1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang

Size: px
Start display at page:

Download "1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang"

Transcription

1 nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution An eqution is nonliner, i it is not liner Liner lgebric eqution: A b Generl (nonliner) lgebric eqution: ( ) O course, liner lgebric eqution is specil cse o the generl/nonliner lgebric eqution Below re some common nonliner lgebric equtions Emple: squre root (which is probbly the very irst nonliner lgebric eqution mny o us got epose to in n lgebr clss) which is equivlent to (or eine s) number, which when squre, results in : Emple: qurtic eqution (which is probbly the secon nonliner lgebric eqution mny o us ever got epose to) b c Emple: cubic root Emple: cubic eqution b c Emple: polynomil n n Emple: logrithmic unction ln( ) log( ) log bse ( ) which is equivlent to (or eine s) number, which when eponentite, results in : e bse Emple: rc sine or inverse sine n bunch o other trigonometric unctions sin( ) which is equivlent to (or eine s) number, whose sine is : sin( ) Emple: when we got beyon lgebr n strte tking clculus, we were introuce to integrls n erivtives in clculus courses g( t) t t g( t) evlute_t_

2 nonlinermc Stnr nottion: ()= To evelop systemtic pproch to solving nonliner lgebric equtions, we use the "stnr" ()= nottion To put given eqution into this stnr orm, it is simple mtter o moving everything to one sie o the eqution, n it oes not mtter which sie (LHS or RHS) Below re some emples or re-writing lgebric equtions in the "stnr" ()= orm squre root: ( ) cubic root: ( ) logrithm: e ( ) e inverse sine sin( ) ( ) sin( ) einite integrl g( t) t ( ) g( t) t Bisection Metho We strt with two points n b to brcket the root, where () n (b) hve ierent signs, ie, () (b)< I () n (b) hve ierent signs, the given unction () must pss through (ie, intersect with the horizontl is) t some points within [ b] We then choose the mi point between n b s the net guess I ( ) hs the sme sign s (), becomes the new let intervl Otherwise, becomes the new right intervl b The sme process is repete or the net guess, n so orth, until the intervl [ b] is suiciently smll Ech itertion hlves the intervl [ b] n gins one signiicnt binry igit Thus, every three itertions reuce the intervl by /8, which is roughly / This correspons to gining one signiicnt eciml igit An vntge o bisection metho is tht it lwys converges to root, lbeit somewht slower thn the other methos (eg, Newton's metho) tht re introuce lter here A chie isvntge is tht we hve to come up with two points o ierent signs to get the itertion process strte net guess is the mi point between n b: b 5 Bisection Metho ter unction evlution t : ter unction evlution t : b ter unction evlution t : b ter unction evlution t 4 : b b () () b

3 nonlinermc Metho o Chors As in bisection metho, we strt with two points n b, where () n (b) hve ierent signs, ie, () (b)< Inste o choosing the mi point between n b s the net guess, we rw chor to connect n b, n the intersection o the chor with the horizontl is becomes the net guess I ( ) hs the sme sign s (), becomes the new let intervl Otherwise, becomes the new right intervl b The sme process is repete or the net guess, n so orth net guess is where the chor intersects with horizontl is: ( ) b ( b) ( ) b ( b) b ( b) ( ) 5 Metho o Chors () () b Metho o Secnt We strt with ny two points n Unlike the metho o chors, these two initil points o not necessrily hve to yiel ierent signs or Inste o rwing the chor to connect n one o the en points (either or b, epening on the sign o the unction), we rw secnt tht connects the lst two points n - An the intersection o the secnt with the horizontl is becomes the net guess + Bsicll we use the lst two points to estimte the slope o the given unction; n bse on this estimte slope, we in where this secnt intersects with the horizontl is net guess is where the secnt intersects with horizontl is:

