IV Approximation of Rational Functions 1. IV.C Bounding (Rational) Functions on Intervals... 4


 Arthur Patterson
 2 years ago
 Views:
Transcription
1 Contents IV Approxiation of Rational Functions 1 IV.A Constant Approxiation IV.B Linear Approxiation IV.C Bounding (Rational) Functions on Intervals IV.D The Extree Value Theore (EVT) for Rational Functions
2 IV Approxiation of Rational Functions IV.1 IV Approxiation of Rational Functions Suppose f(x) is a rational function. An approxiating function, A(x), is another function usually uch sipler than f whose values are approxiately the sae as the values of f. The difference f(x) A(x) = E(x) is the error, and easures how accurate or inaccurate our approxiation is. Thus f(x) = A(x) + E(x) (approxiation) (error) The two siplest functions are constant functions and linear functions and if A(x) is constant or linear we talk about a constant approxiation, or a linear approxiation for f. When we choose our approxiation we want the error to be sall. Now we know what it eans for a nuber to be sall, but our error, E(x), is a function and it is less clear what it eans for a function to be sall. In fact there are different notions of sallness for functions and depending on which one uses, one gets different kinds of approxiations. In this section we shall fix a doain (f) and we shall say our error is sall if E(x) is very sall for all x very close to a. Thus our approxiation will be good if A(x) is very close to f(x) for all x very close to a. Accordingly, we want to find an approxiation for a rational function f(x) on a sall interval [a h, a+h] do(f) by soe sipler function A(x). Then we want to deterine the accuracy of the approxiation by bounding the absolute value of the error E(x) = f(x) A(x). In order to use the approxiation we need to have an idea of how big the error is. (If the error is 5 ties the value of the function, the approxiation is clearly no good!) The MVT and EMVT allow us to get a bound on the error. IV.A Constant Approxiation The best constant approxiation of f is A 0 (x) = f(a). graph of f(x) E0( x) f(a) (a, f( a)) f(x ) graph of A 0 (x) A 0 (x ) a x We write the error ter as E 0 (x) = f(x) A 0 (x) = f(x) f(a). The MVT tells us that f (z) = f(x) f(a) x a for soe z between a and x. Therefore f (z)(x a) = f(x) f(a) = E 0 (x).
3 IV.A Constant Approxiation IV. So to bound the error on soe interval [a h, a + h] around a, we need to bound E 0 (x) = f (z) x a on [a h, a + h]. Observe that x a h and so E 0 (x) ax f (z) h. We can thus bound the error z E 0 (x) as soon as we can bound f (z) on [a h, a + h]. Exaple. Let f(x) = x 3 4x + 5 and a =. Find A 0 (x) and a nuber h so that E 0 (x) 1 0 = δ when h x + h. Solution: A 0 (x) = f() = 5. How to find h when given a and δ? 1. Choose h 0 so that [a h 0, a + h 0 ] do(f). (h 0 is an initial guess for h). Find M so that f (z) M on [a h 0, a + h 0 ]. 3. Choose h in (0, h 0 ] so that M h δ. (Reeber we want h > 0, thus the ( in (0, h 0 ].) If we succeed with 1. 3., we get: If x [a h, a + h], then E 0 (x) = f (z) x a Mh δ. iplications a h x a + h 0<h h 0 Suarized as a diagra we have the z between x and a a h 0 x a + h 0 a h 0 z a + h 0 h x a h as M is upper bound for f (z) on [a h 0, a + h 0 ] f (z) M as 0 x a h and Mh δ x a h E 0 (x) = f (z) x a Mh δ and it is the botto iplication we want to achieve. Coing back to the actual exaple: 1. Choose h 0 = 1, then [a h 0, a + h 0 ] = [1.9,.1].. We want to bound the absolute value of f (z) = 3z 4: For z [1.9,.1], we have 1.9 z.1 = 3.61 z 4.41 =.83 3z 13.3 = z Thus f (z) = 3z = M on [1.9,.1]. 3. Find h in (0, 1 1 ] so that 9.3 h = Mh δ = We want 0 < h 1 Let h =.005. Fro 1.,., 3. we get finally To spell it out: 0. = 0.1 and h = h = x.005 = + h = E 0 (x) M h = = 1 0.
