APPROXIMATING FUNCTIONS
|
|
- Julia Chapman
- 7 years ago
- Views:
Transcription
1 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) APPROXIMATING FUNCTIONS Te main ideas are: Constructing and interpreting difference tables Before te eam you sould know: How to construct and interpret a difference table, and te notation used Te formula for te Newton Interpolating polynomial (also known as Newton Forward Difference Metod) Using a difference table to calculate Newton s interpolating polynomial Lagrange s Polynomials/Metod (Finite) Difference Tables. Suppose tat you are given a collection of data points for some function, for eample: f() Note: we must ave equally spaced -values to use any of te metods in tis capter wic relate to looking at difference tables. Column of First 0 f( ) = f0 + Δ f0 + ( 0)( ) Δ f 0 +! ( 0)( )( )! f... Δ 0 + Lagrange s Polynomials wic fit points witout equally spaced -values. For eample A ( b)( c) B ( c)( a) f( ) = + ( a b)( a c) ( b c)( b a) C ( a)( b) + ( c a)( c b) goes troug (a,a), (b, B) and (c, C) Column of Second Differences Column of Tird Differences Column of Fourt Differences Differences i f i Δf i Δ f i Δ f i Δ 4 f i Δ 5 f i Column of Fift Differences Tis entry is calculated as =. Tis entry is calculated as =.8 As a rule of tumb, if te variation in a column is less tan about 0% of te average value for tat column. we would say tat it is nearly constant Te tird differences column is nearly constant so in Newton s Interpolation formula and Newton s Interpolating polynomial use terms up to and including Δ f 0 for a good approimation, unless you require an eact fit.
2 te Furter Matematics network V 07 Newton s Interpolating Polynomial (a.k.a Newton s Forward Difference Metod) Eample Te cubic f( ) = + passes troug te points (, ), (0, ), (, 0) and (, 5). Sow tat Newton s Interpolating Polynomial for tese four points is te original cubic polynomial. Te four points, (, ), (0, ), (, 0) and (, 5) give te i and te f i values 0 i f i Δf i Δ f i Δ f i f 0 Δf 0 Δ f 0 Δ f 0 So, substituting in te formula and wit =, gives 0 ( 0)( ) ( 0)( )( ) f( ) = f0 + Δ f0 + Δ f 0 + Δ f 0!! ( ( ))( 0) ( ( ))( 0)( ) = + [( ( )) ( ) ] = + ( ) ( + ) ( ) = + = + Langrange s Tecnique Eample For a certain function f, f() = 4.5, f() = 4.9, f(4) = 5.8 and f(7) = 8.4. Use Lagrange s Metod to estimate f(5) by fitting a cubic to te grap of f at te points given. Note tat because te -value we ave are not evenly spaced, we ave no coice but to use Lagrange s metod ere. Te curve passing troug (a,a), (b, B), (c, C) and (d, D) is A( b)( c)( d) B( a)( c)( d) C( a)( b)( d) D( a)( b)( c) y = ( a b)( a c)( a d) ( b a)( b c)( b d) ( c a)( c b)( c d) ( d a)( d b)( d c) Using tis wit te points (, 4.5), (, 4.9), (4, 5.8), (7, 8.4) gives 4.5( )( 4)( 7) 4.9( )( 4)( 7) 5.8( )( )( 7) 8.4( )( )( 4) y = ( )( 4)( 7) ( )( 4)( 7) (4 )(4 )(4 7) (7 )(7 )(7 4) Now you must let = 6 to get an approimation to f(6), notice tat ere tere is no need to simplify te interpolating polynomial itself! 4.5(6 )(6 4)(6 7) 4.9(6 )(6 4)(6 7) 5.8(6 )(6 )(6 7) 8.4(6 )(6 )(6 4) f(6) ( )( 4)( 7) ( )( 4)( 7) (4 )(4 )(4 7) (7 )(7 )(7 4) = = 7.8 (to d.p.)
3 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) ERRORS AND APPROXIMATION Te main ideas are: Significant Figures and Decimal Places Measures of error Propagation of Errors Before te eam you sould know: Te formula for measures of error. Tese are error, relative error, absolute error and absolute relative error. Te rules for propagation of relative errors addition, multiplication and division. Tat care must be taken wen subtracting approimations to nearly equal quantities, many significant figures can be lost in accuracy. Measures of Error Wen a given value is approimated by X : te errorε is given byε = X. te relative error is defined by, error relative error = eact value ε =. te absolute error is defined as te modulus of te error. In oter words, absolute error = ε = X. te absolute relative error is defined as error ε X absolute relative error = = =. eact value Subtraction of Nearly Equal Quantities Note: usage of tese terms on eam papers as not been completely consistent to date, e.g. an approimation of 0.6 to 0.6 may be described as aving an error of 0.0 instead of an error of 0.0. Eample Te values X =.45 and Y = 0.5 are approimations to values and y wic are bot correct to 6 significant figures. Give y correct to as many significant figures as is possible wit tis information. Since X =.45 is correct to 6 significant figures.455 <.455. Similarly, since Y =.5 is correct to 6 significant figures Terefore y < < y< or.0 < y<.0 So y must be.0 to significant figures, but it cannot be given to 4 significant figures from te information available.
