Understanding the Derivative Backward and Forward by Dave Slomer
|
|
- Florence Patterson
- 7 years ago
- Views:
Transcription
1 Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange. Tis activity will define procedures for computing instantaneous rates of cange exactly and, wen exact is impossible or impractical, approximately. To do tis requires a rigorous definition of slope of a curve. Te slope of a curve at a point is defined to be te slope of te line tangent to te curve at tat point (see figure 1a). But te slope formula requires two points to get a slope. Often, as in figure 1a, a tangent line passes troug only one point on a curve. So were do we get anoter point from wic to compute te slope? We look to any oter nearby point. [Wen tey exist, tangent lines can be understood intuitively by zooming in at te point of tangency and noticing tat, after enoug zooms, te tangent and te curve appear to be te same grap. Hence, te curve, being virtually linear in tat window, can be tougt of as aving a definite slope. Before proceeding, exploring tis penomenon via anoter activity migt be advisable.] Fig. 1a Fig. 1b f ( x) If f is any function, ten is te slope of te secant line (see figure 1b) x a troug (a, f (a)) and (x, f (x)). Because it is a quotient in wic bot te numerator and denominator are differences, tat expression is called a difference quotient. Any secant line goes troug (at least) two points on te grap of a function; ence te numerator will always be te difference of two function values. If we want to someow use te secant line to approximate te slope of te tangent, ten x sould be near a. Exercise 1: Te point (x, f (x)) is referred to in te numerator of te difference quotient above. In figure 1b, were is tat point? Is tat point near enoug to P to give a decent approximation of te slope of te tangent at P (sown in fig. 1a)? f Oter forms of difference quotients include (wic is very generic, but concise) and x f ( a + x). If te latter two difference quotients are going to give slopes tat are ( a + x) a close to te slope of te tangent at (a, f (a)), teir denominators are going to ave to be near 0. (Convince yourself of tat.)
2 f ( a + x) Exercise 2: Simplify te difference quotient. (After simplification, ( a + x) a does it still look like a difference of two function values over te difference of two x s?) For tis difference quotient to be a good approximation of te slope of te tangent line at (a, f (a)), x must be near. If x is positive, te point (a + x, f (a + x)) will be to te rigt of (a, f (a)), so te difference quotient in Exercise 2 is called te forward difference quotient. f ( a) f ( a ) Exercise 3: Te difference quotient is te slope of te secant line troug two points on te grap of f. Assume tat is positive and locate te oter point on te sketc below and draw te secant line. [Te function expressions in te numerator tell you wat x s to use in te denominator.] Wat are te coordinates of te two points represented in te difference quotient above? In order for tis secant line to be a good approximation to te slope of te tangent line at P, ten would ave to be near. If is positive, te point (a, f (a )) will be to te left of (a, f (a)), so te difference quotient in Exercise 3 is called te backward difference quotient. Exercise 4: On te grap below, eac tick mark represents one unit and P is te point (a, f (a)). Let be 0.5, wic is not particularly close to 0, and draw te secant line troug te points (a, f (a )) and (a +, f (a + )). Ten express its slope as a difference quotient in terms of f, a, and. Do you see wy tis difference quotient is called te symmetric difference quotient?
3 Using te grid dots on te grap screen, estimate te slope given by te symmetric difference quotient. Draw te tangent line at P. How well does te symmetric difference quotient approximate its slope? π x Exercise 5: Te function in te grap in Exercise 4 is f( x) = sin and P as x- 3 coordinate 1. Grap tat function, zoom in [wit equal zoom factors] at P until te grap looks linear, and estimate te slope. Compute te symmetric difference quotient wit = 0.5. For wat value of are te forward and backward difference quotients as accurate as te symmetric wit = 1? Conclusions? Exercise 6: Sow algebraically [feel free to use your 89] tat te symmetric difference quotient is te average of te forward and backward difference quotients. (For aid, refer to figure 2, in wic te forward difference quotient gives te slope of PR, wile te backward gives te slope of PQ, and te symmetric gives te slope ofqr. You would be wise to let (a, f (a)) represent P and use eiter or x (consider bot to be positive) to determine te coordinates of Q and R.) Fig. 2 Exercise 7: Figure 2 was produced via te following window and function definitions. Matc te tangent line and forward, backward, and symmetric difference quotient secant lines wit te appropriate function in y2 troug y5. Eac equation in y2 troug y5 is a line in point-slope form, but every function defintion is at least partly cut off at te rigt margin. Write te equations for y2 troug y5.
