Review of basic calculus


 Brendan Hoover
 3 years ago
 Views:
Transcription
1 Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below is not fmilir to you, plese consult clculus book nd review the topic in more detil. Limits nd continuity Clculus begins with the definition of limit. In fct, there re two, relted definitions. We sy tht sequence of numbers n pproches the limit L in cse for ll positive ɛ there exists rel number N such tht n L < ɛ whenever n > N. This is the definition of sequentil limit. We write lim n L n s shorthnd nottion to men tht the sequence n pproches the limit L. A second nottion for this is to write n L s n. The second, relted definition is functionl limit: we sy tht f(x) pproches L s x pproches in cse for ll positive ɛ there exists positive δ such tht f(x) L < ɛ whenever 0 < x < δ. As bove, we use the nottion lim f(x) L nd lso the nottion f(x) L s x. x One cn lso define onesided limits: for the lefthnd limit we chnge the definition to require tht f(x) L < ɛ when δ < x < ; for the righthnd limit the requirement holds when < x < + δ. The left nd righthnd limits re denoted lim f(x) nd lim f(x), respectively. x x + It is theorem tht lim f(x) L if nd only if whenever n we hve tht f( n ) L. It is lso x theorem tht f hs limit t if nd only if f hs both left nd righthnd limit nd these limits re equl. lim x The notion of limit is the bsis for the definition of continuity: we sy tht f is continuous t if f(x) f(). Sid nother wy, f is continuous t if the following three conditions re stisfied: f is defined t. f hs limit t. The limit equls f(). We cn interpret limits in the context of pproximtions. To sy tht f(x) L s x is to sy tht f(x) L when x. More precisely, to sy tht f(x) L < ɛ is to sy tht L ɛ < f(x) < L+ɛ, which is to sy tht f(x) L with error less thn ɛ. There re mny wys function cn be discontinuous. One wy the most importnt for us is to hve removble discontinuity. This hppens when f hs limit L t point, but either f is not defined t or else f() L. In this cse, if we redefine f() to be L, then the new definition gives function which is continuous t. If both the left nd righthnd limits exits, but re not equl to one nother, then we sy tht f hs jump discontinuity. Some of the most importnt exmples of functions with jump discontinuity re the unit step functions sometimes clled the Heviside functions which re defined by the rule { 0, if x < c u c (x), if x c A third type of discontinuity is verticl symptote. This hppens when for ll M there is positive δ such tht f(x) > M whenever 0 < x < δ. We sometimes use the nottion lim f(x) or x f(x) s x. There re mny, mny other kinds of discontinuities. Just to give yourself n inkling of how bd discontinuity cn be, look t grph of the function sin(/x) ner 0. One of the min properties of continuous function is the Intermedite Vlue Theorem, which sys tht if f is continuous on intervl contining points nd b, nd if z is ny vlue between f() nd f(b), then there is point c between nd b such tht f(c) z. This is often used in the following specil cse: if f is continuous on some intervl, nd if f tkes positive vlue somewhere on the intervl nd negtive vlue t nother, then somewhere in between f must equl 0.
2 The Intermedite Vlue Theorem leds to method for solving equtions defined by continuous functions. If f is continuous on the intervl [, b] nd if f chnges sign between nd b tht is, f() nd f(b) re of opposite sign then either f chnges sign somewhere between nd the midpoint ( + b)/2, or else between this midpoint nd b. In the former cse we replce b by the midpoint; in the ltter we replce by the midpoint. In either cse we hve replced the intervl by one hlf s long, nd so hve nrrowed by hlf the possible error in knowing where f(x) 0. This is clled the bisection method. It is very slow, but hs the dvntge tht it includes esily predictble error estimte. One of the min theorems is the continuity of lgebr, which sys tht ny function defined by lgebric opertions ddition, subtrction, multipliction, nd division is continuous t every point of the domin (tht is, t ny point where you don t try to divide by 0). In prticulr, ll polynomil functions re continuous everywhere. It is lso theorem tht if g is the inverse function of onetoone function f, nd if f is continuous t, then g is continuous t f(). In prticulr, the frctionl powers x /n re ll continuous wherever they re defined. Slope nd derivtive A nonverticl line hs constnt slope. In other words, if points (, b) is ny point of the line nd if ( + x, b + y) is ny second point, then the rtio y/ x lwys yields the sme vlue, no mtter wht two points we choose. (Of course, when we chnge the line then the vlue of this slope chnges.) The slope is the most importnt chrcteristic of the line. A positive slope mens the line represents n incresing function; negtive slope corresponds to decresing function; nd constnt function hs slope 0. Lines cn be defined by mny types of equtions. For exmple 2x 3y 7, x 9, y 8 5(x + 4) re ll equtions which define lines. The lst is n exmple of pointslope formul: if the line psses through the point (, b) nd hs slope m then one eqution for the line is y b m(x ). Every nonverticl line hs welldefined slope, nd hence every point on it yields n eqution