4 te Furter Matematics network V 07 Propagation of relative error upon multiplying or dividing approimations Eample Suppose tat X =. is used as an approimation to =.478 and Y = 0.0 is used an approimation to y = 0.097, i) ii) iii) Wat is te relative error in eac of tese approimations? Wat is te relative error wen XY is used an approimation of y? X Wat is te relative error wen Y is used as an approimation of y? Find any relationsips between te values you ave calculated? i) X..478 r = = = (to 5 d.p.).478. Y y r = = = (to 5 d.p.) y ii) Te relative error wen XY is used to approimate y is: XY y (. 0.0) ( ) r = = y = = (to 5 d.p.) a. A possible relationsip is tat r + r = is approimately equal to tis value. iii) X Te relative error wen Y is used to approimate y is X..478 Y y r = = = = (to 5 d.p.) a. 4 y b. A possible relationsip is tat r r = = (to 5 d.p.) approimately equal to tis value. In fact tese observations old in general: is If X is an approimation of wit relative error r and Y is an approimation of y wit relative error r ten: te relative error in XY as an approimation of y is approimately r + r X te relative error in Y as an approimation of y is approimately r r.
5 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) NUMERICAL DIFFERENTIATION Te main ideas are: Using te forward and central difference formulae Knowing about te order of convergence for eac metod and being able to etrapolate from approimations you ave calculated An appreciation of ow te gradient of a function influences te error in f() wen tere is an error in. Using te forward and central difference formulae. Before te eam you sould know: Know te formulae for te forward and central difference approimations to te derivative (or gradient) of a function. Te forward difference approimation to te derivative of a function f at a value is given by f( + ) f( ) f( ) were is small. Te central difference approimation to te derivative of a function f at a value is given by f( + ) f( ) f( ) were is small. If X is used as an approimation to and te error is (so tat X = ) ten te error wen f(x) is used as an approimation to f() is approimately f( ) You sould know tat te central difference metod is a second metod and te forward difference metod is a first order metod (and wat tis means!). Eample For te function f() = sin, calculate te forward difference approimations to = 0.,0.05,0.05,0.05. Te forward difference approimation to f( ) is given by f() wit In eac case below =. Wit = 0., tis gives f( + ) f( ) f( ). f ( 0.) f (). sin. sin + = f () =.6956 (to 4 d.p.) 0. Wit = 0.05, f ( ) f () =.05 sin.05 sin f () =.890 (to 4 d.p.) 0.05 Tese approimations appear to be approacing a value sligtly less tan.
6 te Furter Matematics network V 07 Eample Values of a function, f, for various values of are given in te following table....4 f() 0 Find an approimation to f (.) using forward difference approimations = 0.. : Te central difference approimation wit = 0. is, f (.) f (.) f (.) = = Remembering te formulae Probably te easiest way to remember te formula for forward difference approimation to te derivative and central difference approimation to te derivative is to commit te following diagrams to memory. y-ais Forward Difference Approimation y-ais Central Difference Approimation f(+) f() + f(+)-f() -a gradient of cord cange in y coord = cange in coord f( + )-f( ) = f(+) f() f(-) - f(+)-f(-) + -a gradient of cord cange in y-coordinate = cange in -coordinate f( + ) f( ) = How te gradient of a function influences te error in f() wen tere is an error in. Eample Calculate te error incurred if te approimations X = 6.0 and X = 5.9 are used in te function obtain approimations to f(5.9). Relate tese to te derivative of f. f( ) = to Te eact value of f(5.9) is approimation of f(5.9) is 5.9 = Using X = 6.0 gives f(x) = 6. Te error in tis f (6) f (5.9) = = Te relationsip wit te derivative of f is tat f( X ) f( ) f ( ) were = X. In fact f ( ) = = ( ) Using X = 5.9 gives f(x) = Te error in tis approimation of f(5.9) is f (5.9) f (5.9) = = f ( ) = = Tis time = and ( )
7 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) NUMERICAL INTEGRATION Te main ideas are: Using te midpoint rule, trapezium rule and Simpson s rule to approimate integrals. Knowing ow accurate eac of tese metods is so tat you are able to estimate error. Te Midpoint Rule y a m m m m 4 m 5 m 6 m 7 m 8 b Before te eam you sould know: You sould be familiar wit all te standard notation used in Numerical Integration. You sould know te formulae for M n, te midpoint rule wit n strips, T n te trapezium rule wit n strips and S n, Simpson s rule. You sould know te following formulae, and use tem to your advantage in te eam. T n = ( Tn + Mn ) Sn = ( Tn + Mn) You sould know tat te midpoint and trapezium rule are second order metods. Tis means tat alving te strip widt gives you a rougly four times more accurate approimation. You sould know tat Simpson s rule is a fourt order metod. Tis means tat alving te strip widt gives you a siteen times better approimation. You sould be able to display some knowledge of ow tese facts can allow you to estimate te error in a given approimation by looking at te difference between tat and a previous approimation, see te last section on tis seet for more details. Remember tat eac time you apply tese rules, canges and te values of f, f, cange according to te particular strip widt you are using. Te composite form of te mid-point rule, using n strips, eac of widt gives b f( ) (f( ) f( )... f( )) d m + m + + mn = M n, a were m, m,..., mn are te values of at te midpoints of te strips.
