John Riley 30 August 2016

Size: px
Start display at page:

Download "John Riley 30 August 2016"

Transcription

1 Joh Riley 3 August 6 Basic mathematics of ecoomic models Fuctios ad derivatives Limit of a fuctio Cotiuity 3 Level ad superlevel sets 3 4 Cost fuctio ad margial cost 4 5 Derivative of a fuctio 5 6 Higher order derivatives 3 7 Partial derivatives 3 8 Cocave fuctios 5 Techical Appedi pages

2 Fuctios ad limits Cosider a firm that uses quatities of m differet iputs to produce a sigle output q A particular iput pla (or iput choice) is the a vector (,, m ) The set of all possible m -dimesioal vectors is writte as m Sice iputs caot be egative the feasible set of iput vectors is the set of positive vectors i m This is writte as m For ay iput vector there is some maimum techologically feasible output q Such a mappig q f () from a vector i m (the domai) to a sigle umber i (the rage) is called a fuctio Ecoomists call this maimum output fuctio a productio fuctio More geerally, suppose that the domai of the fuctio S is a subset of Uless both the domai of the fuctio is clear we write f : S where S vector Typically, the fuctios used i ecoomics are cotiuous Iformally, a small chage i the results i a small chage i the mappig f() While we will ot dwell much o mathematical formalities i this course, cotiuity plays such a importat role that we ow provide a formal defiitio We first eed to defie the limit of a fuctio f( ) for some value Here is the idea Cosider first a mappig of oe variable, f : Pick ay, as small as you like The there eists a such that if lies i the ope iterval (, ), the f( ) lies i the ope iterval ( f ( ), f ( ) ) For fuctios of variables the defiitio is essetially the same We simply replace the iterval (, ) by the ope -ball of radius, that is N (, ) { } Defiitio: Limit of a fuctio f : The fuctio f( ) has a limit L at if, for ay there eists a such that f( ) lies i the iterval ( L, L ) for all i the ope -ball N(, )

3 Cotiuity A fuctio f is cotiuous at f ( ) Epressed i mathematical shorthad if the limit of the fuctio at is equal to the value of the fuctio Defiitio: Cotiuity of f : at f( ) is cotiuous at if f ( ) lim f ( ) Eample: a, f( ) g( ), where g ( ) Note that g ( ) is ot defied at However for all, ( )( ) g( ) Thus g ( ) has a limit of at It follows that the fuctio f( ) is cotiuous at if ad oly if a Defiitio: Cotiuity of f : o S f is cotiuous o S if it is cotiuous for all S

4 3 Level set, superlevel set ad sublevel set The set of all vectors i the domai of the fuctio that yield the same value of the fuctio is called a level set m { f ( ) q} The set of all vectors for which f() is at least q is called a superlevel set m { f ( ) q} The set of all vectors for which f() is at most q is called a sublevel set m { f ( ) q} Eample : Level ad superlevel sets of the productio fuctio f ( ) The set of iput vectors for which f() (the maimum output) is satisfies the equatio Level ad Superlevel sets of a productio fuctio These are also called cotour, upper cotour ad lower cotour sets 3

5 Dividig by yields the mappig This level set is depicted above For all the vectors to the right of this curve the maimum output is higher Thus the shaded regio is a superlevel set Note that i this eample the two aes form the level set { f ( ) } Eample : Level ad superlevel sets of the utility fuctio f () / / The set of cosumptio vectors for which utility is u are the vectors (, ) satisfyig the equatio f () u / / Rearragig this equatio yields the mappig ( u ) The level set with / u is depicted below For all the vectors to the right of this curve the maimum output is higher Thus the shaded regio is a superlevel set Note that i cotrast with eample, f (, ) ad f(,) are both strictly icreasig Thus each level set itersects with both aes Level ad Superlevel sets of a productio fuctio 4