4 IV.B Linear Approxiation IV.3 The constant approxiation A 0 (x) = 5 to f(x) = x 3 4x + 5 is accurate to within an error E 0 (x) of at ost 1 0 as long as x [ 0.05, ]. IV.B Linear Approxiation The best linear approxiation for f near a is A 1 (x) = f(a) + f (a)(x a) the function whose graph is the tangent line to the graph of f through (a, f(a)). It is called the linear approxiation to f at a and its error is E 1 (x) = f(x) f(a) f (a)(x a) graph of A 1 (x) E 1 ( x) graph of f(x) (a, f(a)) A 1 ( x) f(x ) a x The EMVT tells us that f (z)(x a) = f(x) f(a) f (a)(x a) = f(x) A 1 (x) = E 1 (x), for soe z between a and x. Therefore we can control the error E 1 (x) as soon as we can find an upper bound, f (z) M, for the second derivative. Exaple. Let f(x) = x 3 4x + 5 and a =. Find A 1 (x) and a nuber h so that E 1 (x) 1 0 = δ when h x + h. Solution. A 1 (x) = f() + f ()(x ) = 5 + 8(x ) = 8x 11. How to find h? 1. Choose h 0 so that [a h 0, a + h 0 ] do(f). (h 0 is again an initial guess for h.). Find M so that f (z) M on [a h 0, a + h 0 ]. 3. Choose h in (0, h 0 ] so that Mh δ. If we succeed with 1. 3., we obtain: If x [a h, a + h], then E 1 (x) = f (z) (x a) Mh δ.
5 IV.C Bounding (Rational) Functions on Intervals IV.4 In our exaple: 1. Choose h = 1 ; [a h, a + h] = [1.9,.1] (the sae guess as above why not?).. We want to bound the absolute value of f (z) = 6z: For z [1.9,.1], we have 1.9 z.1 = f (z) 6(.1) = 1.6 = M Thus f (z) 1.6 = M on [1.9,.1]. 3. Find h in (0, 0.1] so that 1.6 h = Mh δ = 1 0. Therefore we want 0 < h 0.1 and h δ M = = or h Let h = We get finally that for h = 1.9 x.08 = + h = E 1 (x) 1 0. If you copare this linear approxiation with the constant one, to achieve the sae accuracy the interval has widened fro [1.995,.005] for the constant approxiation to [1.9,.08] for the linear approxiation. In other words, the linear approxiation is better than the constant approxiation as should be expected. Indeed, if f (z) M 1 and f (z) M on soe interval [a h, a + h], then on that interval E 0 (x) M 1 x a and E 1 (x) M x a. Both E 0 (x), E 1 (x) go to 0 as x a, but E 1 (x) goes uch faster than E 0 (x): x a 1/ 1/0 1/00 1/000 Bound for E 0 M 1 / M 1 /0 M 1 /00 M 1 /000 Bound for E 1 M /00 M /0000 M / M / IV.C Bounding (Rational) Functions on Intervals In order to ake the preceding ethods work, we need soe way of getting the constants M 1 and M. These are to be chosen so that f (z) M 1, z [a h, a + h] and f (z) M, z [a h, a + h]. As you will see below, we also want soeties to find a constant such that 0 < f (z), z [a h, a+h]. Since f and f are rational functions, we need soe way of solving the following. General Proble. M such that Given an interval [a, b] in the doain of a rational function g, find constants and g(x) M, for all x in [a, b]. Soe Techniques: 1. (Rules for absolute values; see also... ) If a, b R, then i) a b = a b
6 IV.C Bounding (Rational) Functions on Intervals IV.5 ii) a b = a b iii) a b = a n b n for every n 1. iv) a + b a + b (triangle inequality) v) a b a b (second version of the triangle inequality). Exaple. Find an upper bound for x 7 6x 4 + x on [, 1]. Solution: x [, 1] x 1 = x x, and for x one has x 7 6x 4 + x (iv) x 7 6x 4 + x (iv) x 7 + 6x 4 + x (i) = = x x 4 + x = x x 4 + x (iii) = 6.. (Bounds for reciprocals) 0 < n Q(x) N = 0 < 1 N 1 Q(x) 1 n. Exaple. Find an upper bound for 1 x 4 +x +6 on R. Solution: x 4 + x + 6 = x 4 + x = x 4 +x , for every x R. 3. (Bounding fractions) If P (x) M and 0 < n Q(x) N on [a, b], then N P (x) Q(x) M n every x [a, b]. Exaple. Find an upper bound for P (x) Q(x) = x7 6x 4 +x x 4 +x +6 on [, 1]. for Solution: P (x) Q(x) M n = 6 6 = (Critical points) If f(x) is a rational function defined on [a, b], then f (x) is defined on [a, b]. Furtherore, the extreal values that is, inia or axia of f(x) on [a, b] can occur only at those points where x = a, x = b, or f (x) = 0. Exaple. Bound f(x) = x 4 + x 3 x on [.5, 0]. f (x) = 4x 3 + 6x 4x = (x + )x(x 1). Therefore the roots of f (x) are, 0, 1. We don t care about x = 1 as it is not in the interval we are interested in. At the reaining points x =.5, x = 0 (the endpoints) and x = (the only root of f in (.5, 0)), we need to evaluate f. We find f(.5) = f(0) = 0 f( ) = 8 Therefore 8 f(x) 0 on [.5, 0], and so 0 f(x) 8 on [.5, 0]. 