Lecture 3. Limits of Functions and Continuity


 Darlene Wilkins
 4 years ago
 Views:
Transcription
1 Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live without more definitions (unless you pln to go to grd school in mth) Briefly n open set in the plne hs fuzzy boundry like the open disc {x x < 1} A closed set is the complement of n open set So the boundry is hrd not fuzzy; eg, like the closed disc {x x 1} You cn use open sets to eliminte εδ nd limit from the definition of continuous function See Lng, Theorem 56, p 156 See Figure 1 for pictures of open nd closed sets So tht brings us to Lng, Chpter 7  the story of limits of functions in nd on normed vector spces Why re we interested in this question? We wnt to know when we cn interchnge limit nd integrl, limit nd derivtive We wnt to know whether series of functions such s Tylor series or Fourier series converges In fct, we need to know precisely wht we men by convergence of series of functions We wnt to think of the definite integrl s function on the spce C[, b] of continuous functions on finite closed intervl [, b] Wht sort of function is it? Liner? Continuous? In fct proving things bout limits of functions into/on normed vector spces is no hrder thn it ws for functions into/on the rel line We will bsiclly copy the proofs from Lecture 3 of Mth 142 Suppose tht V nd W re normed vector spces I will use the sme symbol for the norm on V nd the norm on W Suppose S V nd f : S W Before defining the limit lim f(x), we need to mke sure tht there re points which re rbitrrily close to V We mke the x sme sort of definition s before Definition 1 Let V be normed vector spce with norm nd S V We sy the point V is dherent to S iff for every δ > there is point such tht x < δ You should picture n dherent point to set s point sticking to the set See Figure 2 Exmples 1) Any point in S is dherent to S 2) If V = R 2, using ny of our fvorite norms, ny point in the closed bll { x R 2 x r } is dherent to the open bll S = { x R 2 x < r } Definition 2 Suppose S V nd f : S W Here V nd W re normed vector spces We will denote both norms by though they my be different Assume V is dherent to S Define lim f(x) = L W to men tht for every ε >, there exists δ > (depending on ε) such tht nd x < δ implies f(x) L < ε Exmples 1) Let V = C[, b], the spce of continuous relvlued functions on the finite intervl [, b] Define I : V R by I(f) = f(x)dx Suppose our norms re f 1 = f(x) dx on V nd the usul bsolute vlue on R Does Answer: Yes In fct, here δ = ε, since using the fct tht integrls preserve inequlities, we see tht lim I(f) =? f 1
2 Figure 1: Pictures of n open set (top) nd closed set (bottom) Figure 2: Picture of red dherent point to purple set 2
3 f 1 < ε implies I(f) I() = f(x)dx f(x) dx = f 1 < ε 2) Set f(x, y) = y2 x 2 y 2 +x, for (x, y) (, ) in R 2 Does lim f(x, y) =? Here you cn use ny of our fvorite 2 (x,y) (,) norms on R 2 nd ordinry bsolute vlue on R Figure 1 below shows the grph of z = f(x, y) Answer No Set y = kx Then f(x, kx) = k2 1 k 2 +1 Thus f(x, y) hs different vlue on vrious lines through the origin In prticulr, it is 1 on the xxis nd +1 on the yxis There is no limit s (x, y) pproches the origin There re more exmples in homework 2 from Mth 142A The properties of limits re similr to those stted in prt 3 of the lectures Properties of Limits in Normed Vector Spces We ssume tht V, W re normed vector spces, is dherent to S V, f, g : S W 1) Uniqueness Suppose lim 2) Linerity Suppose α, β R nd f(x) = L nd lim lim f(x) = M Then L = M f(x) = L nd lim g(x) = M Then 3) Composite Suppose we hve 3 normed vector spces V, W, Z nd functions f : S T, is dherent to the set S V nd the vector L is dherent to the set T W If then lim g (f(x)) = M 4) Inequlities Suppose f, g : S R nd f(x) g(x) for ll If lim 5) Independence of Which Equivlent Norm is Used Using equivlent norms on either V or W leds to the sme definition of lim (αf(x) + βg(x)) = αl + βm g : T Z, plus we know tht the vector lim f(x) = L nd lim g(y) = M, y L y T f(x) = L nd lim lim f(x) = L g(x) = M, then L M Some Proofs 1) L M = L f(x) + f(x) M L f(x) + f(x) M For ny ε > δ 1 such tht x < δ 1 implies L f(x) < ε 2 And δ 2 such tht x < δ 2 implies f(x) M < ε 2 Thus tking δ = min{δ 1, δ 2 } we see tht x < δ implies L M L f(x) + f(x) M < ε Since ε ws rbitrry this mens L M = (see Lng, p 3 if you don t believe this) By the first xiom of norms, this implies L M = 2) We know tht given ε > δ 1 such tht y L < δ 1 implies g(y) M < ε (αf(x) + βg(x)) (αl + βm) = αf(x) αl + βg(x) βm αf(x) αl + βg(x) βm = α f(x) L + β g(x) M 3