4 4 nonlinermc 5 Secnt Metho () () Newton's Metho We ssume tht () n its erivtive '() re both given We strt with ny one point initil guess Unlike the secnt metho, the slope t the present point is not estimte by connecting the lst two points, but nlyticlly evlute through '( ), which is given We rw tngent through the unction n the intersection o the tngent with the horizontl is becomes the net guess + slope ' verticl_rise horizontl_run ' 5 Newton's Metho () ()

5 Emple: squre root ( ) '( ) ' i strt with = 55 = 659 = 96 5 nonlinermc 4 = 4 6 compre = 6 Moiictions There re mny moiictions to the Newton's metho For emple, the bove squre root emple shows bit o oscilltion prior to convergence We sometimes tepily tke smll step α, inste o ull step Emple: squre root As beore, strt with iterte or times: i α 5 i i α ' i i Convergence is slower, but there is less oscilltion

6 6 nonlinermc Emple: ivision without oing ivision opertion At the CPU level, some computers cn perorm ition n multipliction opertions but not ivision opertions ( ) ' strt with 5 ( ) i = 5 = = 4 4 = '( ) = converge! strt with 8 = = 94 = = = 58 4 iverge! In generl, some initil guesses converge to n swer, while others iverge Or, when there re multiple solutions, the converge nswer epens on the initil guess Rigorous Derivtion o Newton's Metho or sclr cse We strt with n initil guess o, n we epn () roun with Tylor's series '' ( ) '! '''! We mke n pproimtion by keeping only the st orer term n ignoring the n orer n higher orer terms, n we hope to mke ( )= ' ' Once we hve, We repet the sme process to in Then rom, we in, etc Thus, we rrive t the ollowing itertion ormul ' or i=,,,, until convergence

7 nonlinermc Rigorous Derivtion o Newton's Metho or vector cse Let us strt with n lgebric equtions (, n ) contining n unknown vribles ( ) These equtions re couple n cnnot be solve iniviully in sequentil mnner Anlogy: A =b, n couple liner lgebric equtions contining n unknown vribles ( ) : nlogy :,, n b,,, n b n n, n,, vector nottion: ( ) A b, n n We epn ech o these n unctions roun <> with Tylor's series We hve n equtions : + < > + n_erivtive_n_higher_orer_terms + < > + n_erivtive_n_higher_orer_terms b n n n n n + < > + n_erivtive_n_higher_orer_terms n We mke n pproimtion by keeping only the st orer terms n ignoring the n orer n higher orer terms, n we hope to choose =(,,, ) to orce (,,, )=, (,,, )=, n (,,, )= simultneously + < n>

8 : n nonlinermc n n n In mtri-vector nottion, the bove n sclr equtions become: n n n n n n n < > < > < > It urther simpliies to Newton's ormul or couple set o nonliner lgebric equtions (linerize represention o ()): ( ) compre to liner eqn A b b Δ Δ b compre to liner eqn A b We wnt to choose <> such tht the bove eqution is stisie < > where ( ) n Newton's itertion ormul < i > n compre to liner eqn A b n n Jcobin mtri

9 9 nonlinermc Quick-Dirty Derivtion o Newton's Metho or vector cse Let us strt with how much chnges by Δ s chnges by Δ Δ Δ < i > < i> Sme s bove with some vector epne < i > Δ Δ Δ n Δ Δ Δ where j j j how ech unction chgnes s j chnges j n n n n n n n < i > < i> < i >

10 Emple: couple pretor-prey ynmics rt( r, ) r b r t( r, ) c r vector nottion rt( r, ) t( r, ) r r b r c r g( ) rbbit (prey) o (pretor) g rt( r, ) g t( r, ) nonlinermc b c g ( ) g g g g ( rt) r ( t) r ( rt) ( t) b b r c r b b c Apply Newton's itertion ormul mnully or one itertion < i > g g r r b b r c r r b r c r with b c provie initil guess r r r 4 I we go or nother itertion, we substitute <> n in <> with the sme Newton's itertion ormul r We cn see tht, we hve lrey converge to root, becuse <> = <>