8 Te Trapezium Law y te Furter Matematics network V 07 Te composite form of te trapezium law using n strips, eac b 0 n n. a of widt gives f ( d ) [ f + (f + f + f f ) + f ] Note te notation ere, f 0 is te value of te function at te left and end of te first strip, f is te value of te function at te left and end of te second strip (or te rigt and end of te first strip) and so on. Finally, f n is te value of te function at te rigt and end of te n t strip. a b Simpson s Rule In general, over any interval (a, b) divided into an even number, n, of strips, of widt, te composite Simpson s rule gives, b f ( d ) Sn = f0 + 4(f+ f + f f n ) + (f + f f n ) + fn a [ ] Simpson s rule approimates te function by a series of quadratics, one for eac pair of neigbouring strips. Note tat te above approimation can be calculated as Sn = ( Tn + Mn ) How accurate are my estimates? In te table below and to te rigt. we ave a series of trapezium rule estimates to sin d. Immediately to te rigt is te grap of tis function How accurate can we give an estimate to tis integral based on tese results? Te first ting to say is tat T 8 and T 6 agree to two decimal places, tey bot round to To tree decimal places tey disagree, T 8 rounds to wereas T 6 rounds to As te trapezium rule is a second order metod we can safely say tat to two decimal places sin( d= ) We cannot safely say wat te estimate is to tree decimal places. Estimates get about four times better (four times closer to te true value) upon eac alving of te strip widt. Tis means tat te ratio of differences will be about 0.5 as is sown in te table on te rigt. We may wis to etrapolate to obtain furter estimates of T6 T8 T6 T8 T6 T8 " T " = T6 +," T64 " = T6 + + etc etc. 4 4 In fact you may wis to look at te number T T 6 T8 T 6 T8 T6 T8 T wic can be found by summing an appropriate geometric progression. y Trapezium Differences Ratio of Estimates T i+ -T i Differences T = T = T 4 = T 8 = T 6 = T 8 T 4 T 6 True Value
9 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) RATE OF CONVERGENCE IN NUMERICAL PROCESSES Te main ideas are: Using te forward and central difference formulae. Knowing about te order of convergence for eac metod and being able to etrapolate from approimations you ave calculated An appreciation of ow te gradient of a function influences te error in f() wen tere is an error in. Detecting First Order Convergence by looking at Ratio of Differences Eample Sow, by considering ratios of differences, tat te following sequence as first order convergence. Before te eamination you sould know: A converging sequence is said to ave first order convergence if, for some fied positive number k, absolute error in absolute error in k, k,.. absolute error in 0 absolute error in A converging sequence is said to ave second order convergence if, for some fied positive number k, absolute error in absolute error in k, k,... 0 ( absolute error in ) ( absolute error in ) For metods depending on, like tose found in Numerical Integration and Differentiation, a metod is said to be an n t order metod if te absolute error in an approimation is proportional to n. Te forward difference approimation is a first order metod. Te central difference approimation, trapezium rule and midpoint rule are second order metods. Simpson s Rule is a fourt order metod Te ratios of differences are given in te table below. You can calculate te ratios of differences very quickly using a spreadseet program. n n n+- n ( n+- n)/( n+- n+) Ratios of differences Te ratio of differences column, being nearly constant, provides evidence of first order convergence. Detecting Second Order Convergence Second or iger order convergence is muc faster tan first order and so you can get a very good approimation to te limit wit only a few iterations. Since suc an approimation is very close to te actual value of te limit you can use it to estimate closely te absolute error in eac term.
