Matrices and Systems of Equations

Size: px
Start display at page:

Download "Matrices and Systems of Equations"

Transcription

1 Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m n n n mn We will use the double subscript nottion for ech element of the mtri. For emple, the element of mtri A in the ith row nd the jth column is denoted b ij. The dimensions of mtri re given s rows columns. This is red m b n. A mtri is squre if it is n n; we s it hs order n. The min digonl of squre mtri is ll the elements ij for which i = j.

2 Emple: Determine the dimensions of ech mtri. CHAT Pre-Clculus Section 8. ) b) nswer: nswer: c) d) 8 8 nswer: nswer: A mtri tht hs onl one row is clled row mtri. A mtri tht hs onl one column is clled column mtri.

3 CHAT Pre-Clculus Section 8. Look t the sstem If we write the sstem of equtions without the vribles, ddition signs, nd equl signs, nd put it into mtri, we get wht is clled n ugmented mtri. or or (Different tetbooks will write them in vrious ws.) The coefficient mtri would be Note: An time term is missing from sstem of equtions, ero is put in its plce in the mtri.

4 CHAT Pre-Clculus Section 8. Elementr Row Opertions We need to trnslte the elementr row opertions from our lst unit into the lnguge of mtrices. Note tht these opertions will trnsform the ugmented mtri of sstem of equtions into the ugmented mtri of n equivlent sstem of equtions. If one mtri hs been obtined from nother mtri b using elementr row opertions, then the two mtrices re sid to row-equivlent. Here re the opertions. Elementr Row Opertions. Interchnge two rows.. Replce n row b nonero multiple of itself.. Replce n row b the sum of itself nd multiple of n other row in the mtri. We use the mtri version of the Gussin Elimintion method to solve sstems of liner equtions. Emple: Solve the sstem using the mtri version of Gussin Elimintion.

5 CHAT Pre-Clculus Section 8. Solution: Strt with the ugmented mtri. Multipl the st row b - nd dd to the nd row, replcing the nd row with our sum eqution. ) ( Multipl the st row b - nd dd to the rd row, replcing the rd row with our sum eqution. ) (

6 CHAT Pre-Clculus Section 8. Multipl the nd row b nd dd to the rd row, replcing the rd row with our sum eqution. () Replce the bottom row with - times itself. ) ( This finl mtri is sid to be in row-echelon form.

7 Row-Echelon Form nd Reduced Row-Echelon Form A mtri is in row-echelon form if it hs the following properties. CHAT Pre-Clculus Section 8.. All rows consisting entirel of eros occur t the bottom of the mtri.. The first nonero element of n row is, clled leding.. For two successive nonero rows, the leding in the upper row is frther to the left thn the leding in the lower row. A mtri is in reduced row-echelon form if the column with leding hs eros everwhere else in tht column. Emple: Continue working with the bove ugmented mtri until it is in reduced row-echelon form. Multipl the rd eqution b nd dd to the nd eqution, replcing the nd eqution with the sum. ()

8 CHAT Pre-Clculus Section 8. 8 Multipl the rd eqution b - nd dd to the st eqution, replcing the st eqution with the sum. ) ( Multipl the nd eqution b - nd dd to the st eqution, replcing the st eqution with the sum. ) ( The finl mtri is in reduced row-echelon form.

9 CHAT Pre-Clculus Section 8. Look t the row-echelon form nd the reduced row-echelon form. Use bck substitution to solve the sstems tht ech represents. First the row-echelon form: () () The solution is (-,, ). Now the reduced row-echelon form. The solution is (-,, ). **Both mtrices gve us the sme solution.

10 CHAT Pre-Clculus Section 8. Solving sstem of equtions b Gussin elimintion is tking the ugmented mtri of the sstem of equtions nd trnsforming it into row-echelon form, nd then using bcksubstitution to find the vlues of the vribles. Solving sstem of equtions b Guss-Jordn elimintion is tking the ugmented mtri of the sstem of equtions nd trnsforming it into reduced row-echelon form. The lst column gives the vlues of the vribles. Tip: Work column b column. Get the first column the w ou wnt it, then move to the second column, nd so on. Emple: Which of the following mtrices re in reduced row-echelon form? If it is not in reduced row-echelon form, stte whether it is in row-echelon form. ) b) 8 reduced row-echelon form reduced row-echelon form

11 CHAT Pre-Clculus Section 8. c) d) just row-echelon form reduced row-echelon form Emple: Solve the sstem b Gussin elimintion. 8 The solution is (, -, ).

12 CHAT Pre-Clculus Section 8. Emple: Solve b Guss-Jordn elimintion. The solution is (, -). Emple: Solve the sstem b Guss-Jordon elimintion. This sstem hs no solution.