6 4 Cost fuctio of a firm Suppose that the firm is a price taker i all markets Rather tha cosider the full profit-maimiatio problem directly, it ofte proves helpful to break the problem ito two stages Stage : Solve for the least cost iput vector * ( q, r) for every output q The resultig miimied cost * C( q, r) r ( q, r) is called the cost fuctio of the firm Stage : Solve for the output q that maimies TR( q) C( q) Profit-maimiig output Cosider the secod stage of the maimiatio problem Sice the firm is also a price taker i the output market, total reveue is TR( q) pq As ecoomists, we are traied to look o the margi ad compare margial beefit ad margial cost of switchig from oe pla to aother Sice we are ot cosiderig iput price chages, we write the cost fuctio as Cq ( ) The the cost of producig oe more uit is MC( q) C( q ) C( q) We will assume that the margial cost is higher whe more uits are produced Eample: C( q) q Sice ( a b) a ab b it follows that ( ) q q q Hece MC( q) C( q ) C( q) 4q We use the Greek letter to idicate profit The for a price takig firm the profit is TR TC pq C( q) For the price takig firm the margial reveue from sellig oe more uit is the price Therefore the margial profit is M MR MC( q) p MC( q) 5

7 Both MR ad MC are depicted below for the eample whe the output price is For all outputs of 4 or less, the margial profit is strictly positive ad for all higher outputs, the margial profit is strictly egative Thus the profit-maimiig output is * q 5 Margial cost ad margial reveue Cosider ay p satisfyig 8 p That is, p is i the closed iterval [8,] Note that MC(4) 8 ad MC(5) Thus for ay price p [8,], * q 5 maimies the profit of the firm If p 8 the MR MC(4) so the firm makes the same profit if it produces 4 or 5 uits Similarly, if p the MR MC(5) so the firm makes the same profit if it produces 5 or 6 uits The complete mappig from prices to profit-maimiig outputs is depicted below 6

8 Mappig from price to profit-maimiig output Sice the mappig is multi-valued for the prices {,6,,4,}, it caot be reduced to a simple formula So i ecoomic modellig ecoomists make a further simplificatio They assume that there is o atural miimum uit Istead the output of a firm is assumed to be ay positive real umber To compute margial cost we cosider some icremetal output which cost chages is q q q The rate at C C q C q q q q ( ) ( ) We the defie margial cost to be the rate at which cost rises as the icremetal output approaches ero For our eample C C( q ) C( q ) ( q ) ( q ) ( q q )( q q ) q q q q q q q q q I the figure below this is the slope of the chord AB for some Thus the margial cost at output q is q I the limit q approaches q MC( q ) q 7

9 I the figure ote that the firm s margial cost is the slope of the lie just touchig the cost curve This is called the taget lie It is ow easy to solve for the profit maimiig output of the firm The firm caot effect the price so MR p Margial profit is therefore M MR( q) MC( q) p q Thus margial profit is positive it p q ad egative if p q Thus the firm maimies profit by choosig the output q p This is the firm s supply fuctio * Eercise: Idustry supply Suppose that the cost fuctio for firm i,, is C ( q ) a q i i i i (a) Solve for firm i 's supply fuctio (b) Hece show that with two firms the idustry supply fuctio is q * ( ) q a a (c) If there are firms show that the idustry supply fuctio is * q ( ) q a i i 8

10 5 Derivative of a fuctio Of course by takig a limit of small icremets i outputs we are simply followig the lead of Newto ad Leibit ad have reiveted calculus For ay fuctio lie touchig the graph of the fuctio as fuctio Cq ( ) dc ( q ) dq Cq ( ) we write the slope of the Mathematicias call this the derivative of the We use the idea of a limit to write dow a formal defiitio of the derivative of a fuctio For ay q q, defie the slope fuctio C C q C q sq ( ) q q q ( ) ( ) I the figure above, this is the slope of the chord AB Defiitio: Derivative of Cq ( ) at The derivative of Cq ( ) at q q is the limit of the slope fuctio sq ( ) at q 9

11 The derivative of a fuctio is typically writte i oe of the followig ways d dq df f ( q) ( q) f ( q) dq We have already established that if As a secod eample, suppose that C( q) C( q) q q 3 dc the ( q ) q dq The C q ( q ) sq ( ) q q q 3 3 As ca be readily checked, 3 3 q ( q ) ( q q )( q qq ( q ) ) Therefore s( q) q qq ( q ) Takig the limit at q, dc ( q ) 3( q ) dq The followig rule for the derivatives of power fuctios is the most used rule i ecoomic modelig Derivative of a power fuctio If d ( Aq ) Abq dq b b q ad q b b are both defied, the logarithm Ecoomists ofte use two other fuctios, the epoetial fuctio ad its iverse, the atural Review: The epoetial ad logarithmic fuctios