4 [FYI figure 3 below was produced by turning off y3 troug y5 and storing {.1,.075,.05,.25} into.] OK, fine. We can find a lot of different types of secant lines wose slope can be expressed as many different-looking difference quotients tat are related in an interesting way (Exercise 6). But ow do we find slopes of tangents? Refer to figure 3 to see several forward secant lines (for =.1,.075,.05,.025) tat approac, as a limit, te tangent line, wic is also included in figure 3. Fig. 3 Exercise 8: In figure 3, were is te point of tangency? Wic line is te tangent? If te point of tangency is (a, f (a)), wat would be te equation of te tangent tere? It looks like te slope of te tangent line in figure 3 is a positive number. Wen it exists [sometimes it doesn t], te slope of te line tangent to te function y = f (x) at (a, f (a)) is f ( a + ) defined as f ( a) = lim. Tis is called te derivative of f at a. 0 Exercise 9: Wic difference quotient is used in te definition of derivative? Could eiter or bot of te oter two also ave been used? Explain. If, for some reason, te limits cannot be computed, te derivative can be approximated by using any of te individual difference quotients, if a small enoug can be used. As figure 2 sows, te backward and forward difference quotients are not likely to very good at estimating te slope of te tangent line unless is very close to 0. But, as figure 2 also sows (at least for te function involved), te symmetric difference quotient can be very close to correct, even if is not so small, or if we ave only a grap to proceed from. Exercise 10: For te function defined grapically and numerically below, estimate its derivative at x = 2.45 using all 3 difference quotients.
5 Te data above came from te function f(x) = 2sin(3x) cos(x). Use te definition of derivative (a limit of a forward difference quotient) to calculate te exact value of te derivative. Your TI-89 can calculate many derivatives exactly. Define f(x)=2sin(3x) cos(x) and ten enter te command d(f(x),x) x=2.45. [d is above te 8 key.] Did you get te same result tat you did in te first bullet above? (Just cecking!) Wic difference quotient came closest? By ow muc did it miss? Wat is te relative error in te calculation? [Relative error is defined as actual value approximate value /actual value.] Wic missed by most? By ow muc? Wat is te relative error? Conclusions? Exercise 11: Consider te points (5,5), (10,25), and (15,10), plotted on 3 axes below. Te fact is tat any one of te 3 difference quotients could give te best estimate of te derivative. Draw a function for wic te FDQ would be te best estimate of te derivative at (10,25). Draw a function for wic te BDQ would be te best estimate of te derivative at (10,25). Draw a function for wic te SDQ would be te best estimate of te derivative at (10,25). Wic of te 3 functions you graped do you tink would be most likely to occur in a real-life problem? Wy? Exercise 12: Te symmetric difference quotient is so good tat it gives exact results for parabolas, even witout taking a limit! Prove tat tis is so. Exercise 13: Unlike te TI-89, some calculators, suc as te TI-83, cannot compute limits, since tey cannot do symbol manipulation. Still, tey ave te built-in capability to find derivatives approximately. Wy do you tink tey use te symmetric difference quotient to numerically approximate derivatives?