11 Emple: couple three equtions sin( ) y ln( nonlinermc y z y z 5 Step First, we epress the given three sclr equtions in stnr vector orm "()=" (, sin( ) y ln( (, z ) y z (, y z 5 Or in vector orm or the unction (, sin( ) y ln( y z y z 5 Or in vector orm or the unknown vrible s well ( vect) sin vect vect vect vect ln vect vect vect vect vect 5 where vect vect vect vect Step Fin the Jcobin mtri (ie, mtri o erivtives o ech unction wrt ech vrible) (, y y y z z z cos( ) y ln( ) y z z Step Provie n initil guess (z= is not goo initil guess becuse o ln( vect y z Step 4 Apply Newton's ormul y z vect y y, y, z, y, z z z

12 nonlinermc cos y sin y ln z vect y z y z ln( ) y z z y z y z 5 vect y z cos( ) ln( ) sin( ) ln( ) 5 vect y ln( ) z 4 4 Procee to the net itertion by pplying the sme Newton's ormul vect y y, y, z, y, z z z cos y sin y ln z vect y z y z ln( ) y z z y z y z 5 vect y z 68 4 cos( 68) ln( ) 4 4 sin( 68 ) ( ) ln( 4) vect y z Iterte likewise until cnovergence

13 nonlinermc Implement Newton's itertion ormul in computer Moel prmeters b c Keep own vribles rt( r, ) r b r t( r, ) c r Combine to orm vector eqution ( ) rbbit (prey) o (pretor) rt, t, ( ), ( r, ) b b r c r Strting with n initil guess o <>, iterte or number o times i 5 reset = < i > It seems we hve reche convergence Disply the lst columns s the solution cols( ) = Implement Newton's itertion ormul in computer (or the -eqution problem) Keep own vribles (, sin( ) y ln( (, z ) y z (, (, y z 5 cos( ) y ln( ) y z z Combine to orm vector eqution vect, vect, vect ( vect) vect, vect, vect ( ) vect, vect, vect vect vect, vect, vect Strting with n initil guess o vect <>, iterte or number o times i 5 vect vect i vect < i > vect i vect i vect = It seems we hve reche convergence Disply the lst columns s the solution 5 vect cols( ) =

x dx does exist, what does the answer look like? What does the answer to

x dx does exist, what does the answer look like? What does the answer to Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl

More information

Instantaneous Rate of Change of at a :

Instantaneous Rate of Change of at a : AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim

More information

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics

More information

Basic Derivative Properties

Basic Derivative Properties Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

Chapter 3 Solving Nonlinear Equations

Chapter 3 Solving Nonlinear Equations Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,

More information

ax bx c (2) x a x a x a 1! 2!! gives a useful way of approximating a function near to some specific point x a, giving a power-series expansion in x

ax bx c (2) x a x a x a 1! 2!! gives a useful way of approximating a function near to some specific point x a, giving a power-series expansion in x Elementr mthemticl epressions Qurtic equtions b b b The solutions to the generl qurtic eqution re (1) b c () b b 4c (3) Tlor n Mclurin series (power-series epnsion) The Tlor series n n f f f n 1!! n! f

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

More information

f a L Most reasonable functions are continuous, as seen in the following theorem:

f a L Most reasonable functions are continuous, as seen in the following theorem: Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f

More information

VII. The Integral. 50. Area under a Graph. y = f(x)

VII. The Integral. 50. Area under a Graph. y = f(x) VII. The Integrl In this chpter we efine the integrl of function on some intervl [, b]. The most common interprettion of the integrl is in terms of the re uner the grph of the given function, so tht is

More information

5.3 The Fundamental Theorem of Calculus

5.3 The Fundamental Theorem of Calculus CHAPTER 5. THE DEFINITE INTEGRAL 35 5.3 The Funmentl Theorem of Clculus Emple. Let f(t) t +. () Fin the re of the region below f(t), bove the t-is, n between t n t. (You my wnt to look up the re formul

More information

Section 6.3 The Fundamental Theorem, Part I

Section 6.3 The Fundamental Theorem, Part I Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt

More information

x ) dx dx x sec x over the interval (, ).

x ) dx dx x sec x over the interval (, ). Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

More information

Numerical Analysis: Trapezoidal and Simpson s Rule

Numerical Analysis: Trapezoidal and Simpson s Rule nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!! Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

Matrix & Vector Basic Linear Algebra & Calculus

Matrix & Vector Basic Linear Algebra & Calculus Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

More information

Appendix 3, Rises and runs, slopes and sums: tools from calculus

Appendix 3, Rises and runs, slopes and sums: tools from calculus Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full

More information

and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

and that at t = 0 the object is at position 5. Find the position of the object at t = 2. 7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable. Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd

More information

5.2 Exponent Properties Involving Quotients

5.2 Exponent Properties Involving Quotients 5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

Math 211A Homework. Edward Burkard. = tan (2x + z)

Math 211A Homework. Edward Burkard. = tan (2x + z) Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

More information

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but... Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship

5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship 5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

1 Functions Defined in Terms of Integrals

1 Functions Defined in Terms of Integrals November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider

More information

If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as

If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x

More information

1 Part II: Numerical Integration

1 Part II: Numerical Integration Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble

More information

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

dt. However, we might also be curious about dy

dt. However, we might also be curious about dy Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For

More information

5.5 The Substitution Rule

5.5 The Substitution Rule 5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

Homework Problem Set 1 Solutions

Homework Problem Set 1 Solutions Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:

More information

Chapter 8: Methods of Integration

Chapter 8: Methods of Integration Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION

SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION Physics 8.06 Apr, 2008 SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION c R. L. Jffe 2002 The WKB connection formuls llow one to continue semiclssicl solutions from n

More information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

More information

1 Linear Least Squares

1 Linear Least Squares Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

More information

The Algebra (al-jabr) of Matrices

The Algebra (al-jabr) of Matrices Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

Calculus - Activity 1 Rate of change of a function at a point.

Calculus - Activity 1 Rate of change of a function at a point. Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

More information

ES.182A Topic 30 Notes Jeremy Orloff

ES.182A Topic 30 Notes Jeremy Orloff ES82A opic 3 Notes Jerem Orloff 3 Non-independent vribles: chin rule Recll the chin rule: If w = f, ; nd = r, t, = r, t then = + r t r t r t = + t t t r nfortuntel, sometimes there re more complicted reltions

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Math 231E, Lecture 33. Parametric Calculus

Math 231E, Lecture 33. Parametric Calculus Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

More information

1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics

1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics 0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = y-intercept, (x 0, y 0 ) = some given point ) slope-intercept point-slope There re two importnt

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for

LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.

More information

RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties

RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII

More information

Section 17.2 Line Integrals

Section 17.2 Line Integrals Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

Calculus II: Integrations and Series

Calculus II: Integrations and Series Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5 Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

CHAPTER 4a. ROOTS OF EQUATIONS

CHAPTER 4a. ROOTS OF EQUATIONS CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment

More information

The Fundamental Theorem of Calculus Part 2, The Evaluation Part

The Fundamental Theorem of Calculus Part 2, The Evaluation Part AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt

More information

( x) ( ) takes at the right end of each interval to approximate its value on that

( x) ( ) takes at the right end of each interval to approximate its value on that III. INTEGRATION Economists seem much more intereste in mrginl effects n ifferentition thn in integrtion. Integrtion is importnt for fining the expecte vlue n vrince of rnom vriles, which is use in econometrics

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

M 106 Integral Calculus and Applications

M 106 Integral Calculus and Applications M 6 Integrl Clculus n Applictions Contents The Inefinite Integrls.................................................... Antierivtives n Inefinite Integrls.. Antierivtives.............................................................

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

Introduction to Group Theory

Introduction to Group Theory Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements

More information

Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Valley High School Calculus Summertime Fun Packet Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

7. Indefinite Integrals

7. Indefinite Integrals 7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

B Veitch. Calculus I Study Guide

B Veitch. Calculus I Study Guide Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

More information

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information