5. (Cobined ethods) To bound a rational function f(x) = P (x) Q(x), one can cobine the above ethods. For exaple, we can find a lower bound for f(x) by finding a lower bound for P (x) and an upper bound N for Q(x) using the ethod of critical points (3.) for each of P and Q and then using (4.) to bound the fraction f(x) fro below by N. Alternatively, one ight use (1.) for P and (3.) for Q or whatever cobination is suitable. Exaple. It would be foolish to try to bound f(x) = x 7 6x 4 +x x 4 +x +6 fro above by exaining the roots of f (x) if you don t believe it, calculate f (x)! Indeed, the solution given in (3.) above is optial here. Iportant Notes:
7 IV.C Bounding (Rational) Functions on Intervals IV.6 If one wants to bound f(x) on an interval that is not closed, say on an open interval (a, b), it is of course sufficient to bound the function on [a, b] as f(x) M for x [a, b] = f(x) M for x (a, b). You just have to ake sure that f is not only defined on (a, b) but on [a, b] as well. Make sure the function is defined where you want to bound it! Otherwise you ight end up with an arguent like 1 x 1 for x [ 1, 1] as (?) 1 1 = 1 1 = 1 and ( ) 1 x = 1 x is never zero (??). Checking the values of f at the endpoints, x = a, x = b, and at the roots of f on (a, b), tells you what the iniu/axiu of f on [a, b] is but not (directly) what the iniu/axiu of f on [a, b] is! That has to be established separately. Exaple. Assue we know that f is a rational function defined on [a, b], satisfying 8 f(x) for x [a, b], and taking on both extreal values 8 and on [a, b]. Then we know by the IVT that f(x) has a zero in [a, b] and we get on [a, b]: Miniu Maxiu f(x) 8 f(x) 0 8 A istake that happens all too often is of the following sort: Yikes! 8 f(x) = 8 f(x). It is crucial to reeber on which interval you want to bound a function! Exaple. Find lower and upper bounds for f(x) = x 3 9x + 4 on (i) [ 3, 3], (ii) [ 1, 1], (iii) [1, 3], (vi) ( 5, 4), (v) [ 3, 1). What is an upper bound for f on these intervals? Solution: The derivative of f(x) is f (x) = 3x 9 = 3(x 3)(x + 3). Evaluating f at the various endpoints and at the roots ± 3 of f, we find the following table where we check ( ) those points relevant to the interval in question and finally read off a lower () and upper bound (M): x f(x) [ 3, 3] [ 1, 1] [1, 3] ( 5, 4) [ 3, 1) M M 1 1 M M 4 3 M lower bound for f(x) upper bound upper bound for f
8 IV.D The Extree Value Theore (EVT) for Rational Functions IV.7 As a picture tells ore than a thousand words: (a) f(x)=x^39x (c) f(x) on [1,1] 15 M (e) f(x) on (5,4) 40 M (b) f(x) on [3,3] 15 M (d) f(x) on [1,3] 4 M (f) f(x) on [3,1) M IV.D The Extree Value Theore (EVT) for Rational Functions We have just discussed techniques for finding bounds. Are there always such bounds? The answer is no : On (0, 1), the rational function f(x) = x 1/ x(x 1) has neither a lower nor an upper bound. If x 0 fro inside the interval (0, 1), then f(x) +, whereas if x 1 fro inside the interval (0, 1), then f(x). The situation is fortunately different for closed intervals: The Extree Value Theore (EVT for rational functions): Let f be a rational function defined on [a, b]. Then f takes on an absolute iniu and an absolute axiu M. There are thus c, d [a, b] such that f(c) = and f(d) = M, and f(x) M for all x [a, b].
