1 Numerical Solution to Quadratic Equations


 Brenda Scott
 1 years ago
 Views:
Transcription
1 cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll from lst lecture tht we wnted to find numericl solution to qudrtic eqution of the form x 2 + bx = c. () One obvious method for solving the eqution is to use the fmilir qudrtic formul: x,2 = b ± b 2 + 4c. (2) 2 Now, the formul does not provide us with solution tht is written in deciml form; to get such solution we need to evlute the bove expression. To this end, since most computers perform dditions, subtrctions, multiplictions, nd even divisions very fst, we should focus on the need to tke squreroot. In short, the explicit solution for the qudrtic eqution ctully reduces our problem to finding n lgorithm for computing squreroots. Even if we hve such n lgorithm, we hve to sk if using the qudrtic formul is the most efficient wy to solve this eqution; in other words, whether the best lgorithm for finding squreroots is superior to the best lgorithm tht solves the qudrtic eqution without using the formul. As we shll show now, we cn extend the powerful squre root lgorithm we proposed in the lst lecture so tht it solves generl qudrtic equtions, mking the use of the qudrtic formul (2) unnecessry (nd, in fct, inefficient). 2 Finding Squre Roots nd Solving Qudrtic Equtions 2. Finding Squre Roots As we discussed lst time, there is simple scheme for pproximting squre roots to ny given precision. More formlly, we cn find nonnegtive solution to the qudrtic eqution: x 2 = c, x 0, c 0 (3) using n itertive method tht llows us to control the precision of the solution. The ide is tht we find function tht, given n pproximtion of the solution s input (x old ), outputs more precise pproximtion (x new ). If we use this function itertively by recycling the vlues it produces on ech itertion, we will rrive t better nd better solutions. We cn continue this process until we rech ny level of precision we wnt. Recll tht the itertion we used ws: x new = x2 old + c 2x old (4)
2 An interesting property of this lgorithm is tht, roughly speking, the precision of the output (x new ) is doubled with ech intertion. As we shll see lter, this speed of convergence of the lgorithm will be n importnt fctor in evluting numeric lgorithms. 2.2 Solving Qudrtic Equtions Now tht we hve scheme for solving restricted kind of qudrtic eqution, cn we use the scheme to solve our originl problem? The nswer is yes. To solve equtions of the form x 2 + bx = c (5) We simply need to dd nother term to the denomintor of the formul: x new = x2 old + c 2x old + b We cn use this new formul itertively to rrive t numericl solutions of the qudrtic eqution tht re rbitrrily precise. Ech itertion will tke only 5 opertions (2 multiplictions, 2 dditions, nd division). Conclusion: the explicit qudrtic eqution reduced our problem from iterting with (6) to iterting with (4). While pedntic student my rgue (correctly!) tht the ltter itertion involves fewer opertions, we should ll gree tht the thoughttobewonderful qudrtic formul hs ctully only fringe vlue (t most). So, it should now be cler tht lgorithms tht re good for finding numericl solutions re often quite different from the methods we use to find nlytic solutions. 3 Clculting Definite Integrls Another good exmple of the difference between numericl computtion nd nlytic solutions is in clculting definite integrls. Recll tht the definition of the definite integrl of continuous (nd we will ssume positive) function f(t) over the intervl [, b], b f(t)dt is the re under the curve, s shown in Figure 3. (6) b Figure : The Definite Integrl of f(t) over [, b] 2
3 Wht is the best wy to clculte the definite integrl? Our experience in mth clsses suggests tht we might use the ntiderivtive pproch. We find function F such tht F = f, nd then b But this brings up two importnt questions:. How do we determine n ntiderivtive function? f(t)dt = F(b) F(). (7) 2. How difficult is it to evlute the ntiderivtive function t nd b? Sometimes the nswer to both questions is it s esy. For exmple, () consider 0 t3 dt. We know how to compute n ntiderivtive, nmely 4 t4, nd since this ntiderivtive is polynomil, we know how to evlute it efficiently using few rithmetic opertions. So we evlute 4 t4 0 = 4 0 = 4. Unfortuntely, most exmples re not so nice. For exmple, (b) consider x tdt. We know how to compute n ntiderivtive, nmely log e t, so we cn rewrite the integrl s x t dt = log e x log e = log e x. (8) To get numeric solution, we now need to evlute log e t x. Unfortuntely this is difficult  much more complicted thn evluting t. Any lgorithm for evluting log will be more complicted thn one for evluting t, while the re tht we re fter is completely determined by t. Some exmples re even