fig. 6).5 Solution: f() = INT() is not continuous t = in the intervl [.5,.7] so the Funmentl Theorem of Clculus cn not be use. We cn, however, use our unerstning of the mening of n integrl to get.7 INT() = (re for between.5 n ) + (re for between n.7).5 Prctice : Evlute. = (bse)(height) + (bse)(height) = (.5)() + (.7)() =.9..4 INT().
5 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 5 Clculus is the stuy of erivtives n integrls, their menings n their pplictions. The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re closely relte n tht integrtion is relly ntiifferentition, the inverse of ifferentition. Applictions The Future Clculus is importnt for mny resons, but stuents re usully require to stuy clculus becuse it is neee for unerstning concepts n oing pplictions in vriety of fiels. The Funmentl Theorem of Clculus is very importnt to both pursuits. Most pplie problems in integrl clculus require the following steps to get from the problem to numericl nswer: Applie Riemnn efinite number problem sum (or re) integrl In some cses, the pth from the problem to the nswer my be bbrevite, but the three steps re commonly use. Step is bsolutely vitl. If we cn not trnslte the ies of n pplie problem into n re or Riemnn sum or efinite integrl, then we cn not use integrl clculus to solve the problem. For few types of pplie problems, we will be ble to go irectly from the problem to n integrl, but usully it will be esier to first brek the problem into smller pieces n to buil Riemnn sum. Section 4.8 n ll of chpter 5 will focus on trnslting ifferent types of pplie problems into Riemnn sums n efinite integrls. Computers n clcultors re selom of ny help with Step. n Step is usully esy. If we hve Riemnn sum f(ck ) k on the intervl [,b], then the limit of the k= b f(). sum is simply the efinite integrl Step cn be hnle in severl wys. If the function f is reltively simple, there re severl wys to fin n ntierivtive of f (sections 4.6, prts of chpter 6 n others), n then Prt of the Funmentl Theorem of Clculus cn be use to get numericl nswer. If the function f is more complicte, then integrl tbles (section 4.8) or computers (symbolic mnipultors such s Mple or Mthemtic) cn be use to fin n ntierivtive of f. Then Prt of the Funmentl Theorem of Clculus cn be use to get numericl nswer.
6 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 6 If n ntierivtive of f cnnot be foun, pproimte numericl nswers for the efinite integrl cn be foun by vrious summtion methos (section 4.9). These summtion methos re typiclly one on computers, n progrm listings re inclue in n Appeni. Usully the ifficulties in solving n pplie problem come with the st n r steps, n the most time will be spent working with them. There re techniques n etils to mster n unerstn, but it is lso importnt to keep in min where these techniques n etils fit into the bigger picture. The net Emple illustrtes these steps for the problem of fining volume of soli. Problems of fining volumes of solis will be emine in more etil in Section 5.. Emple 5: Fin the volume of the soli in Fig. 7 for. (Ech perpeniculr "slice" through the soli is squre.) Solution: Step : Going from the figure to Riemnn sum. If we brek the soli into n "slices" with cuts perpeniculr to the is, t,,,..., n (like cutting lof of bre), then the volume of the originl soli is the sum of the volumes of the "slices" (Fig. 8): n Totl Volume = (volume of the i th slice ). i= The volume of the i th slice is pproimtely equl to the volume of bo: (height of the slice). (bse of the slice). (thickness) ( c i + ). ( c i + ). i where c i is ny vlue between i n i. Therefore, n Totl Volume ( c i + ). ( c i + ). i i= which is Riemnn sum.
7 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 7 Step : Going from the Riemnn sum to efinite integrl. The Riemnn sum pproimtion of the totl volume in Step is improve by tking thinner slices (mking ll of the i smll), n Totl volume = = lim mesh" & N ( *( '%(c i +) # (c i +) # $ i + )( i=,( ( + )( + ) = ( + + ). Step : Going from the efinite integrl to numericl nswer. We cn use Prt of the Funmentl Theorem of Clculus to evlute the integrl. F() = + + is n ntierivtive of + + (check by ifferentiting F() ), so ( + + ) = F() F() = { + + } { + + } = { 6 } { } = 6 = 8. The volume of the soli shpe in Fig. 7 is ectly 8 cubic inches. Prctice 4: Fin the volume of the soli shpe in Fig. 9 for. (Ech "slice" through the soli perpeniculr to the is is squre.) Leibniz' Rule For Differentiting Integrls If the enpoint of n integrl is function of rther thn simply, then we nee to use the Chin Rule together with prt of the Funmentl Theorem of Clculus to clculte the erivtive of the integrl. Accoring to the Chin Rule, if A() = f(), then A( ) = f( ). n, pplying the Chin Rule to the erivtive of the integrl, g() ( f(t)t ) = A( g() ) = f( g() ). g'().