9 IV.D The Extree Value Theore (EVT) for Rational Functions IV.8 Note: There ay be several arguents z [a, b] such that f(z) = and there ay be as well several arguents u [a, b] such that f(u) = M. The absolute iniu or axiu ay be taken on at one of the endpoints. The EVT akes no prediction at which arguent in [a, b] the function will take on either iniu or axiu. Cobining the IVT and the EVT, one obtains the following stateent. Corollary: If f is defined on [a, b], the iage of this closed interval under f is again a closed interval, f ([a, b]) = [, M]. The nuber is the absolute iniu of f on [a, b], the nuber M is the absolute axiu of f on [a, b]. Proof: The EVT says that there are, M R such that f ([a, b]) [, M] and that and M are taken on as values. The IVT says that any value between and M is taken on as well. Thus there are no gaps and f ([a, b]) = [, M].
Factor Model. Arbitrage Pricing Theory. Systematic Versus NonSystematic Risk. Intuitive Argument
Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit.
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationLesson 13: Voltage in a Uniform Field
Lesson 13: Voltage in a Unifor Field Most of the tie if we are doing experients with electric fields, we use parallel plates to ensure that the field is unifor (the sae everywhere). This carries over to
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More information4.3 The Graph of a Rational Function
4.3 The Graph of a Rational Function Section 4.3 Notes Page EXAMPLE: Find the intercepts, asyptotes, and graph of + y =. 9 First we will find the intercept by setting the top equal to zero: + = 0 so =
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 453167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationReconnect 04 Solving Integer Programs with Branch and Bound (and Branch and Cut)
Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, Reconnect 04 Solving Integer Progras with Branch and Bound (and Branch and Cut) Cynthia Phillips (Sandia National
More informationThe Mathematics of Pumping Water
The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationPreferencebased Search and Multicriteria Optimization
Fro: AAAI02 Proceedings. Copyright 2002, AAAI (www.aaai.org). All rights reserved. Preferencebased Search and Multicriteria Optiization Ulrich Junker ILOG 1681, route des Dolines F06560 Valbonne ujunker@ilog.fr
More informationMINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS
MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3unifor hypergraph on n vertices with iniu
More informationChapter 13 Simple Harmonic Motion
We are to adit no ore causes of natural things than such as are both true and sufficient to explain their appearances. Isaac Newton 13.1 Introduction to Periodic Motion Periodic otion is any otion that
More informationA Gas Law And Absolute Zero
A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationRECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure
RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University
More informationProblem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)
Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x
More informationGuide to SRW Section 1.7: Solving inequalities
Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is 65 43 21 0
More informationON SELFROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 778433128
ON SELFROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 7788 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute
More informationThis paper studies a rental firm that offers reusable products to price and qualityofservice sensitive
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523464 eissn 5265498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed
More informationMachine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationConstruction Economics & Finance. Module 3 Lecture1
Depreciation: Construction Econoics & Finance Module 3 Lecture It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationPlane Trusses. Section 7: Flexibility Method  Trusses. A plane truss is defined as a twodimensional
lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at
More information5.7 Chebyshev Multisection Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multisection Matching Transforer Reading Assignent: pp. 555 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationSOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting
More informationExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network
2013 European Control Conference (ECC) July 1719, 2013, Zürich, Switzerland. ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona
More informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and nonconservative forces, with soe
More informationSINGLE PHASE FULL WAVE AC VOLTAGE CONTROLLER (AC REGULATOR)
Deceber 9, INGE PHAE FU WAE AC OTAGE CONTROER (AC REGUATOR ingle phase full wave ac voltage controller circuit using two CRs or a single triac is generally used in ost of the ac control applications. The
More informationEvaluating Inventory Management Performance: a Preliminary DeskSimulation Study Based on IOC Model
Evaluating Inventory Manageent Perforance: a Preliinary DeskSiulation Study Based on IOC Model Flora Bernardel, Roberto Panizzolo, and Davide Martinazzo Abstract The focus of this study is on preliinary
More informationAlgorithmica 2001 SpringerVerlag New York Inc.