worse. (c) Consider b e t2 dt, n integrl tht is useful in sttistics. The integrnd does hve n ntiderivtive, sy F (t) = e t2. But it is not esy to compute the ntiderivtive, or even to write down simple closedform. In fct, n efficient wy for evluting F t x is to compute the integrl x 0 f(t)dt. Finlly, (d) consider the integrl of the step function shown in Figure 3. Computing this integrl Figure 2: Approximting the Definite Integrl Using Subintervls is trivil. But no ntiderivtive exists. The bove discussion highlights fundmentl issue: the re we need to compute depends on the function f. Our ttempt to use the ntiderivtive pproch brings to the scene new function F whose properties my not be in tndem with those of the originl f. So is there method for clculting the definite integrl tht uses only informtion bout the function f? The nswer is yes, we cn use the definition of the definite integrl to pproximte its vlue, s shown in Figure 3. We strt by splitting up the intervl into subintervls of equl size h, then estimting the definite 3
4 integrl over ech subintervl, then dding them ll together. If we split the intervl [, b] into N subintervls of equl width h = b N, we cn clculte the definite integrl s: b f(t)dt = N j= +jh +(j )h f(t)dt. (9) + 2h + h b Figure 3: Approximting the Definite Integrl Using Subintervls We hve reduced the problem to estimting the definite integrl over smller subintervls. Here re two possible strtegies for estimting the smller definite integrls: Rectngle Rule +h f(t)dt f()h (0) In the rectngle rule, we clculte the re of the rectngle with width h nd height determined by evluting the function t the left endpoint of the intervl. Midpoint Rule +h f(t)dt f( + h )h () 2 In the midpoint rule, we clculte the re of the rectngle with width h nd height determined by evluting the function t the midpoint of the intervl. These two methods re depicted in Figure 3. Using these methods, we cn clculte the definite integrl by simply evluting the originl function t definite points, which we should lwys be ble to do (the previling ssumption is tht we hve n lgorithm for evluting f. In the bsence of such lgorithm, we ctully do not know wht f is). We cn improve the precision in ech cse by chnging the size of the subintervls h  the smller the subintervl, the greter the precision. The next question is, how cn we decide which of these lgorithms to use? Intuitively, it seems tht the midpoint rule would be better choice, but how cn we quntify this intuition? 4
5 Rectngle Rule Midpoint Rule Figure 4: The Rectngle nd Midpoint Rules 3. Evluting Numeric Algorithms In generl, we evlute numeric lgorithms using 3 mjor criteri: Cost  The mount of time (usully clculted s the number of opertions) the computtion will tke. Frequently we cn trdeoff cost versus ccurcy by chnging prmeter like the number of subintervls. Accurcy  How quickly the lgorithm pproches our desired precision. If we define the error s the difference between the ctul nswer nd the nswer returned by the numeric lgorithm, we often mesure the rte t which the error pproches 0 s cost is incresed. Robustness  How the correctness of the lgorithm is ffected by different types of functions nd different types of inputs. The lgorithm for clculting squre roots turns out to be extremely robust: it will work for ny squre root, even if we give it terrible initil vlue. 3.2 Evlution of Algorithms for Definite Integrls Applying these criteri to the midpoint nd rectngle rules: Cost Ech lgorithm performs single function evlution to estimte the definite integrl over subintervl, so the cost is identicl. The totl cost over the entire computtion grows linerly with the number of intervls N in ech cse, mening tht there is some fixed constnt c, not dependent on the number of intervls, such tht the totl cost is c N = O(N). For exmple, if the number of intervls doubles, the cost will double s well. Note tht we cn trdeoff between cost nd ccurcy by chnging the number of intervls N. Accurcy Clerly s the size of ech subintervl h gets smller, the error gets smller s well. Is there ny difference between the rte t which the rectngle nd midpoint rules error pproches 0? Rectngle Rule  For the rectngle rule, the error pproches 0 in direct proportion to the rte t which h pproches 0. In other words, the error ǫ = c h for some fixed constnt c, or ǫ = O(h). For exmple, hlving the size of h will hlve the mount of error. Midpoint Rule  For the midpoint rule, the error pproches 0 in proportion to the rte t which the squre of h pproches 0. Tht is the error ǫ = c 2 h 2 for some fixed constnt c 2, or ǫ = O(h 2 ). For exmple, hlving the size of h will divide the error by 4. 5