12 Suppose that a bak offers a aual iterest rate of per dollar of savigs If this is computed aually, the at the ed of the year the each dollar grows to If the iterest is computed semiaually the dollar grows to If the compoudig takes place case i which the iterest rate is As rises the ed of year value, after si moths ad ( )( ) ( ) by the ed of the year times the ed of year amout is ( ) ( ) Cosider first the special rises more ad more slowly ad approaches a limitig value of approimately 78 The actual limit is slightly larger ad is writte as e The ed of your value for other iterest rates ca also be computed For ay defie m / where is the umber of times that iterest is computed The m ad so m m ( ) ( ) [( ) ] m m r The epressio i brackets has a limitig value of e so the limitig value of ( ) is e The graph of the epoetial fuctio y e is depicted below The iverse of this fuctio, that is, the mappig from y to is the logarithmic fuctio It is ot too difficult to show that the logarithmic fuctio has the followig very ice properties:

13 a l l l (ii) l al (i) Epoetial fuctio Logarithmic fuctio Derivative of the epoetial fuctio ad atural logarithm a If f ( ) e the df d a ad if g ( ) l ae the dg d You are epected to review the basics of sigle variable calculus prior to takig this course Here are some importat thigs to review Derivative of a sum If h( q) f ( q) g( q) the dh ( q) df ( q) dg ( q) dq dq dq You ca always check your work by goig to

14 Derivative of a product dh df dg If h( q) f ( q) g( q) the ( q) ( q) g( q) f ( q) ( q) dq dq dq The et rule follows directly Derivative of a ratio If df dg ( q) g( q) f ( q) ( q) f( q) dh dq dq hq ( ) the ( q) gq ( ) dq g( q) Chai Rule dh dg d If h( q) g( f ( q)) the ( q) where dq d dq f ( q) Eample : q q df dg h( q) qe Let f ( q) q ad g( q) e The ( q) ad ( q ) e q dq dq Therefore dh df dg ( q) ( q) g( q) f ( q) ( q) e qe ( q) e dq d dq q q q Eample : q df dg hq ( ) Let f ( q) q ad g( q) q The ( q) ( q) q dq dq Therefore df dg ( q) g( q) f ( q) ( q) dh dq dq ( q) q ( q) dq g( q) ( q) ( q) Eample 3: h q Defie h( ) l ad ( ) l( q ) ( q) q 3

15 The dh ( ) dh d q q q dq d dq q 6 Higher order derivatives If a fuctio is differetiable for all S df, the the derivative ( ) d ca therefore use the defiitio of a derivative to defie the secod derivative is a fuctio defied o S We d df ( ( )) d d Ecoomists typically use oe of the followig short-had for the secod derivative: d df d f ( ( )) f ( ) ( ) d d d Higher order derivatives are defied i the same maer 7 Partial derivatives of f : If the fuctio f() compoet of the vector is a mappig of vectors oto the real lie we ca cosider chagig just the ad defiig the partial derivative with respect to this variable j-th We defie the slope fuctio as follows s ( ) j j f (,,,,,, ) f (,,,,,, ) j j j m j j j m j j Defiitio: The j -th partial derivative of f() at j -th partial derivative of f() at s at is the limit of the slope fuctio j( j) 4

16 The vector of all the partial derivatives of a fuctio is called the gradiet vector Gradiet vector f f f ( ) ( ( ),, ( )) m Eample : Cosumer s utility fuctio U ( ) l l Sice d U j l q if follows that, j,, dq q j j Eample : Productio fuctio of a firm f () A Sice d ( q ) q dq if follows that f the partial derivative with respect to A f () Usig a idetical argumet for f it follows that the gradiet vector is (, ) f( ) 5

17 Cocave fuctios For ay pair of -vectors ad, a cove combiatio of these two vectors is a weighted sum where the weights are both strictly positive ad sum to oe Thus we ca write the cove combiatios of two vectors as follows: Defiitio: Cove combiatio of ad ( ), where Defiitio : Cocave fuctio For ay set S the fuctio f : S is cocave o S if for all ad S, ad every cove combiatio f f f ( ) ( ) ( ) ( ) I words, the value of fuctio f( ) eceeds ( ) f ( ) f ( ), that is the correspodig weighted sum of the values fuctio at ad Oe dimesioal case f : We begi by cosiderig the oe dimesioal case i which is a poit o the real lie the iterval The graph of a cocave fuctio is as depicted below As you ca check, [, ] ito three subitervals of equal legth 3 /3 ad /3 partitio Cosider the taget lie at taget lie The slope of the taget lie is df y f d ( ) ( )( ) As is ituitively clear, the graph of the fuctio must lie below this df d ( ), thus the formula for the taget lie is 3 Cofirm that /3 /3 3 6