8 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 8 If f is continuous function n A() = f(t)t then ( f(t)t ) = A() = f() (Funmentl Theorem, Prt I) n, if g is ifferentible, ( g() f(t)t ) = A( g() ) = f( g() ). g'() (Leibniz' Rule) Emple 6: Clculte 5 ( t t ), ( cos(u) u ), w ( sin(w) z z ). Solution: ( 5 t t ) = (5). 5 = 5. ( cos(u) u ) = cos( ). =. cos( ) ( w sin(w) z z ) = (sin(w)). cos(w) = sin (w)cos(w). Prctice 5: Fin ( sin(t) t ). PROBLEMS:. A() = t t () Use prt of the Funmentl Theorem to fin formul for A() n then ifferentite A() to obtin formul for A'(). Evlute A'() t =,, n. (b) Use prt of the Funmentl Theorem to evlute A'() t =,, n.. A() = ( + t ) t () Use prt of the Funmentl Theorem to fin formul for A() n then ifferentite A() to obtin formul for A'(). Evlute A'() t =,, n. (b) Use prt of the Funmentl Theorem to evlute A'() t =,, n.
9 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 9 In problems 8, evlute A'() t =,, n. t t. A() = t t 4. A() = t t 5. A() = 6. A() = t t ( t ) t 7. A() = sin(t) t 8. A() = In problems 9, A() = f(t) t for the functions in Figures 4. Evlute A'(), A'(), A'(). 9. f in Fig.. f in Fig.. f in Fig.. f in Fig. In problems, verify tht F() is n ntierivtive of the integrn f() n use Prt of the Funmentl Theorem to evlute the efinite integrls. 4, F() =., F() = , F() = ( + 4 ), F() = , F() = ln( ) 8.. 5, F() = ln( ) , F() = ln( )., F() = ln( ) + / π/ π, F() = / cos(), F() = sin( ). sin(), F() = cos(). 4 4., F() = / 5. 7, F() = / 6. 4, F() =
10 4.5 The Funmentl Theorem of Clculus Contemporry Clculus , F() = 9. e, F() = e, F() = 8. +, F() = ln( + ). π/4 sec (), F() = tn(). e ln(), F() =. ln(). +, F() = ( + ) / For problems 4 48, fin n ntierivtive of the integrn n use Prt of the Funmentl Theorem to evlute the efinite integrl. 5 e ( + 4 ) 7. π/ 8. sin() 9. π/ " e " e 47. sin(). ln() π/ π/6 ( ) sec () In problems 49 54, fin the re of ech she region. 49. Region in Fig Region in Fig Region in Fig. 6.
11 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 5. Region in Fig Region in Fig Region in Fig. 9. Leibniz' Rule 55. If D( A() ) = tn(), then fin D( A() ), D( A( ) ), n D( A( sin() ) ). 56. If D( B() ) = sec(), then fin D( B() ), D( B( ) ), n D( B( sin() ) ) ( + t t ) 58. ( + t t ) 59. sin() ( + t t ) 6. + ( t + 5 t ) 6. ( t + t ) 6. ( 9 t + t ) 6. ( π cos(t) t ) 64. ( 7 π cos(t) t ) 65. ( tn(t) t ) 66. ( π cos(t) t ) 67. ( ln() 5t. cos(t) t) ) 68. ( π tn(7t) t ) Very Optionl Problems ice. Wht clculus stuent puts in rink. b. cbin Where Abe Lincoln ws born. cerely jm c. cos() How the clculus stuent ene letter.. Wht forester puts on tost.
12 4.5 The Funmentl Theorem of Clculus Contemporry Clculus Section 4.5 PRACTICE Answers Prctice : A() =, A() =.5, A() =, A(4) =.5 Prctice : A '() = f() so A '() = f() =, A '() = f() =, A '() =, A '(4) =. F() = is one ntierivtive of f() = ( F ' = f ) so = = ( ) ( ) = 4. F() = + 7 is nother ntierivtive of f() = so = + 7 = ( + 7) ( + 7) = 4. No mtter which ntierivtive of f() = you use, the vlue of the efinite integrl is 4. Prctice :..4 INT() =.9. Since f() = INT() is not continuous on the intervl [.,.4] so we cn not use the Funmentl Theorem of Clculus. Inste, we cn think of the efinite integrl s n re (Fig. ). Prctice 4: Totl volume = lim mesh" = = ' N ) + ) (&(# c i ) $ (# c i ) $ % i, *) i= ) ( )( ) (9 6 + ) = 9 + = (8 + 8 ) ( +) = 8. Prctice 5: ( sin(t) t ) = sin( ) =. sin( ).