Algorithica 2001) 30: 101 139 DOI: 101007/s0045300100030 Algorithica 2001 SpringerVerlag New York Inc Optial Search and OneWay Trading Online Algoriths R ElYaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationA Gas Law And Absolute Zero Lab 11
HB 040605 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution
More informationFixedIncome Securities and Interest Rates
Chapter 2 FixedIncoe Securities and Interest Rates We now begin a systeatic study of fixedincoe securities and interest rates. The literal definition of a fixedincoe security is a financial instruent
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis  Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationInformation Processing Letters
Inforation Processing Letters 111 2011) 178 183 Contents lists available at ScienceDirect Inforation Processing Letters www.elsevier.co/locate/ipl Offline file assignents for online load balancing Paul
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationTrading Regret for Efficiency: Online Convex Optimization with Long Term Constraints
Journal of Machine Learning Research 13 2012) 25032528 Subitted 8/11; Revised 3/12; Published 9/12 rading Regret for Efficiency: Online Convex Optiization with Long er Constraints Mehrdad Mahdavi Rong
More informationF=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu
Chapter 4 F=a Fro Probles and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, orin@physics.harvard.edu 4.1 Introduction Newton s laws In the preceding two chapters, we dealt
More informationEndogenous Market Structure and the Cooperative Firm
Endogenous Market Structure and the Cooperative Fir Brent Hueth and GianCarlo Moschini Working Paper 14WP 547 May 2014 Center for Agricultural and Rural Developent Iowa State University Aes, Iowa 500111070
More informationFactored Models for Probabilistic Modal Logic
Proceedings of the TwentyThird AAAI Conference on Artificial Intelligence (2008 Factored Models for Probabilistic Modal Logic Afsaneh Shirazi and Eyal Air Coputer Science Departent, University of Illinois
More informationPREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW
PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 66 2 Nonprobability Sapling and Bias 66 Stratified Rando Sapling 62 6 4 Cluster Sapling 64 6 5 Systeatic Sapling 69 6 6 Nonresponse 62 6 7 Suary and Review of
More informationSolving epsilondelta problems
Solving epsilondelta problems Math 1A, 313,315 DIS September 29, 2014 There will probably be at least one epsilondelta problem on the midterm and the final. These kind of problems ask you to show 1 that
More informationIntroduction to Unit Conversion: the SI
The Matheatics 11 Copetency Test Introduction to Unit Conversion: the SI In this the next docuent in this series is presented illustrated an effective reliable approach to carryin out unit conversions
More informationENZYME KINETICS: THEORY. A. Introduction
ENZYME INETICS: THEORY A. Introduction Enzyes are protein olecules coposed of aino acids and are anufactured by the living cell. These olecules provide energy for the organis by catalyzing various biocheical
More informationManaging Complex Network Operation with Predictive Analytics
Managing Coplex Network Operation with Predictive Analytics Zhenyu Huang, Pak Chung Wong, Patrick Mackey, Yousu Chen, Jian Ma, Kevin Schneider, and Frank L. Greitzer Pacific Northwest National Laboratory
More informationMedia Adaptation Framework in Biofeedback System for Stroke Patient Rehabilitation
Media Adaptation Fraework in Biofeedback Syste for Stroke Patient Rehabilitation Yinpeng Chen, Weiwei Xu, Hari Sundara, Thanassis Rikakis, ShengMin Liu Arts, Media and Engineering Progra Arizona State
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationExperiment 2 Index of refraction of an unknown liquid  Abbe Refractometer
Experient Index of refraction of an unknown liquid  Abbe Refractoeter Principle: The value n ay be written in the for sin ( δ +θ ) n =. θ sin This relation provides us with one or the standard ethods
More informationThe students will gather, organize, and display data in an appropriate pie (circle) graph.