13 CHAT Pre-Clculus Section 8. Emple: Solve the sstem b Guss-Jordon elimintion. 8 Note: With onl equtions, it is impossible to get rid of both nd in the top eqution. Let =. Then = - nd = 8, which mens = + 8 when we substitute for. The solution to the sstem is (+8, -, ). Emple: Solve the sstem. The solution is (-,, ).

14 CHAT Pre-Clculus Section 8. Mtri Opertions on the Grphing Clcultor Entering Mtri. Press [ nd ] [MATRX] [EDIT]. Choose [A] to enter our mtri s mtri A.. On the top of the edit screen ou must enter the dimensions (i.e. order) for our mtri. Move the cursor to ech spot nd enter the number of rows nd then columns.. Move the cursor down to the row, column entr spot. You will see the row, column reference t the bottom of the screen. Tpe in our entr. Press [ENTER]. Your cursor goes to the net element s spot.. Continue entering ll of the elements of the mtri. When finished, press [ nd ] [Quit].. To see the mtri on our min screen, press [ nd ][MATRX] [A] [ENTER]. Emple: Enter the ugmented mtri into our grphing clcultor s mtri A for the following sstem.

15 CHAT Pre-Clculus Section 8. Chnging Mtri to Row-Echelon Form. Press [ nd ] [MATRX] [MATH] [ref( ]. ( ref stnds for row-echelon form. ). On our min screen ou will see ref(. You need to tell the clcultor the nme of the mtri tht ou wnt it to put in row-echelon form. To do this, press [ nd ] [MATRX] [A] to enter mtri A. Press [ENTER]. Note: Row-echelon form mtrices re not unique. It depends on the sequence of row opertions chosen. Emple: Using the bove sstem nd mtri entered into our clcultor s mtri A, find the row-echelon form on our grphing clcultor Notice how it writes the frctions s decimls. To solve this sstem, we would hve to bck-substitute.

16 CHAT Pre-Clculus Section 8. Chnging Mtri to Reduced Row-Echelon Form. Press [ nd ] [MATRX] [MATH] [rref( ]. ( rref stnds for reduced row-echelon form. ). On our min screen ou will see rref(. You need to tell the clcultor the nme of the mtri tht ou wnt it to put in row-echelon form. To do this, press [ nd ] [MATRX] [A] to enter mtri A. Press [ENTER]. Note: Reduced Row-echelon form mtrices re unique. The will be the sme regrdless of the choices mde in row opertions. Emple: Using the bove sstem nd mtri entered into our clcultor s mtri A, find the reduced row-echelon form on our grphing clcultor. This tells us tht the solution to the sstem is (, -).

Solution of Linear Algebraic Equations using the Gauss-Jordan Method

Solution of Linear Algebraic Equations using the Gauss-Jordan Method Solution of Liner Algebric Equtions using the Guss-Jordn Method Populr pproch for solving liner equtions The Guss Jordn method depends on two properties of liner equtions: Scling one or more of ny of the

More information

Hyperbolas. Definition of Hyperbola

Hyperbolas. Definition of Hyperbola CHAT Pre-Clculus Hyperols The third type of conic is clled hyperol. For n ellipse, the sum of the distnces from the foci nd point on the ellipse is fixed numer. For hyperol, the difference of the distnces

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

1.1 Lines AP Calculus

1.1 Lines AP Calculus . Lines AP Clculus. LINES Notecrds from Section.: Rules for Rounding Round or Truncte ll finl nswers to 3 deciml plces. Do NOT round before ou rech our finl nswer. Much of Clculus focuses on the concept

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

Study Guide for Exam 3

Study Guide for Exam 3 Mth 05 Elementry Algebr Fll 00 Study Guide for Em Em is scheduled for Thursdy, November 8 th nd ill cover chpters 5 nd. You my use "5" note crd (both sides) nd scientific clcultor. You re epected to no

More information

Section 5.3 : Finding Area Between Curves

Section 5.3 : Finding Area Between Curves MATH 9 Section 5. : Finding Are Between Curves Importnt: In this section we will lern just how to set up the integrls to find re etween curves. The finl nswer for ech emple in this hndout is given for

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus MATH 6 The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus (FTC) gives method of finding the signed re etween the grph of f nd the x-xis on the intervl [, ]. The theorem is: FTC: If f is

More information

Graphing Conic Sections

Graphing Conic Sections Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where

More information

MSTH 236 ELAC SUMMER 2017 CP 1 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

MSTH 236 ELAC SUMMER 2017 CP 1 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. MSTH 236 ELAC SUMMER 2017 CP 1 SHORT ANSWER. Write the word or phrse tht best completes ech sttement or nswers the question. Find the product. 1) (8y + 11)(4y 2-2y - 9) 1) Simplify the expression by combining