6 Exercise 14: Explain wy te symmetric difference quotient (and terefore te TI-83, for example) gives 0 as an approximate value of te derivative of x at 0. Wat do you get if you take te limit of te symmetric difference quotient for x at 0? (Hint: take te leftand rigt-and limits wen you compute te derivative from te definition.) Conclusions? Since te derivative is a limit, x can approac a from bot te rigt (x > a; see figure 1b) and te left (x < a), so it makes sense to ten talk about left- and rigt-and derivatives. If f ' (a) exists, te left- and rigt-and derivatives at a are equal, and all 3 derivatives equal te slope of te tangent line at (a, f (a). (See figure 1.) Sometimes x cannot approac a from bot sides. If ( ) 4 f( x) = x + x, te derivative only exists at 0 from te left. (Wy?) Oter times, te function may not even ave a derivative, aving different left- and rigt-and derivatives. (Can you tink of an example or two of suc a function?) And even if a function as a derivative for all x, if te function is piecewise-defined, it would be necessary to compute te left- and rigt-and derivatives separately at any point were te function definition canges from one piece to te next. Tus, te following definitions are necessary: Te rigt-and derivative of f at a, often designated as f ( a ) +, is defined as f( a+ ) f( a) lim. Similarly, te left-and derivative, f ( a), can be defined by eiter + 0 f( a+ ) f( a) f( a) f( a ) lim or lim Exercise 15: Compute te left-and derivative for ( ) ( ) 4 f x = x + x at 0. ln( x),0 < x< 1 Exercise 16: For f( x) = 2, find te left- and rigt-and derivatives at 1 ( x 2) 1, x 1 and let tem tell you weter te derivative of f exists at 1. Draw te grap of te derivative. Describe its beavior at 1. Conclusions? Calculus Generic Scope and Sequence Topics: Derivatives NCTM Standards: Number and operations, Algebra, Geometry, Measurement, Connections, Communication, Representation
Tangent Lines and Rates of Change
Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationf(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationChapter 11. Limits and an Introduction to Calculus. Selected Applications
Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications
More informationAverage and Instantaneous Rates of Change: The Derivative
9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationNew Vocabulary volume
-. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationCompute the derivative by definition: The four step procedure
Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationWriting Mathematics Papers
Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More informationCan a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationAn inquiry into the multiplier process in IS-LM model
An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn
More informationGuide to Cover Letters & Thank You Letters
Guide to Cover Letters & Tank You Letters 206 Strebel Student Center (315) 792-3087 Fax (315) 792-3370 TIPS FOR WRITING A PERFECT COVER LETTER Te resume never travels alone. Eac time you submit your resume
More information- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz
CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger
More informationEC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution
EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat
More informationCHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0
CHAPTER TWO 2.1 SOLUTIONS 99 Solutions for Section 2.1 1. (a) Te average rate of cange is te slope of te secant line in Figure 2.1, wic sows tat tis slope is positive. (b) Te instantaneous rate of cange
More information4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a
Capter 4 Real Analysis 281 51. Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.
Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But
More informationPressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:
Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force
More informationShell and Tube Heat Exchanger
Sell and Tube Heat Excanger MECH595 Introduction to Heat Transfer Professor M. Zenouzi Prepared by: Andrew Demedeiros, Ryan Ferguson, Bradford Powers November 19, 2009 1 Abstract 2 Contents Discussion
More informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationHow To Ensure That An Eac Edge Program Is Successful
Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More information2.23 Gambling Rehabilitation Services. Introduction
2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority
More informationStrategic trading in a dynamic noisy market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral
More informationChapter 10: Refrigeration Cycles
Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators
More informationStrategic trading and welfare in a dynamic market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationComputer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit
More informationCyber Epidemic Models with Dependences
Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas
More informationGrade 12 Assessment Exemplars
Grade Assessment Eemplars Learning Outcomes and. Assignment : Functions - Memo. Investigation: Sequences and Series Memo/Rubric 5. Control Test: Number Patterns, Finance and Functions - Memo 7. Project:
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals
More informationBinary Search Trees. Adnan Aziz. Heaps can perform extract-max, insert efficiently O(log n) worst case
Binary Searc Trees Adnan Aziz 1 BST basics Based on CLRS, C 12. Motivation: Heaps can perform extract-max, insert efficiently O(log n) worst case Has tables can perform insert, delete, lookup efficiently
More informationFor Sale By Owner Program. We can help with our for sale by owner kit that includes:
Dawn Coen Broker/Owner For Sale By Owner Program If you want to sell your ome By Owner wy not:: For Sale Dawn Coen Broker/Owner YOUR NAME YOUR PHONE # Look as professional as possible Be totally prepared
More informationSchedulability Analysis under Graph Routing in WirelessHART Networks
Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,
More informationPre-trial Settlement with Imperfect Private Monitoring
Pre-trial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire Jee-Hyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial
More informationEnvironmental Policy and Competitiveness: The Porter Hypothesis and the Composition of Capital 1
Journal of Environmental Economics and Management 7, 65 8 Ž 999. Article ID jeem.998.6, available online at ttp: www.idealibrary.com on Environmental Policy and Competitiveness: Te Porter ypotesis and
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More information13 PERIMETER AND AREA OF 2D SHAPES
13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a two-dimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter
More information2.12 Student Transportation. Introduction
Introduction Figure 1 At 31 Marc 2003, tere were approximately 84,000 students enrolled in scools in te Province of Newfoundland and Labrador, of wic an estimated 57,000 were transported by scool buses.