6 This difference is substntil. For exmple, if we wnt to compute the integrl of function between 0 nd, nd we use N = 000 subintervls, ech subintervl will hve size h = 000, so the rectngle rule s error will be on the order of 000, while the midpoint rule s error will be on the order of 0 6. Robustness It turns out tht for functions tht re resonbly wellbehved, both the endpoint rule nd the midpoint rule re quite robust. There do exist situtions, however, in which it is possible to evlute function t the left endpoint of n intervl, nd not t the midpoint. Through this evlution, we hve shown tht, in generl, the midpoint rule is fr better choice thn the rectngle rule for pproximting the definite integrl, since, for the sme cost, it will give us much more precision. 4 Loss of Significnce One finl wy tht numericl computtion is different thn the wy you hve done mthemtics to this point is the potentil loss of significnce. Consider two rel numbers stored with 0 digits of precision beyond the deciml point: π = b = The ctul numbers π nd b contin dditionl digits tht we hve not stored. So our stored numbers re ctully good, cceptble pproximtions of the true π, b. Now, we wnt to compute π b, nd wnt to hve similrly good pproximte representtion: 0 significnt digits, i.e., once the zeros end nd the number begins. However, ll we cn do is subtrct the given pproximtion to obtin : only two significnt digits! So, we hve lost most of the precision of the two originl numbers. One might try to solve this problem by incresing the precision of the originl numbers, but this is not solution: For ny finite precision storge, numbers tht re close enough will be indistinguishble. There is no universl wy to void loss of precision! The only generl solution is to void subtrcting numbers tht re lmost equl. An exmple of the effect of loss of significnce occurs in clculting derivtives. Exmple 4. (Clculting Derivtives). Given the function f(t) = sint, wht is f (2)? We ll know from clculus tht the derivtive of sint is cos t, so one could clculte cos(2) in this cse. However, just s with clculting definite integrls, it is not lwys possible, or efficient, to evlute the derivtive of function. Insted, recll tht the vlue of the derivtive of function f t fixed point t 0 is the slope of the line tngent to f t t 0. Recll, lso, tht we cn pproximte this slope by clculting the slope of secnt line tht intersects f t t 0 nd t point very close to t 0, sy t 0 + h, s shown in Figure 4. We lerned in clculus, tht s h 0, the slope of the secnt line pproches the slope of the tngent line, or: f f(t 0 + h) f(t 0 ) (t 0 ) = lim. (2) h 0 h 6
7 Tngent line to sin(2) Secnt line through sin(2), sin(2+h) h Figure 5: Tngent nd Secnt Lines on f(t) = sint Just s with the definite integrl, we could pproximte the derivtive t t 0 by performing this clcultion for smll vlues of h. However, notice tht s h gets very smll, the numertor of Eqution 2 will become 0, due to loss of significnce, so tht we will erroneously clculte ll derivtives s 0. Preventing loss of significnce in our clcultions will be n importnt prt of this course. 5 Topic I: Solving Equtions Now tht we hve some ide of wht lgorithms for computtion re, we will discuss lgorithms for performing one of the most bsic numericl tsks, solving equtions in one vrible. sin(x) x Figure 6: Grphicl Solution for x 3 = sinx Consider the following eqution, with solution depicted in Figure 5. x 3 = sinx. (3) There is no nlytic solution to this eqution. Insted, we will present n itertive method for finding numericl solution. 5. Itertive Method As in our solution to finding squre roots, we would like to find function g, such tht if we input n initil guess t solution, g will output better pproximtion of the ctul solution. More 7
8 formlly, we would like to define sequence x 0, x, x 2,... with x n+ g(x n ) such tht (x n ) 0 converges to the solution. We cn continue clculting vlues x i until we rech two vlues with x j = x j+. Clerly, one property of g is tht g(r) = r, where r is the rel solution to Eqution 3. A trnsformtion of Eqution 3 (e.g. x = sinx x 3 + x) will stisfy this property, the trick is to choose the trnsformtion tht produces converging series of vlues x, x 2,.... 8
Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More information5.2 The Definite Integral
5.2 THE DEFINITE INTEGRAL 5.2 The Definite Integrl In the previous section, we sw how to pproximte totl chnge given the rte of chnge. In this section we see how to mke the pproximtion more ccurte. Suppose
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationArea Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
More informationSection 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that
Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the longpromised connection between differentition nd integrtion, between res nd tngent lines.