18 Thus if a fuctio f : is cocave, the for all cove combiatios df f ( ) f ( ) ( )( ) d It is ot difficult to show that the coverse is also true If a fuctio is differetiable we have a secod equivalet defiitio of a cocave fuctio Defiitio : Cocave fuctio f : S For ay set S the differetiable fuctio f : S is cocave o S if for all every cove combiatio, S ad df f ( ) f ( ) ( ) ( ) d 7

19 From the diagram, ote that icreases 4 The coverse is also true If at some depicted below) the fuctio caot be cocave df f ( ) f ( ) ( )( ) if the slope of the fuctio decreases as d ab the slope is icreasig (as i the iterval [, ] Thus we have a third equivalet defiitio for differetiable fuctios Defiitio 3: Cocave fuctio f : S For ay set S the differetiable fuctio f : S is cocave o S if for all S, the derivative of f is decreasig, that is f( ) 4 Techical footote: From the defiitio of a itegral, derivative is decreasig, f ( ) f ( ) f ( ) d f ( )( ) ( ) ( ) ( ) f f f d Therefore if the 8

20 The geeral case: f : S where S While we shall ot prove it here 5, the followig iequality holds for higher dimesios as well f f f ( ) ( ) ( ) ( ) It is ot difficult to show that the coverse is also true Thus for the geeral case we have a secod equivalet defiitio of a cocave fuctio Defiitio: Cocave fuctio For ay set S the differetiable fuctio f : S is cocave o S if for all ad S ad every cove combiatio f f f ( ) ( ) ( ) ( ) Sufficiet coditios for a fuctio to be cocave Give the great simplificatios for characteriig a maimum for fuctios that are cocave, we ow summarie sufficiet coditios for a fuctio to be cocave S A fuctio f : S is cocave if f( ) S The sum of cocave fuctios is cocave S3 A cocave fuctio of a icreasig cocave fuctio is cocave S4: If f( ) is homogeeous of degree (that is f ( ) f ( ) ad the superlevel sets of f are cove the f( ) is cocave We have already discussed the first sufficiet coditio The secod follows directly from the first defiitio of a cocave fuctio 6 The third required the applicatio of Defiitio twice Eample : f ( ) l j j j 5 A proof is sketched i the Techical Appedi The proof is NOT required readig 6 Simply write dow the defiitio for the cocave fuctios ad the add the iequalities 9

21 Each term i the summatio has a egative secod derivative By S each term is cocave The by S the sum is cocave The derivatio of the fourth set of sufficiet coditios is sigificatly more subtle Eample : f ( ) ( ) where j j ad Each term i the summatio has a egative secod derivative By S each term is cocave The by S the sum is cocave Sice the sum s is a icreasig fuctio, the by S3, s is cocave For may fuctios commoly used i ecoomics, the followig result ca be sued to show that superlevel sets are cove Sufficiet coditio for cove superlevel sets f The superlevel sets of are cove if, for some icreasig fuctio g :, the fuctio h( ) g( f ( ) is cocave Eample: f ( ) ( ) a / Defie g( y) / y The g( f ( )) Each term o the right had side has a egative secod derivative sice Thus by S each term is cocave The by S the sum is cocave Thus the superlevel sets of f are cove Fially we ote that f ( ) (( ) ( ) ) ( ) ( ) f ( ) a / a / a /

22 TECHNICAL APPENDIX Propositio: The differetiable fuctio f is cocave if ad oly if for ay, Proof (oly if): f f f ( ) ( ) ( ) ( ) Defie ( ) ( ) ( ), ad y( ) f ( ( )) Note that d j j j d so d d Therefore dy f d f d d ( ) ( ( )) ( ( )) ( ) (*) Also

23 y( ) y() f ( ( )) f ( ()) Sice f( ) is cocave, for all i the ope iterval (,) Therefore f f f f f f ( ( )) ( ) ( ) ( ) ( ) ( ( ) ( )) y( ) y() ( f ( ) f ( )) We have argued that y( ) y() ( f ( ) f ( )) Divide both sides by (,) y( ) y() f Take the limit as dy dt ( ) f ( ) () f ( ) f ( ) Appealig to (*), settig ad otig that dy d Therefore f f () ( ) ( ) ( ) ( ) f ( ) f ( ) () it follows that