in pointslope form. For us this will be the most useful formul for lines. The derivtive of f t point is defined to be y lim x 0 x lim x 0 f( + x) f(), x provided the limit exists. If it does we use either of the nottions f () or df () to denote this limit, nd refer to it s the instntneous slope of f t. When the derivtive exists we sy tht the function is differentible. Another wy to think bout the derivtive is to consider the slope function m( x), which mesures the slope of the secnt line thru (, f()) nd ( + x, f( + x)). This function is not defined when x 0, but when it hs removble discontinuity there the vlue we define when x 0 is the derivtive of f t. A differentible function cn be pproximted by liner function on sufficiently smll intervls. So, if you grph differentible function on grphing clcultor nd then zoom in on smll intervls then wht you see is indistinguishble from stright line. This is becuse if y/ x f (), then for ny positive ɛ we hve tht y f () x, with error less thn ɛ x, whenever x is sufficiently smll. One of the most importnt consequences of this interprettion is the theorem tht if f is differentible then f is continuous. When we hve function of more thn one vrible, f(x, y,...) nd if we hold ll but one vrible constnt, nd differentite with respect to the remining vrible, we obtin the prtil derivtive of f. For exmple, if f is function of x nd y, then it hs two prtil derivtives, which we denote f/ x nd f/ y. We will mke only slight use of prtil derivtives this semester. The higher order derivtives of function re defined by iterting the opertion of the differentition. So, the second derivtive is the derivtive of the derivtive, nd is denoted either f or d2 f. The nth 2 derivtive of f is denoted either f (n) or dn f n. 2
3 The Men Vlue Theorem If f is continuous on closed intervl [, b] then there is point c in tht intervl such tht f(c) f(x) for every x in [, b]. Such vlue f(c) is clled the mximum of f on [, b]. Note tht there my be severl points t which f tkes its mximum vlue. Similrly, if f is continuous on [, b] the f tkes minimum vlue somewhere in [, b]. Fermt s Theorem is the observtion tht if the f tkes on mximum (or minimum) vlue t some point c in the interior (, b), nd if f is differentible t c, then f (c) 0. When f (c) 0 we sy tht c is criticl point for f. So, if f hs mximum (or minimum) t c then either c is criticl point, n endpoint, or point where f is not differentible (sometimes clled singulrity). Using Fermt s Theorem we cn prove the Men Vlue Theorem for Derivtives, which sys tht if f is continuous on [, b], nd differentible on the interior (, b), then there is point c in the interior such tht f f(b) f() (c). b The righthnd side of this eqution is the verge slope of f on [, b], whence the nme of the theorem. We will stte generliztion of this theorem to higher order derivtives when we come to power series. The Men Vlue Theorem hs n importnt consequence for the study of differentil equtions the uniqueness of solutions to differentil equtions. The simplest form of this sys tht of f nd g re differentible on (, b), nd if f (x) g (x) for ll x in (, b), then in fct f(x) g(x) is constnt on the intervl. In geometric lnguge this sys tht if the grphs of f nd g hve the sme (instntneous) slope t every point, then the grph of one cn be shifted up or down to mtch the grph of the other. A third wy to stte this theorem is to sy tht if we re given the vlue of the derivtive of f t every point, nd the vlue of f t one point, then there is t most one possibility for f. When we hve covered integrtion we will see in fct tht there is lwys exctly one possibility for f, ltho it my be difficult or even impossible to find net formul for it. Antidifferentition From the Men Vlue Theorem for Derivtives we lerned tht if we re given function g(x) is defined on [, b] nd rel number C then there is t most one differentible function F(x) such tht F (x) g(x) nd F() C. Such solution F(x) is clled n ntiderivtive of g(x), nd the process of finding the ntiderivtive is clled ntidifferentition. Is there lwys n ntiderivtive? How would we find it? In bsic clculus we nswered these questions with Riemnn integrls nd the Fundmentl Theorem of Clculus: if g(x) is continuous then stisfies the given initil vlue problem. Riemnn integrls F(x) C + g(t) dt Recll tht derivtive, slope, velocity, nd rte of chnge re ll essentilly synonyms. We use different words in different contexts, but mthemticlly they re ll the sme concept. We cn use this to motivte our solution the simple initil problem F (x) g(x), F() C. Here, g(x) nd C re given; F(x) is the unknown. The ide is to think of F(x) s the position of prticle moving long line with velocity g(x) t time x. In this interprettion C represents the position t time. If the velocity g(x) is continuous then on very smll time intervl, sy from t to t 2, the velocity is pproximtely constnt, roughly equl to g(t ), nd so we expect tht on the intervl [t, t 2 ] we would compute the chnge in F s follows: F g(t ) t where t t 2 t. This is the fmilir distnce rte time formultion, which works when the rte is constnt. Now if we divide the intervl [, x] into series of smll subintervls [t, t 2 ], [t 2, t 3 ],... 3