More information

Are You Ready for Algebra 3/Trigonometry? Summer Packet **Required for all Algebra 3/Trig CP and Honors students**

Are You Ready for Algebra 3/Trigonometry? Summer Packet **Required for all Algebra 3/Trig CP and Honors students** Are You Red for Algebr /Trigonometr? Summer Pcket **Required for ll Algebr /Trig CP nd Honors students** Pge of The Algebr /Trigonometr course prepres students for Clculus nd college science courses. In

More information

Definition of Regular Expression

Definition of Regular Expression Definition of Regulr Expression After the definition of the string nd lnguges, we re redy to descrie regulr expressions, the nottion we shll use to define the clss of lnguges known s regulr sets. Recll

More information

)

) Chpter Five /SOLUTIONS Since the speed ws between nd mph during this five minute period, the fuel efficienc during this period is between 5 mpg nd 8 mpg. So the fuel used during this period is between

More information

1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)

1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric

More information

8.2 Areas in the Plane

8.2 Areas in the Plane 39 Chpter 8 Applictions of Definite Integrls 8. Ares in the Plne Wht ou will lern out... Are Between Curves Are Enclosed Intersecting Curves Boundries with Chnging Functions Integrting with Respect to

More information

Subtracting Fractions

Subtracting Fractions Lerning Enhncement Tem Model Answers: Adding nd Subtrcting Frctions Adding nd Subtrcting Frctions study guide. When the frctions both hve the sme denomintor (bottom) you cn do them using just simple dding

More information

Section 3.1: Sequences and Series

Section 3.1: Sequences and Series Section.: Sequences d Series Sequences Let s strt out with the definition of sequence: sequence: ordered list of numbers, often with definite pttern Recll tht in set, order doesn t mtter so this is one

More information

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών ΕΠΛ323 - Θωρία και Πρακτική Μταγλωττιστών Lecture 3 Lexicl Anlysis Elis Athnsopoulos elisthn@cs.ucy.c.cy Recognition of Tokens if expressions nd reltionl opertors if è if then è then else è else relop

More information

LU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw

LU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw LU Decomposition Mechnicl Engineering Mjors Authors: Autr Kw Trnsforming Numericl Methods Eduction for STEM Undergrdutes // LU Decomposition LU Decomposition LU Decomposition is nother method to solve

More information

Introduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1)

Introduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1) POLAR EQUATIONS AND GRAPHS GEOMETRY INU4/54 (MATHS ) Dr Adrin Jnnett MIMA CMth FRAS Polr equtions nd grphs / 6 Adrin Jnnett Objectives The purpose of this presenttion is to cover the following topics:

More information

Unit 5 Vocabulary. A function is a special relationship where each input has a single output.

Unit 5 Vocabulary. A function is a special relationship where each input has a single output. MODULE 3 Terms Definition Picture/Exmple/Nottion 1 Function Nottion Function nottion is n efficient nd effective wy to write functions of ll types. This nottion llows you to identify the input vlue with

More information

MTH 146 Conics Supplement

MTH 146 Conics Supplement 105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points

More information

Constrained Optimization. February 29

Constrained Optimization. February 29 Constrined Optimiztion Februry 9 Generl Problem min f( ) ( NLP) s.. t g ( ) i E i g ( ) i I i Modeling nd Constrints Adding constrints let s us model fr more richer set of problems. For our purpose we

More information

Summer Review Packet For Algebra 2 CP/Honors

Summer Review Packet For Algebra 2 CP/Honors Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review

More information

Topics in Analytic Geometry

Topics in Analytic Geometry Nme Chpter 10 Topics in Anltic Geometr Section 10.1 Lines Objective: In this lesson ou lerned how to find the inclintion of line, the ngle between two lines, nd the distnce between point nd line. Importnt

More information

Integration. September 28, 2017

Integration. September 28, 2017 Integrtion September 8, 7 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my

More information

Integration. October 25, 2016

Integration. October 25, 2016 Integrtion October 5, 6 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my hve

More information

5/9/17. Lesson 51 - FTC PART 2. Review FTC, PART 1. statement as the Integral Evaluation Theorem as it tells us HOW to evaluate the definite integral

5/9/17. Lesson 51 - FTC PART 2. Review FTC, PART 1. statement as the Integral Evaluation Theorem as it tells us HOW to evaluate the definite integral Lesson - FTC PART 2 Review! We hve seen definition/formul for definite integrl s n b A() = lim f ( i )Δ = f ()d = F() = F(b) F() n i=! where F () = f() (or F() is the ntiderivtive of f() b! And hve seen

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting the

More information

The notation y = f(x) gives a way to denote specific values of a function. The value of f at a can be written as f( a ), read f of a.