Algebra/Geoetry Institute Suer 2005 Lesson Plan 3: Pie Graphs Faculty Nae: Leslie Patten School: Cypress Park Eleentary Grade Level: 5 th grade PIE GRAPHS 1 Teaching objective(s) The students will gather,
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationTHREEPHASE DIODE BRIDGE RECTIFIER
Chapter THREEPHASE DIODE BRIDGE RECTIFIER The subject of this book is reduction of total haronic distortion (THD) of input currents in threephase diode bridge rectifiers. Besides the reduction of the
More informationEfficient Key Management for Secure Group Communications with Bursty Behavior
Efficient Key Manageent for Secure Group Counications with Bursty Behavior Xukai Zou, Byrav Raaurthy Departent of Coputer Science and Engineering University of NebraskaLincoln Lincoln, NE68588, USA Eail:
More informationAmplifiers and Superlatives
Aplifiers and Superlatives An Exaination of Aerican Clais for Iproving Linearity and Efficiency By D. T. N. WILLIAMSON and P. J. WALKE ecent articles, particularly in the United States, have shown that
More informationThe Benefit of SMT in the MultiCore Era: Flexibility towards Degrees of ThreadLevel Parallelism
The enefit of SMT in the MultiCore Era: Flexibility towards Degrees of ThreadLevel Parallelis Stijn Eyeran Lieven Eeckhout Ghent University, elgiu Stijn.Eyeran@elis.UGent.be, Lieven.Eeckhout@elis.UGent.be
More informationn is symmetric if it does not depend on the order
Chapter 18 Polynoial Curves 18.1 Polar Fors and Control Points The purpose of this short chapter is to show how polynoial curves are handled in ters of control points. This is a very nice application of
More informationSupport Vector Machine Soft Margin Classifiers: Error Analysis
Journal of Machine Learning Research? (2004)??? Subitted 9/03; Published??/04 Support Vector Machine Soft Margin Classifiers: Error Analysis DiRong Chen Departent of Applied Matheatics Beijing University
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationOnline Bagging and Boosting
Abstract Bagging and boosting are two of the ost wellknown enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationREQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES
REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES Charles Reynolds Christopher Fox reynolds @cs.ju.edu fox@cs.ju.edu Departent of Coputer
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationBudgetoptimal Crowdsourcing using Lowrank Matrix Approximations
Budgetoptial Crowdsourcing using Lowrank Matrix Approxiations David R. Karger, Sewoong Oh, and Devavrat Shah Departent of EECS, Massachusetts Institute of Technology Eail: {karger, swoh, devavrat}@it.edu
More informationESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET
ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More informationSimple Harmonic Motion MC Review KEY
Siple Haronic Motion MC Review EY. A block attache to an ieal sprin uneroes siple haronic otion. The acceleration of the block has its axiu anitue at the point where: a. the spee is the axiu. b. the potential
More informationSearching strategy for multitarget discovery in wireless networks
Searching strategy for ultitarget discovery in wireless networks Zhao Cheng, Wendi B. Heinzelan Departent of Electrical and Coputer Engineering University of Rochester Rochester, NY 467 (585) 75{878,
More informationMultiClass Deep Boosting
MultiClass Deep Boosting Vitaly Kuznetsov Courant Institute 25 Mercer Street New York, NY 002 vitaly@cis.nyu.edu Mehryar Mohri Courant Institute & Google Research 25 Mercer Street New York, NY 002 ohri@cis.nyu.edu
More informationInsurance Spirals and the Lloyd s Market
Insurance Spirals and the Lloyd s Market Andrew Bain University of Glasgow Abstract This paper presents a odel of reinsurance arket spirals, and applies it to the situation that existed in the Lloyd s
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationThe Lagrangian Method
Chapter 6 The Lagrangian Method Copyright 2007 by David Morin, orin@physics.harvard.edu (draft version In this chapter, we re going to learn about a whole new way of looking at things. Consider the syste