4 then the new position t time x is pproximtely equl to the strting position C plus the sum of the chnges F over ll of the subintervls: F(x) C + n g(t i ) t. Riemnn s Theorem is tht the error in this pproximtion vnishes s the length of ll the subintervls shrinks to 0. More precisely, if g(x) is continuous then the limit g(t)dt i lim mx t 0 i n g(t i ) t does not depend on how you divide up the intervl [, x] into subintervls now how you choose the representtive point t i from the ith subintervl. No mtter how you mke these choices, the limit lwys exists, nd lwys gives the sme result, provided only tht the lengths of ll these subintervls shrink to 0. This limit is clled the Riemnn integrl of g on the intervl [.x]. There re mny wys to interpret the Riemnn integrl, nd hence mny pplictions. If we drw grph of g then we see tht its integrl lso mesures the re between the grph nd the horizontl xis, t lest where g(t) > 0: if g(t) > 0 then g(t) t is the re of rectngle of height g(t) nd width t. If g(t) < 0 then this differentil mesures the negtive of the re. The Men Vlue Theorem for Integrls Another importnt interprettion of Riemnn integrl is s n verging process. It is simplest to explin this when we divide up the intervl [, b] into n equlsize subintervls, hence of size (b )/n. Since the common fctor (b ) is constnt, we cn fctor it out of the Riemnn sum nd lso out of the limit: n g (b ) lim g(t i ). n i Since the sum inside the limit is simply the men (verge) vlue of the g(t i ), it is resonble to interpret b g men vlue of g on [, b]. The men vlue of the g(t i ) lies between the minimum nd mximum vlues of g on [, b]. By the Intermedite Vlue Theorem we deduce tht if g is continuous then g tkes its men vlue somewhere in [, b]. Tht is, if g is continuous on [, b] then for some c in [, b] we hve tht g(c) b The Fundmentl Theorem of Clculus We now hve ll of the tools we need to see tht the Riemnn integrl does indeed solve our simple initil vlue problem tht is, tht g is n ntiderivtive of g. To see this we compute the difference quotient nd pply the Men Vlue Theorem for Integrls: Fundmentl Theorem of Clculus If g is continuous nd we define F(x) C + g then proof. df df lim F x 0 x lim x 0 x g. g(x), F() C. + x x g(t)dt lim g(c) g(x). x 0 This lst equlity is becuse of the Squeeze Lw: since c is between x nd x + x nd since x 0 we conclude tht c x. 4
5 There re two wys (t lest!) tht we pply the Fundmentl Theorem: to define functions s solutions to initil vlue problems, nd to evlute integrls when we know n ntiderivtive by some other mens (usully rote memory). Let s look t exmples of ech of these pplictions. None of the bsic rules for derivtives Power Rule, Chin Rule, Product Rule,... combine to give formul for the ntiderivtive of /x. But /x is continuous, t lest when x 0, nd so the Fundmentl Theorem gurntees tht it hs n ntiderivtive. Since this ntiderivtive is importnt in pplictions, we give it nme: logrithm. (The word nd the definition re due to John Npier, who lived before Clculus ws invented!) More precisely, dt log(x) ln(x) t. We use the nottion log(x) nd ln(x) interchngebly. The nturl logrithm is in essence the only logrithm: ll other logrithms re sclr multiples of it. So, whenever we sy logrithm in this clss we men the nturl logrithm, unless we explicitly sy otherwise. So, the nturl logrithm is the unique solution to the initil vlue problem d log(x), log() 0. x Hence it is n incresing function on (0, + ). Therefore it hs n inverse, which we cll the exponentil function: exp(log(x)) x, log(exp(x)) x. If we pply the Chin Rule to this second reltion then we obtin the derivtive of the exponentil function: exp(x) d exp(x), whence d exp(x) exp(x). We sometimes use the nottion e x for exp(x). This is becuse it obeys the usul lws for exponents. To see this, we strt with the identities log(b) log() + log(b), log( n ) n log(). The first cn be proved by replcing b by x nd then observing tht both sides hve the sme derivtive nd the sme vlue t. By the Men Vlue Theorem for Derivtives they must be the sme everywhere. The second reltion (for positive integers n) is simply repeted ppliction of the first. If we trnslte these reltions using the definition of exp(x) s the inverse function for log(x) we find tht e x e y e x+y, e nx (e x ) n. More generlly, we define logrithms nd exponentils to ny positive bse, other thn, by the rules log b (x) log(x)/ log(b), b x exp(x log(b)). Note tht log(2 n ) n log(2) + s n +. Since log(x) is strictly incresing we conclude tht log(x) + s x +. Similrly log(x) s x 0 + ; e x + s x + ; nd e x 0 s x. Sometimes we know the ntiderivtive lredy, nd wnt to evlute the integrl. The Fundmentl Theorem tells us how to do this: if F (x) g(x) then For exmple, if n then g F(b) F(). x n bn+ n+. n + 5