More informationCatalogue no. 12-001-XIE. Survey Methodology. December 2004
Catalogue no. 1-001-XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationA Multigrid Tutorial part two
A Multigrid Tutorial part two William L. Briggs Department of Matematics University of Colorado at Denver Van Emden Henson Center for Applied Scientific Computing Lawrence Livermore National Laboratory
More informationChapter 6 Tail Design
apter 6 Tail Design Moammad Sadraey Daniel Webster ollege Table of ontents apter 6... 74 Tail Design... 74 6.1. Introduction... 74 6.. Aircraft Trim Requirements... 78 6..1. Longitudinal Trim... 79 6...
More informationRISK ASSESSMENT MATRIX
U.S.C.G. AUXILIARY STANDARD AV-04-4 Draft Standard Doc. AV- 04-4 18 August 2004 RISK ASSESSMENT MATRIX STANDARD FOR AUXILIARY AVIATION UNITED STATES COAST GUARD AUXILIARY NATIONAL OPERATIONS DEPARTMENT
More informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
More information3 Ans. 1 of my $30. 3 on. 1 on ice cream and the rest on 2011 MATHCOUNTS STATE COMPETITION SPRINT ROUND
0 MATHCOUNTS STATE COMPETITION SPRINT ROUND. boy scouts are accompanied by scout leaders. Eac person needs bottles of water per day and te trip is day. + = 5 people 5 = 5 bottles Ans.. Cammie as pennies,
More informationSWITCH T F T F SELECT. (b) local schedule of two branches. (a) if-then-else construct A & B MUX. one iteration cycle
768 IEEE RANSACIONS ON COMPUERS, VOL. 46, NO. 7, JULY 997 Compile-ime Sceduling of Dynamic Constructs in Dataæow Program Graps Soonoi Ha, Member, IEEE and Edward A. Lee, Fellow, IEEE Abstract Sceduling
More informationThe Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios
Te Dynamics of Movie Purcase and Rental Decisions: Customer Relationsip Implications to Movie Studios Eddie Ree Associate Professor Business Administration Stoneill College 320 Wasington St Easton, MA
More informationImproved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands
More informationThe use of visualization for learning and teaching mathematics
Te use of visualization for learning and teacing matematics Medat H. Raim Radcliffe Siddo Lakeead University Lakeead University Tunder Bay, Ontario Tunder Bay, Ontario CANADA CANADA mraim@lakeeadu.ca rsiddo@lakeeadu.ca
More informationAn Intuitive Framework for Real-Time Freeform Modeling
An Intuitive Framework for Real-Time Freeform Modeling Mario Botsc Leif Kobbelt Computer Grapics Group RWTH Aacen University Abstract We present a freeform modeling framework for unstructured triangle
More informationANALYTICAL REPORT ON THE 2010 URBAN EMPLOYMENT UNEMPLOYMENT SURVEY
THE FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA CENTRAL STATISTICAL AGENCY ANALYTICAL REPORT ON THE 2010 URBAN EMPLOYMENT UNEMPLOYMENT SURVEY Addis Ababa December 2010 STATISTICAL BULLETIN TABLE OF CONTENT
More informationSAT Math Must-Know Facts & Formulas
SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationOperation go-live! Mastering the people side of operational readiness
! I 2 London 2012 te ultimate Up to 30% of te value of a capital programme can be destroyed due to operational readiness failures. 1 In te complex interplay between tecnology, infrastructure and process,
More informationTHE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present. Prepared for public release by:
THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present Prepared for public release by: Tom Scroeder Kimberly Ault Division of Hazard and Injury Data Systems U.S. Consumer Product Safety Commission
More informationAverage rate of change of y = f(x) with respect to x as x changes from a to a + h:
L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