More informationSoftware Quality Characteristics Tested For Mobile Application Development
Thesis no: MGSE201502 Software Quality Characteristics Tested For Mobile Application Developent Literature Review and Epirical Survey WALEED ANWAR Faculty of Coputing Blekinge Institute of Technology
More informationFuzzy Sets in HR Management
Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,
More informationOSCILLATION OF DIFFERENCE EQUATIONS WITH SEVERAL POSITIVE AND NEGATIVE COEFFICIENTS
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Hasan Öğünez and Özkan Öcalan OSCILLATION OF DIFFERENCE EQUATIONS WITH SEVERAL POSITIVE AND NEGATIVE COEFFICIENTS Abstract. Our ai in this paper is to
More informationA magnetic Rotor to convert vacuumenergy into mechanical energy
A agnetic Rotor to convert vacuuenergy into echanical energy Claus W. Turtur, University of Applied Sciences BraunschweigWolfenbüttel Abstract Wolfenbüttel, Mai 21 2008 In previous work it was deonstrated,
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationAngles formed by 2 Lines being cut by a Transversal
Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa. 1 2 3 4 t 5 6 7 8 If I asked you
More informationINTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS
Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE
More informationAnswer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This printout should have 4 questions. Multiplechoice questions ay continue on the next colun or page find all choices before aking your selection.
More informationKeywords: Educational Timetabling; Integer Programming; Shift assignment
School Tietabling for Quality Student and Teacher Schedules T. Birbas a ( ), S. Daskalaki b, E. Housos a a Departent of Electrical & Coputer Engineering b Departent of Engineering Sciences University of
More informationA CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS
A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS Isaac Zafrany and Sa BenYaakov Departent of Electrical and Coputer Engineering BenGurion University of the Negev P. O. Box
More informationI Can Mathematics Descriptors
I Can Matheatics Descriptors Year 1 to Year 6 and beyond; Supporting the 2014 National Curriculu EdisonLearning s Precision Pedagogy Based on research into current approaches in UK schools that have had
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationDensity and Center of Mass
Density and Center of Mass Objectives Integrate a density function to find Total Population or Total Mass Integrate or use Suation to find Center of Mass of a Substance with Certain Density Density Density
More informationAlgebra (Expansion and Factorisation)
Chapter10 Algebra (Expansion and Factorisation) Contents: A B C D E F The distributive law Siplifying algebraic expressions Brackets with negative coefficients The product (a + b)(c + d) Geoetric applications
More informationSolutions of Equations in One Variable. FixedPoint Iteration II
Solutions of Equations in One Variable FixedPoint Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationEquivalent Tapped Delay Line Channel Responses with Reduced Taps
Equivalent Tapped Delay Line Channel Responses with Reduced Taps Shweta Sagari, Wade Trappe, Larry Greenstein {shsagari, trappe, ljg}@winlab.rutgers.edu WINLAB, Rutgers University, North Brunswick, NJ
More informationInternational Journal of Management & Information Systems First Quarter 2012 Volume 16, Number 1
International Journal of Manageent & Inforation Systes First Quarter 2012 Volue 16, Nuber 1 Proposal And Effectiveness Of A Highly Copelling Direct Mail Method  Establishent And Deployent Of PMOSDM Hisatoshi
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationElectric Forces between Charged Plates
CP.1 Goals of this lab Electric Forces between Charged Plates Overview deterine the force between charged parallel plates easure the perittivity of the vacuu (ε 0 ) In this experient you will easure the
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
More information