24 Proof: (if) For ay y, f f ( y) f ( ) ( ) ( y ) (i) Set ad (ii) Set ad y y The The f f f f f ( ) f ( ) ( ) ( ) ( ) ( ) ( ) ( ) Eercise: Complete the proof Hit: Multiply the first iequality by ( ) iequalities ad the secod by The add the resultig 3

A.1 Algebra Review: Polynomials/Rationals. Definitions:

A.1 Algebra Review: Polynomials/Rationals. Definitions: MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole

More information

Mathematical Foundations -1- Sets and Sequences. Sets and Sequences

Mathematical Foundations -1- Sets and Sequences. Sets and Sequences Mathematical Foudatios -1- Sets ad Sequeces Sets ad Sequeces Methods of proof 2 Sets ad vectors 13 Plaes ad hyperplaes 18 Liearly idepedet vectors, vector spaces 2 Covex combiatios of vectors 21 eighborhoods,

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

Castiel, Supernatural, Season 6, Episode 18

Castiel, Supernatural, Season 6, Episode 18 13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio

More information

INTRODUCTORY MATHEMATICAL ANALYSIS

INTRODUCTORY MATHEMATICAL ANALYSIS INTRODUCTORY MATHEMATICAL ANALYSIS For Busiess, Ecoomics, ad the Life ad Social Scieces Chapter 4 Itegratio 0 Pearso Educatio, Ic. Chapter 4: Itegratio Chapter Objectives To defie the differetial. To defie

More information

Maximum and Minimum Values

Maximum and Minimum Values Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f

More information

PRACTICE FINAL/STUDY GUIDE SOLUTIONS

PRACTICE FINAL/STUDY GUIDE SOLUTIONS Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)

More information

MATH 10550, EXAM 3 SOLUTIONS

MATH 10550, EXAM 3 SOLUTIONS MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

More information

ECONOMICS 100ABC MATHEMATICAL HANDOUT

ECONOMICS 100ABC MATHEMATICAL HANDOUT ECONOMICS 00ABC MATHEMATICAL HANDOUT 03-04 Professor Mark Machia A. CALCULUS REVIEW Derivatives, Partial Derivatives ad the Chai Rule* You should already kow what a derivative is. We ll use the epressios

More information

The Firm and the Market

The Firm and the Market Chapter 3 The Firm ad the Market Exercise 3.1 (The pheomeo of atural moopoly ) Cosider a idustry i which all the potetial member rms have the same cost fuctio C. Suppose it is true that for some level

More information

ECONOMICS 100A MATHEMATICAL HANDOUT

ECONOMICS 100A MATHEMATICAL HANDOUT ECONOMICS A MATHEMATICAL HANDOUT Fall 3 abd β abc γ abc µ abc Mark Machia A. CALCULUS REVIEW Derivatives, Partial Derivatives ad the Chai Rule You should already kow what a derivative is. We ll use the

More information

CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.

CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve

More information

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE. NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

Honors Calculus Homework 13 Solutions, due 12/8/5

Honors Calculus Homework 13 Solutions, due 12/8/5 Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y

More information

1. By using truth tables prove that, for all statements P and Q, the statement

1. By using truth tables prove that, for all statements P and Q, the statement Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Diploma Programme. Mathematics HL guide. First examinations 2014

Diploma Programme. Mathematics HL guide. First examinations 2014 Diploma Programme First eamiatios 014 33 Topic 6 Core: Calculus The aim of this topic is to itroduce studets to the basic cocepts ad techiques of differetial ad itegral calculus ad their applicatio. 6.1

More information

MATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.)

MATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.) MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral

More information

MAS111 Convergence and Continuity

MAS111 Convergence and Continuity MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

Mathematics Extension 1

Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart. MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a

More information

Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Example 2. Find the upper bound for the remainder for the approximation from Example 1. Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

More information

MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of

MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial

More information

Name: Math 10550, Final Exam: December 15, 2007

Name: Math 10550, Final Exam: December 15, 2007 Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder

More information

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x

More information

( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.

( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let. Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B-: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which

More information

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This

More information

Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 105: Review for Final Exam, Part II - SOLUTIONS Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum

More information

Objective Mathematics

Objective Mathematics 6. If si () + cos () =, the is equal to :. If <

More information

On Random Line Segments in the Unit Square

On Random Line Segments in the Unit Square O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4 . If f = e ta -, the f = e e p e e p e p+ 4 f = e ta -, so f = e ta - + e, so + f = e p + e = e p + e or f = e p + 4. The slope of the lie taget to the curve - + = at the poit, - is - 5 Differetiate -

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n. COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where

More information

Frequency Domain Filtering

Frequency Domain Filtering Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform

More information

Chapter 6 Infinite Series

Chapter 6 Infinite Series Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

More information

Continuous Functions

Continuous Functions Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

Different kinds of Mathematical Induction

Different kinds of Mathematical Induction Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute

1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006 MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the fuctio give by g ftdt. (a) Fid g (, )

More information

HOMEWORK #10 SOLUTIONS

HOMEWORK #10 SOLUTIONS Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous

More information

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i) Math PracTest Be sure to review Lab (ad all labs) There are lots of good questios o it a) State the Mea Value Theorem ad draw a graph that illustrates b) Name a importat theorem where the Mea Value Theorem

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

Section 1 of Unit 03 (Pure Mathematics 3) Algebra

Section 1 of Unit 03 (Pure Mathematics 3) Algebra Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course

More information

Fundamental Concepts: Surfaces and Curves

Fundamental Concepts: Surfaces and Curves UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

Sequences I. Chapter Introduction

Sequences I. Chapter Introduction Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

More information

8. Applications To Linear Differential Equations

8. Applications To Linear Differential Equations 8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.

More information

MTH Assignment 1 : Real Numbers, Sequences

MTH Assignment 1 : Real Numbers, Sequences MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a

More information

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet

More information

CHAPTER 5 SOME MINIMAX AND SADDLE POINT THEOREMS

CHAPTER 5 SOME MINIMAX AND SADDLE POINT THEOREMS CHAPTR 5 SOM MINIMA AND SADDL POINT THORMS 5. INTRODUCTION Fied poit theorems provide importat tools i game theory which are used to prove the equilibrium ad eistece theorems. For istace, the fied poit

More information

10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.

10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term. 0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece

More information

ECONOMICS 200A MATHEMATICAL HANDOUT

ECONOMICS 200A MATHEMATICAL HANDOUT ECONOMICS 00A MATHEMATICAL HANDOUT Fall 07 A. CALCULUS REVIEW Mark Machia Derivatives, Partial Derivatives ad the Chai Rule You should already kow what a derivative is. We ll use the epressios ƒ() or dƒ()/

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j. Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

Pre-calculus Guided Notes: Chapter 11 Exponential and Logarithmic Functions

Pre-calculus Guided Notes: Chapter 11 Exponential and Logarithmic Functions Name: Pre-calculus Guided Notes: Chapter 11 Epoetial ad Logarithmic Fuctios Sectio 2 Epoetial Fuctios Paret Fuctio: y = > 1 0 < < 1 Domai Rage y-itercept ehavior Horizotal asymptote Vertical asymptote

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5 Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

Section 7 Fundamentals of Sequences and Series

Section 7 Fundamentals of Sequences and Series ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),

More information

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4

More information

Student s Printed Name:

Student s Printed Name: Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide

More information

The Growth of Functions. Theoretical Supplement

The Growth of Functions. Theoretical Supplement The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that

More information

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

More information

Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points

Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso

More information

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent. 06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)

More information

Subject: Differential Equations & Mathematical Modeling-III

Subject: Differential Equations & Mathematical Modeling-III Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso

More information

INEQUALITIES BJORN POONEN

INEQUALITIES BJORN POONEN INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad

More information

s = and t = with C ij = A i B j F. (i) Note that cs = M and so ca i µ(a i ) I E (cs) = = c a i µ(a i ) = ci E (s). (ii) Note that s + t = M and so

s = and t = with C ij = A i B j F. (i) Note that cs = M and so ca i µ(a i ) I E (cs) = = c a i µ(a i ) = ci E (s). (ii) Note that s + t = M and so 3 From the otes we see that the parts of Theorem 4. that cocer us are: Let s ad t be two simple o-egative F-measurable fuctios o X, F, µ ad E, F F. The i I E cs ci E s for all c R, ii I E s + t I E s +

More information

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer. 6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

Beurling Integers: Part 2

Beurling Integers: Part 2 Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers

More information

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows

More information

1 Approximating Integrals using Taylor Polynomials

1 Approximating Integrals using Taylor Polynomials Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................

More information