4 Figure 3: Picture of the h(x) vlues (red circles to the right of ) on the rel xis, long with the limiting vlue K ( blue str to the left of ) No wy cn this hppen since the red circles cn never get close to the blue str ε δ 1 so tht x < δ 1 implies f(x) L < 2(1+ α ) ε And δ 2 so tht x < δ 2 implies g(x) M < Then let δ = min{δ 1, δ 2 } so tht x < δ implies 2(1+ β ) (αf(x) + βg(x)) (αl + βm) α f(x) L + β g(x) M < ε α 2 (1 + α ) + ε β 2 (1 + β ) < ε 3) ε δ 1 such tht y L < δ 1 implies g(y) M < ε And δ 2 such tht x < δ 2 implies f(x) L < δ 1 Then x < δ 2 implies, setting y = f(x), g(y) M < ε since y L < δ 1 4) Define h(x) = g(x) f(x) Then h(x) for ll Let K = M L If K <, we cn get contrdiction We know from Property 1 of Limits tht K = M L = lim h(x) See Figure 3 The red circles re the vlues of h(x) to the right of nd the blue str is K, to the left of Tke ε = K K 2 Then δ such tht x < δ implies h(x) K < 2 Then, since K is negtive, h(x) K < K 2 nd, dding K to both sides, h(x) < K 2 <, contrdiction to h(x) We refer you to Lng, p1623 for the generl story of limits of products You get the joy of considering specil cse in Homework 2, problem 2 There re lots of other cses one could look t; eg, sclr vlued function times vector vlued function, mtrix vlued function times mtrix vlued function, If you hte εδ stuff, you will love the following theorem, which llows you to think bout limits of sequences insted Theorem 3 Sequentil Definition of Limits Suppose S V nd f : S W, where V nd W re normed vector spces Let V be dherent to S Then the existence of lim f(x) = L is equivlent to sying tht for every sequence of vectors {x n } in S such tht lim n =, we hve lim n) = L exists n n 4
5 Proof = We leve this prt s n extr credit exercise = Proof by Contrdiction Suppose {x n } in S st lim lim x n =, we hve lim f(x n) = L If, by contrdiction, n n f(x) does not equl L Then using the rules for negting sttement involving lots of, we see tht ε > st n Z +, x n S with x n < 1 n nd f(x n) L ε Since then lim x n =, this is contrdiction to n lim f(x n) = L n Mybe we should try to drw picture of the definition of limit in higher dimensions The problem is tht it is hrd to drw the grph of function unless it mps subset of the plne into the rels Here the grph of z = f(x, y) is 3dimensionl Just plot the points (x, y, f(x, y)) in 3spce Of course in the infinite dimensionl cse good luck drwing pictures Even drwing the grph of function from the plne to the plne requires 4 dimensionl pictures You cn still project them down to 2 dimensions s you would in the cse of rel vlued function of 2 vribles Or you cn mke movie of the grph being rotted Figure 1 shows 3D grph (drwn using Scientific Workplce) of the function z = f(x, y) = y2 x 2 y 2 +x 2 from our erlier exmple It is hrd to see how bdly the function fils to hve limit s (x, y) pproches (, ) z 2 1 x y D plot of z = f(x, y) = y2 x 2 y 2 +x Continuous Functions Suppose tht V, W re normed vector spces nd U is subset of V Definition 4 f : U W is continuous t c U iff lim x c f(x) = f(c) When V = W = R, we view continuity to men tht the grph of y = f(x) does not brek up t x = c When V = R 2, you cn think similr thing bout the surfce z = f(x, y) in 3spce But reclling Figure 1 of the function f(x, y) = y2 x 2 y 2 +x, 2 it is even hrd to see the brek up for function of 2 vribles It is esier to recll tht we sw tht f(x, y) hs different vlue on vrious lines through the origin It is 1 on the yxis nd 1 on the xxis, for exmple Weird or Perhps Ridiculous Fct Define c to be n isolted point of U to men tht there exists δ > such tht the bll of rdius δ bout c contins no points of U; ie the set {x U x c < δ } is empty If c is isolted thn ny function defined t c is continuous there If V = W = R, the point (c, f(c)) would be disconnected from the rest of the grph of y = f(x) This seems to be bd choice of terminology, but it ppers to be the usul one We cn use the properties of limits to deduce the following properties of continuous functions 5
6 Properties of Continuous Functions 1) Linerity Suppose tht f, g : U W, where U V nd V, W re normed vector spces Let α, β be (rel) sclrs Then f nd g continuous t c U implies tht (αf + βg) is continuous t c 2) Composition Suppose tht V, W, Z re normed vector spces with U V nd T W Let c U Suppose tht f : U T nd g : T Z Suppose f is continuous t c nd g is continuous t f(c) Then g f is continuous t c 3) Sequentil Version Assume V, W re normed vector spces with U V The function f : U W is continuous t c U iff sequence {v n } of vectors in V such tht lim n v n = c, we hve lim n f (v n) = f (c) For the proofs, you just hve to look t the corresponding properties of limits We leve it to you nd Serge Exmples (the sme s those in the section on limits) 1) Let V = C[, b], the spce of continuous relvlued functions on the finite intervl [, b] Define I : V R by I(f) = f(x)dx Suppose we use f 1 = f(x) dx on V nd the usul bsolute vlue on R We showed erlier tht the liner function I(f) is ctully continuous t f = Now I clim I(f) is continuous on V ; ie, continuous everywhere Why? Using properties of the integrl on continuous functions tht we proved lst qurter, I(f) I(g) = I(f g) I ( f g ) = f g 1 This mens tht given ε >, we cn tke δ = ε nd then f g 1 < ε implies I(f) I(g) < ε (which is the εδ definition of continuity t g (or f) In fct, since δ depends only on ε nd not on f or g, we hve proved tht the function I(f) is uniformly continuous  concept we re bout