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationNumerical integration
Chpter 4 Numericl integrtion Contents 4.1 Definite integrls.............................. 4. Closed NewtonCotes formule..................... 4 4. Open NewtonCotes formule...................... 8 4.4
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationNotes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information6 Numerical Integration
D. Levy 6 Numericl Integrtion 6.1 Bsic Concepts In this chpter we re going to explore vrious wys for pproximting the integrl of function over given domin. There re vrious resons s of why such pproximtions
More informationNotes #5. We then define the upper and lower sums for the partition P to be, respectively, U(P,f)= M k x k. k=1. L(P,f)= m k x k.
Notes #5. The Riemnn Integrl Drboux pproch Suppose we hve bounded function f on closed intervl [, b]. We will prtition this intervl into subintervls (not necessrily of the sme length) nd crete mximl nd
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationLesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationVECTORVALUED FUNCTIONS
VECTORVALUED FUNCTIONS MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the book: Sections 13.1. 13.2. Wht students should definitely get: Definition of vectorvlued function, reltion with prmetric
More informationScalar Line Integrals
Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationAMTH247 Lecture 16. Numerical Integration I
AMTH47 Lecture 16 Numericl Integrtion I 3 My 006 Reding: Heth 8.1 8., 8.3.1, 8.3., 8.3.3 Contents 1 Numericl Integrtion 1.1 MonteCrlo Integrtion....................... 1. Attinble Accurcy.........................
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationNumber Systems & Working With Numbers
Presenting the Mths Lectures! Your best bet for Qunt... MATHS LECTURE # 0 Number Systems & Working With Numbers System of numbers.3 0.6 π With the help of tree digrm, numbers cn be clssified s follows
More informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
More informationm, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the xxis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationRational Expressions
C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy,
More informationQuadratic Functions. Analyze and describe the characteristics of quadratic functions
Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationSo there are two points of intersection, one being x = 0, y = 0 2 = 0 and the other being x = 2, y = 2 2 = 4. y = x 2 (2,4)
Ares The motivtion for our definition of integrl ws the problem of finding the re between some curve nd the is for running between two specified vlues. We pproimted the region b union of thin rectngles
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationAe2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl wordforword with my lectures which will
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More information5.6 Substitution Method
5.6 Substitution Method Recll the Chin Rule: (f(g(x))) = f (g(x))g (x) Wht hppens if we wnt to find f (g(x))g (x) dx? The Substitution Method: If F (x) = f(x), then f(u(x))u (x) dx = F (u(x)) + C. Steps:
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationContinuous probability distributions
Chpter 8 Continuous probbility distributions 8.1 Introduction In Chpter 7, we explored the concepts of probbility in discrete setting, where outcomes of n experiment cn tke on only one of finite set of
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information1 SuperBrief Calculus I Review.
CALCULUS MATH 66 FALL 203 (COHEN) LECTURE NOTES For the purposes of this clss, we will regrd clculus s the study of limits nd limit processes. Without yet formlly reclling the definition of limit, let
More informationUsing Definite Integrals
Chpter 6 Using Definite Integrls 6. Using Definite Integrls to Find Are nd Length Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: How
More informationChapter 4: Dynamic Programming
Chpter 4: Dynmic Progrmming Objectives of this chpter: Overview of collection of clssicl solution methods for MDPs known s dynmic progrmming (DP) Show how DP cn be used to compute vlue functions, nd hence,
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationUnit 6 Solving Oblique Triangles  Classwork
Unit 6 Solving Oblique Tringles  Clsswork A. The Lw of Sines ASA nd AAS In geometry, we lerned to prove congruence of tringles tht is when two tringles re exctly the sme. We used severl rules to prove
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI
ON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI MICHAEL RAND 1. Introduction The Tower of Hnoi puzzle ws creted over century go by the number theorist Edourd Lucs [, 4], nd it nd its vrints hve chllenged
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More information1. A BRIEF SUMMARY OF CALCULUS
1. A BRIEF SUMMARY OF CALCULUS Clculus is one of the gretest intellectul chievements of humnkind. It llows us to solve mthemticl problems tht cnnot be solved by other mens, nd tht in turn llows us to mke
More informationRational Numbers and Decimal Representation
0 CHAPTER The Rel Numbers nd Their Representtions. Rtionl Numbers nd Deciml Representtion Properties nd Opertions The set of rel numbers is composed of two importnt mutully exclusive subsets: the rtionl
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More information