10 You sould know te following facts: Metod te Furter Matematics network V 07 Effect of alving or doubling n Ratio of differences between suc estimates Forward Difference Approimation Te estimate gets twice as close Central Difference Approimation Te estimate gets four times as close 4 Midpoint Rule Te estimate gets four times as close 4 Trapezium Rule Te estimate gets four times as close 4 Simpson s Rule Te estimate gets siteen times as close 6 Eample Below are Trapezium rule estimates to some integral. By considering te ratio between differences in successive estimates, give te value of te integral to te number of decimal places you feel is justified. T T 4 T T 8 T 4 T Te ratio of differences is = 0.58 (to 4 d.p.). 0.9 Te etrapolated value of T 6 is Tis is very close to te value of 4 predicted by te teory = Notice ow we need to subtract ere because clearly T 6 is epected to be less tan T 8. Te etrapolated value of T is = Tis sequence of values converges to = = = 4 = (to 6 d.p.) Comparing tese values, it looks as toug te estimate of.75 is correct to d.p.
11 te Furter Matematics network V 07 REVISION SHEET NUMERICAL METHODS (MEI) THE SOLUTION OF EQUATIONS Te main ideas are: Using te bisection metod, te metod of false position (also known as linear interpolation), fied point iteration, te Newton Rapson Metod, and te secant metod to approimate te solution of an equation. Knowing ow accurate eac of tese metods is so tat you are able to estimate error. Te Bisection Metod Tis is probably te easiest of all te metods to apply. Te key points are: Before te eam you sould know: And be totally familiar wit using te five metods of approimating solutions to equations used in tis capter. Tese are, te bisection metod, te metod of false position, fied point iteration, te Newton Rapson Metod, and te secant metod. For every one of te metods above, you sould know ow to judge wen you ave te approimation to te required degree of accuracy and ten ceck tat tis is indeed te case. You sould know te order of convergence of te Newton-Rapson metod. It as second order convergence you sould know wat tis means. You sould know te requirements on te function g to find a fied point using fied point iteration. You must begin wit two points tat straddle te solution of f() = 0. Tese are usually called a and b Te reason tat tey are cosen is tat te sign of f(a ) is te opposite of te sign of f(b ). Tis means tat we epect te function to cross te -ais somewere between tem. a+ b Te sign of f is calculated. Depending on tis we know tat te cange of sign as to occur a+ b a+ b eiter between a and or between and b. Te appropriate pair become our a and b and te wole process is repeated. Note tat at every stage in tis process you can say tat te solution satisfies an < < b n. Tis means tat if an and b n are close enoug you can give te solution to any degree of accuracy. Te Metod of False Position (also known as Linear Interpolation) f( ) Given two values, a and b tat straddle a root of f() = 0 an approimation to te root is given by af( b) bf( a) c = f( b) f( a) Tis is te point were te straigt line between ( a,f( a)) and (, b f()) b crosses te -ais. Tis metod can be iterated by testing te sign of f(c) to find weter te root is between a and c or between c and b. f(b) f(a) a c α b
12 Fied Point Iteration One of te main skills you need to acquire for fied point iteration is to get from an equation like f() = 0 to an equation of te form = g(). Here are some eamples = 0 =, te Furter Matematics network V sin = 0 = + 4sin y = If te fied point iteration metod produces values,,,... were n+ = g( n),converging to te fied point of g, ten we epect error in n+ g( ) error in n However we need a little more tan tis if we are to approimate te solution of f() = 0. We need te gradient of te g() to be between and around te fied point. y Gradient of red line negative but >- around te fied point results in a converging series of values, illustrated by te inward moving cobweb in tis diagram. y y = g() Gradient of red line negative and <- around te fied point results in a non-converging series of values, illustrated by te outward moving cobweb in tis diagram. Newton Rapson-Metod To generate a sequence of values converging to a root of f() = 0, near to = 0, use te following iterative f( r ) formulae: = r + r. Tis metod as second order f( r ) convergence. Below te Newton Rapson metod is being used to approimate a solution of + - = 0. So we ave r + r r+ = r + in tis case. Using a starting value of r = gives = 0 = = = = 0.6 = Ten = , 4 = , 5 = , 5 = Notice ow quickly te sequence converges. You can get your calculator to perform tese calculations very quickly using te ANS feature. Tis also applies to Fied Point Iteration so make sure you know ow to do tis. Secant Metod If = 0 and = are approimations to a root of f() = 0, a better approimation to te root will usually be given by 0f( ) f( 0) =. Tis can be f( ) f( 0) repeated wit and replacing 0 and to obtain a value and so on.. Tese calculations can take quite a wile on a pocket calculator! Te main difference between te secant metod and te metod of false position are tat in te secant metod 0 and need not straddle te solution wereas in te metod of false position te first two values, a and b, sould straddle te root. Wen 0 and do straddle te root te point generated, is eactly te same as c in te metod of false position.