to define Extr Credit Is I(f) still continuous when we replce the norm f 1 on V with f 2? Explin your nswer 2) Look gin t the function f(x, y) = y2 x 2 y 2 +x 2, for (x, y) (, ) in R 2 We know tht this function cnnot be continuous t (, ) since it hs no limit s (x, y) (, ) There re more such exmples in the homework Definition 5 Suppose tht V, W re normed vector spces nd U is subset of V We sy tht f : U W is uniformly continuous t on U iff ε > δ > (with δ depending only on ε) such tht u, v U, u v < δ implies f(u) f(v) < ε The point here is tht δ does not depend on u, v U Exmple 1 just considered is n exmple of uniformly continuous function where in fct δ = ε If I were good person nd lectured on the section of Lng bout continuous functions on compct sets, we d hve mny more exmples of uniformly continuous functions, since Theorem 25 on pge 198 of Lng sys the following Theorem 6 Suppose K is compct subset of normed vector spce V nd W is ny normed vector spce A continuous function f : K W must be uniformly continuous on K Wht does "compct set" men? The definition cn be found in Lng, p 193 Let V be finite dimensionl normed vector spce like R n Then K R n is compct iff K is closed nd bounded (This is theorem not the definition of compct) So, for exmple, closed bll of rdius r ; ie, {x R n x r }is compct This is flse in infinite dimensions A bll of rdius r in infinite dimensions is not compct Very inconvenient Anywy this mens tht for f : {x R n x r } W continuity implies uniform continuity More Exmples 3) V =normed vector spce Let f(x) = x Then f is uniformly continuous on V For ll x, y V, the tringle inequlity implies (s it did for ordinry bsolute vlue in n erly homework problem from 142), x y x y 6
7 This sys we cn tke ε = δ gin 4) Suppose tht L : R n R m is liner Then tke e j to be the stndrd bsis vector e j = 1, with 1 in the jth row nd the rest of the entries being Every vector v R n, cn be written uniquely in the form: It follows from linerity of L, tht Write Le j = 1j j 1,j jj mj R m So we see tht v = Lv = v 1 v j 1 v j v n Lv = n j=1 v j = n v j e j j=1 n v j Le j j=1 1j j 1,j jj mj = Av, where we multiply the mtrix A whose entries re ij with the vector v According to homework 2, problem 4, the liner function L is uniformly continuous You cn see this by using the infinity norm on R n nd R m nd showing tht there is constnt C > so tht Lx C x The constnt C depends on the entries ij of the mtrix A If you tke the K = mx ij, then C = nk should work 3 Completeness of C[,b] with respect to the Norm In prt 2 of the Lectures we promised to prove the following Theorem 7 The normed vector spce C[, b] of continuous rel vlued functions on the finite intervl [, b] is complete with respect to the norm f = mx x [,b] f(x) Proof Recll tht V "complete" mens every Cuchy sequence in V converges to n element of V So let {f n } be Cuchy sequence in C[, b] using the norm f This mens for every x [, b], the sequence {f n (x)} of rel numbers is Cuchy; s ε N ε st f n (x) f m (x) f n f m < ε when n, m N ε (1) 7
8 We showed lst qurter tht Cuchy sequences of rel numbers converge to limit in R Thus x [, b] there is function f(x) = lim n f n(x) Now we need to show tht f n converges uniformly to f on [, b] Let ε > be given There is M = M(x, ε) N ε so tht m M implies f m (x) f(x) < ε Then for n N ε we hve the following sneky formul by dding nd subtrcting f m (x) nd using the tringle inequlity: f(x) f n (x) f(x) f m (x) + f m (x) f n (x) < ε + f n f m < 2ε (2) We chose N ε so tht formul (1) holds This implies f f n < 2ε for n N ε which is uniform convergence of f n to f on [, b] since N ε does not depend on x Next we need to show tht f is continuous on [, b] To see this, note tht for x, y [, b], using the tringle inequlity in sneky wy gin (this time dding nd subtrcting f n (x) f n (y)): f(x) f(y) f(x) f n (x) + f n (x) f n (y) + f n (y) f(y) f(x) f n (x) + f n (x) f n (y) + f n (y) f(y) We know tht for n N ε the 1st nd 3rd terms re < 2ε by formul (2) Since f n is continuous, there is positive δ, depending on n, ε nd y such tht x y < δ implies the middle term is lso < ε So the finl result is tht f(x) f(y) < 5ε Replce ε by ε/5, if you like 8
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 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 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 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 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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationMath 61CM  Solutions to homework 9
Mth 61CM  Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
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 informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
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 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 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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
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 informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