The Fundamental Theorem of Calculus Part 2, The Evaluation Part
AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt
More informationVII. The Integral. 50. Area under a Graph. y = f(x)
VII. The Integrl In this chpter we efine the integrl of function on some intervl [, b]. The most common interprettion of the integrl is in terms of the re uner the grph of the given function, so tht is
More informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationM 106 Integral Calculus and Applications
M 6 Integrl Clculus n Applictions Contents The Inefinite Integrls.................................................... Antierivtives n Inefinite Integrls.. Antierivtives.............................................................
More informationBasic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
More informationx dx does exist, what does the answer look like? What does the answer to
Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More information5.3 The Fundamental Theorem of Calculus
CHAPTER 5. THE DEFINITE INTEGRAL 35 5.3 The Funmentl Theorem of Clculus Emple. Let f(t) t +. () Fin the re of the region below f(t), bove the tis, n between t n t. (You my wnt to look up the re formul
More informationAP Calculus AB First Semester Final Review
P Clculus B This review is esigne to give the stuent BSIC outline of wht nees to e reviewe for the P Clculus B First Semester Finl m. It is up to the iniviul stuent to etermine how much etr work is require
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationSection 6.3 The Fundamental Theorem, Part I
Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt
More informationConservation Law. Chapter Goal. 6.2 Theory
Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the
More informationAntiderivatives Introduction
Antierivtives 0. Introuction So fr much of the term hs been sent fining erivtives or rtes of chnge. But in some circumstnces we lrey know the rte of chnge n we wish to etermine the originl function. For
More information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of crosssection or slice. In this section, we restrict
More informationFundamental Theorem of Calculus
Funmentl Teorem of Clculus Liming Png 1 Sttement of te Teorem Te funmentl Teorem of Clculus is one of te most importnt teorems in te istory of mtemtics, wic ws first iscovere by Newton n Leibniz inepenently.
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For , () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationsec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5
Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationUsing integration tables
Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationCHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationES.181A Topic 8 Notes Jeremy Orloff
ES.8A Topic 8 Notes Jeremy Orloff 8 Integrtion: usubstitution, trigsubstitution 8. Integrtion techniques Only prctice will mke perfect. These techniques re importnt, but not the intellectul hert of the
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationIf we have a function f(x) which is welldefined for some a x b, its integral over those two values is defined as
Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is wellefine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationInstantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More information1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics
0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = yintercept, (x 0, y 0 ) = some given point ) slopeintercept pointslope There re two importnt
More informationAP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight
AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationIntroduction. Calculus I. Calculus II: The Area Problem
Introuction Clculus I Clculus I h s its theme the slope problem How o we mke sense of the notion of slope for curves when we only know wht the slope of line mens? The nswer, of course, ws the to efine
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationIntroduction. Calculus I. Calculus II: The Area Problem
Introuction Clculus I Clculus I h s its theme the slope problem How o we mke sense of the notion of slope for curves when we only know wht the slope of line mens? The nswer, of course, ws the to efine
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More information= f (c) f (c) the height of the rectangle guaranteed by the MVT for integrals.
Get Rey: Given (t) = 8t n v() = 6, fin the isplcement n istnce of the oject from t= to t= If () = 4, fin the position of the prticle t t= I. Averge Vlue of Function Wht oes represent? Cn we rw rectngle
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationCourse 2BA1 Supplement concerning Integration by Parts
Course 2BA1 Supplement concerning Integrtion by Prts Dvi R. Wilkins Copyright c Dvi R. Wilkins 22 3 The Rule for Integrtion by Prts Let u n v be continuously ifferentible relvlue functions on the intervl
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More information( x) ( ) takes at the right end of each interval to approximate its value on that
III. INTEGRATION Economists seem much more intereste in mrginl effects n ifferentition thn in integrtion. Integrtion is importnt for fining the expecte vlue n vrince of rnom vriles, which is use in econometrics
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationAppendix 3, Rises and runs, slopes and sums: tools from calculus
Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationChapter 3. Techniques of integration. Contents. 3.1 Recap: Integration in one variable. This material is in Chapter 7 of Anton Calculus.
Chpter 3. Techniques of integrtion This mteril is in Chpter 7 of Anton Clculus. Contents 3. Recp: Integrtion in one vrible......................... 3. Antierivtives we know..............................
More informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationA. Limits  L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. 1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More informationMath 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationWhen e = 0 we obtain the case of a circle.
3.4 Conic sections Circles belong to specil clss of cures clle conic sections. Other such cures re the ellipse, prbol, n hyperbol. We will briefly escribe the stnr conics. These re chosen to he simple
More informationSpecial notes. ftp://ftp.math.gatech.edu/pub/users/heil/1501. Chapter 1
MATH 1501 QUICK REVIEW FOR FINAL EXAM FALL 2001 C. Heil Below is quick list of some of the highlights from the sections of the text tht we hve covere. You shoul be unerstn n be ble to use or pply ech item
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationAP * Calculus Review
AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril.
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More information