Derivatives 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 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 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 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 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 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 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 informationTangent 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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) 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 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 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 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 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 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 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 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 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 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 informationRoots of Equations (Chapters 5 and 6)
Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have
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 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 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 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 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 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 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 informationResearch on the Anti-perspective Correction Algorithm of QR Barcode
Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
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 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 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 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 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 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 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 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 informationRoots of equation fx are the values of x which satisfy the above expression. Also referred to as the zeros of an equation
LECTURE 20 SOLVING FOR ROOTS OF NONLINEAR EQUATIONS Consider the equation f = 0 Roots of equation f are the values of which satisfy the above epression. Also referred to as the zeros of an equation f()
More informationSection 3-3 Approximating Real Zeros of Polynomials
- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationSAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY
ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,
More informationOPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS
OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous
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 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 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 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 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 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 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 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 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 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 informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationFactoring Synchronous Grammars By Sorting
Factoring Syncronous Grammars By Sorting Daniel Gildea Computer Science Dept. Uniersity of Rocester Rocester, NY Giorgio Satta Dept. of Information Eng g Uniersity of Padua I- Padua, Italy Hao Zang Computer
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
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 informationComparison between two approaches to overload control in a Real Server: local or hybrid solutions?
Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor
More informationWork as the Area Under a Graph of Force vs. Displacement
Work as the Area Under a Graph of vs. Displacement Situation A. Consider a situation where an object of mass, m, is lifted at constant velocity in a uniform gravitational field, g. The applied force is
More informationSection 2.3 Solving Right Triangle Trigonometry
Section.3 Solving Rigt Triangle Trigonometry Eample In te rigt triangle ABC, A = 40 and c = 1 cm. Find a, b, and B. sin 40 a a c 1 a 1sin 40 7.7cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A
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 informationTorchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK
News Release Torcmark Corporation 2001 Tird Avenue Sout Birmingam, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK TORCHMARK CORPORATION REPORTS FOURTH QUARTER AND YEAR-END 2004 RESULTS
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 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 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 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 informationRecall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0
1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal
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 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 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 informationa joint initiative of Cost of Production Calculator
a joint initiative of Cost of Production Calculator 1 KEY BENEFITS Learn to use te MAKING MORE FROM SHEEP cost of production calculator to: Measure te performance of your seep enterprise year on year Compare
More informationYale ICF Working Paper No. 05-11 May 2005
Yale ICF Working Paper No. 05-11 May 2005 HUMAN CAPITAL, AET ALLOCATION, AND LIFE INURANCE Roger G. Ibbotson, Yale cool of Management, Yale University Peng Cen, Ibbotson Associates Mose Milevsky, culic
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 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 informationHuman Capital, Asset Allocation, and Life Insurance
Human Capital, Asset Allocation, and Life Insurance By: P. Cen, R. Ibbotson, M. Milevsky and K. Zu Version: February 25, 2005 Note: A Revised version of tis paper is fortcoming in te Financial Analysts
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 informationStaffing and routing in a two-tier call centre. Sameer Hasija*, Edieal J. Pinker and Robert A. Shumsky
8 Int. J. Operational Researc, Vol. 1, Nos. 1/, 005 Staffing and routing in a two-tier call centre Sameer Hasija*, Edieal J. Pinker and Robert A. Sumsky Simon Scool, University of Rocester, Rocester 1467,
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationTraining Robust Support Vector Regression via D. C. Program
Journal of Information & Computational Science 7: 12 (2010) 2385 2394 Available at ttp://www.joics.com Training Robust Support Vector Regression via D. C. Program Kuaini Wang, Ping Zong, Yaoong Zao College
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 informationAn Introduction to Milankovitch Cycles
An Introduction to Milankovitc Cycles Wat Causes Glacial Cycles? Ricard McGeee kiloyear bp 45 4 35 3 5 15 1 5 4 - -4-6 -8 temperature -1 Note te period of about 1 kyr. Seminar on te Matematics of Climate
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 informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
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 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 informationPolynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used
More information