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 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 informationPresentation Problems 5
Presenttion Problems 5 21355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
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 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 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 informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
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 informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
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 informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σfinite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationChapter 6. Infinite series
Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. 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 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 informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
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 informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationProf. Girardi, Math 703, Fall 2012 Homework Solutions: 1 8. Homework 1. in R, prove that. c k. sup. k n. sup. c k R = inf
Knpp, Chpter, Section, # 4, p. 78 Homework For ny two sequences { n } nd {b n} in R, prove tht lim sup ( n + b n ) lim sup n + lim sup b n, () provided the two terms on the right side re not + nd in some
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationFUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (
FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions
More informationa n = 1 58 a n+1 1 = 57a n + 1 a n = 56(a n 1) 57 so 0 a n+1 1, and the required result is true, by induction.
MAS221(21617) Exm Solutions 1. (i) A is () bounded bove if there exists K R so tht K for ll A ; (b) it is bounded below if there exists L R so tht L for ll A. e.g. the set { n; n N} is bounded bove (by
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
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 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 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 informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationMath Lecture 23
Mth 8  Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationAppendix to Notes 8 (a)
Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationRudin s Principles of Mathematical Analysis: Solutions to Selected Exercises. Sam Blinstein UCLA Department of Mathematics
Rudin s Principles of Mthemticl Anlysis: Solutions to Selected Exercises Sm Blinstein UCLA Deprtment of Mthemtics Mrch 29, 2008 Contents Chpter : The Rel nd Complex Number Systems 2 Chpter 2: Bsic Topology
More informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis  January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
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 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 informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information1 Sets Functions and Relations Mathematical Induction Equivalence of Sets and Countability The Real Numbers...
Contents 1 Sets 1 1.1 Functions nd Reltions....................... 3 1.2 Mthemticl Induction....................... 5 1.3 Equivlence of Sets nd Countbility................ 6 1.4 The Rel Numbers..........................
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 informationMath Solutions to homework 1
Mth 75  Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
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 informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
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 informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
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 informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationThe HenstockKurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The HenstockKurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255  Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be onetoone, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
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 informationIMPORTANT THEOREMS CHEAT SHEET
IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
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 informationMTH 5102 Linear Algebra Practice Exam 1  Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm  Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
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 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 information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
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 informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
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 informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationa n+2 a n+1 M n a 2 a 1. (2)
Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside
More information