6 Power series We cn pply this to find power series representtion for log(x). If r < then r + r + r2 + r 3 + r k. This is the geometric series. If we tke r t in the definition of log(x) we find tht log(x) dt t dt ( t) ( t) k dt ( ) k (t ) k dt ( ) k (x )k+ 0 k+ k + (x ) 2 (x )2 + 3 (x )3 4 (x )4 + In prticulr, ln( 0 2 ) (k + ) 2k+ Homework problems: due Tuesdy, 27 June Compute the slope function m( x) for the function x 2 3x t the point (, 4), nd sketch its grph. Use the grph of m( x) to determine the vlue of the derivtive of x 2 3x t. Explin your resoning! 2 Compute the slope function m( x) for the function x+ x t the point (0, 0), nd sketch its grph. Use this grph to explin why x + x is not differentible t 0. Wht kind of discontinuity does the grph of m( x) hve? 3 Use the definition of derivtive to compute f (), for ech of the following. f(x) /(x 3), 2. b f(x) x, 3. c f(x) (2x + ) 7,. 4 Use the Bisection Method to pproximte the root of the eqution x 5 + x 3 0, ccurte to within Let f(x) (x+)x 2 +(x 3)x. Find ll points c in the intervl [, 3] t which the instntneous slope equls the verge slope over the intervl. 6 Use Fermt s Theorem to find the mximum nd minimum vlues of x 3 x on [.5, 0.5]. 7 Drw the grphs of the following integrnds nd evlute the given integrls by interpreting the integrl s re. (Do not find ntiderivtives!) 6 3 (2t 5)dt 6
7 b c d (2t 5)dt (3 t )dt 4 t2 dt 8 Drw grph of f(x) cos(x/2) on [0, 4π], nd use this to find the men vlue of f on this intervl. (Do not find ntiderivtives!) 9 Find power series representtion of f(x) rctn(x) round the point x 0, nd use this to pproximte π/4. 0 Find power series representtion of e x ner x 0, nd use this to pproximte e. 7
The 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 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 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 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 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 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 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 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 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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
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 informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
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 informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
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 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 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 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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
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 information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
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 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 informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
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 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 informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
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 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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
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 information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
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 informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
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 informationSummer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo
Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo yinsu@bufflo.edu Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................
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 informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
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 informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
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 informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
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 informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
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 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 informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
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 information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
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 informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationMAT 168: Calculus II with Analytic Geometry. James V. Lambers
MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd
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 informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
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 informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
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 informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
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 informationMathematics Tutorial I: Fundamentals of Calculus
Mthemtics Tutoril I: Fundmentls of Clculus Kristofer Bouchrd September 21, 2006 1 Why Clculus? You ve probbly tken course in clculus before nd forgotten most of wht ws tught. Good. These notes were put
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 informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
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 informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More information