More informationWelfare, financial innovation and self insurance in dynamic incomplete markets models
Welfare, financial innovation and self insurance in dynamic incomplete markets models Paul Willen Department of Economics Princeton University First version: April 998 Tis version: July 999 Abstract We
More informationKM client format supported by KB valid from 13 May 2015
supported by KB valid from 13 May 2015 1/16 VRSION 1.2. UPDATD: 13.12.2005. 1 Introduction... 2 1.1 Purpose of tis document... 2 1.2 Caracteristics of te KM format... 2 2 Formal ceck of KM format... 3
More informationVolumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m
-5 Wat You ll Learn To find te volume of a pramid To find te volume of a cone... And W To find te volume of a structure in te sape of a pramid, as in Eample Volumes of Pramids and Cones Ceck Skills You
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationA system to monitor the quality of automated coding of textual answers to open questions
Researc in Official Statistics Number 2/2001 A system to monitor te quality of automated coding of textual answers to open questions Stefania Maccia * and Marcello D Orazio ** Italian National Statistical
More informationThe Demand for Food Away From Home Full-Service or Fast Food?
United States Department of Agriculture Electronic Report from te Economic Researc Service www.ers.usda.gov Agricultural Economic Report No. 829 January 2004 Te Demand for Food Away From Home Full-Service
More informationPioneer Fund Story. Searching for Value Today and Tomorrow. Pioneer Funds Equities
Pioneer Fund Story Searcing for Value Today and Tomorrow Pioneer Funds Equities Pioneer Fund A Cornerstone of Financial Foundations Since 1928 Te fund s relatively cautious stance as kept it competitive
More informationON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE
ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park
More informationPredicting the behavior of interacting humans by fusing data from multiple sources
Predicting te beavior of interacting umans by fusing data from multiple sources Erik J. Sclict 1, Ritcie Lee 2, David H. Wolpert 3,4, Mykel J. Kocenderfer 1, and Brendan Tracey 5 1 Lincoln Laboratory,
More informationMultigrid computational methods are
M ULTIGRID C OMPUTING Wy Multigrid Metods Are So Efficient Originally introduced as a way to numerically solve elliptic boundary-value problems, multigrid metods, and teir various multiscale descendants,
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways
More information100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
More informationFree Shipping and Repeat Buying on the Internet: Theory and Evidence
Free Sipping and Repeat Buying on te Internet: eory and Evidence Yingui Yang, Skander Essegaier and David R. Bell 1 June 13, 2005 1 Graduate Scool of Management, University of California at Davis (yiyang@ucdavis.edu)
More informationOn a Satellite Coverage
I. INTRODUCTION On a Satellite Coverage Problem DANNY T. CHI Kodak Berkeley Researc Yu T. su National Ciao Tbng University Te eart coverage area for a satellite in an Eart syncronous orbit wit a nonzero
More informationWarm medium, T H T T H T L. s Cold medium, T L
Refrigeration Cycle Heat flows in direction of decreasing temperature, i.e., from ig-temperature to low temperature regions. Te transfer of eat from a low-temperature to ig-temperature requires a refrigerator
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in- and outflows
More informationMath 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu
Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,
More information