The notation y = f(x) gives a way to denote specific values of a function. The value of f at a can be written as f( a ), read f of a. Chpter Prerequisites for Clculus. Functions nd Grphs Wht ou will lern out... Functions Domins nd Rnges Viewing nd Interpreting Grphs Even Functions nd Odd Functions Smmetr Functions Defined in Pieces Asolute

More information

Dynamic Programming. Andreas Klappenecker. [partially based on slides by Prof. Welch] Monday, September 24, 2012

Dynamic Programming. Andreas Klappenecker. [partially based on slides by Prof. Welch] Monday, September 24, 2012 Dynmic Progrmming Andres Klppenecker [prtilly bsed on slides by Prof. Welch] 1 Dynmic Progrmming Optiml substructure An optiml solution to the problem contins within it optiml solutions to subproblems.

More information

Essential Question What are some of the characteristics of the graph of a rational function?

Essential Question What are some of the characteristics of the graph of a rational function? 8. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..G A..H A..K Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions

More information

12-B FRACTIONS AND DECIMALS

12-B FRACTIONS AND DECIMALS -B Frctions nd Decimls. () If ll four integers were negtive, their product would be positive, nd so could not equl one of them. If ll four integers were positive, their product would be much greter thn

More information

Geometric transformations

Geometric transformations Geometric trnsformtions Computer Grphics Some slides re bsed on Shy Shlom slides from TAU mn n n m m T A,,,,,, 2 1 2 22 12 1 21 11 Rows become columns nd columns become rows nm n n m m A,,,,,, 1 1 2 22

More information

1 The Definite Integral

1 The Definite Integral The Definite Integrl Definition. Let f be function defined on the intervl [, b] where

More information

Section 9.2 Hyperbolas

Section 9.2 Hyperbolas Section 9. Hperols 597 Section 9. Hperols In the lst section, we lerned tht plnets hve pproimtel ellipticl orits round the sun. When n oject like comet is moving quickl, it is le to escpe the grvittionl

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY Joe McBride/Stone/Gett Imges Air resistnce prevents the velocit of skdiver from incresing indefinitel. The velocit pproches it, clled the terminl velocit. The development of clculus

More information

Fundamentals of Engineering Analysis ENGR Matrix Multiplication, Types

Fundamentals of Engineering Analysis ENGR Matrix Multiplication, Types Fundmentls of Engineering Anlysis ENGR - Mtri Multiplition, Types Spring Slide Mtri Multiplition Define Conformle To multiply A * B, the mtries must e onformle. Given mtries: A m n nd B n p The numer of

More information

Section 10.4 Hyperbolas

Section 10.4 Hyperbolas 66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol

More information

Stained Glass Design. Teaching Goals:

Stained Glass Design. Teaching Goals: Stined Glss Design Time required 45-90 minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Rational Numbers---Adding Fractions With Like Denominators.

Rational Numbers---Adding Fractions With Like Denominators. Rtionl Numbers---Adding Frctions With Like Denomintors. A. In Words: To dd frctions with like denomintors, dd the numertors nd write the sum over the sme denomintor. B. In Symbols: For frctions c nd b

More information

9.1 apply the distance and midpoint formulas

9.1 apply the distance and midpoint formulas 9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the

More information

CS201 Discussion 10 DRAWTREE + TRIES

CS201 Discussion 10 DRAWTREE + TRIES CS201 Discussion 10 DRAWTREE + TRIES DrwTree First instinct: recursion As very generic structure, we could tckle this problem s follows: drw(): Find the root drw(root) drw(root): Write the line for the

More information

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html

More information

What do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers

What do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers Wht do ll those bits men now? bits (...) Number Systems nd Arithmetic or Computers go to elementry school instruction R-formt I-formt... integer dt number text chrs... floting point signed unsigned single

More information

Rigid Body Transformations

Rigid Body Transformations igid od Kinemtics igid od Trnsformtions Vij Kumr igid od Kinemtics emrk out Nottion Vectors,,, u, v, p, q, Potentil for Confusion! Mtrices,, C, g, h, igid od Kinemtics The vector nd its skew smmetric mtri

More information

Lecture 5: Spatial Analysis Algorithms

Lecture 5: Spatial Analysis Algorithms Lecture 5: Sptil Algorithms GEOG 49: Advnced GIS Sptil Anlsis Algorithms Bsis of much of GIS nlsis tod Mnipultion of mp coordintes Bsed on Eucliden coordinte geometr http://stronom.swin.edu.u/~pbourke/geometr/

More information

Lecture 7: Integration Techniques

Lecture 7: Integration Techniques Lecture 7: Integrtion Techniques Antiderivtives nd Indefinite Integrls. In differentil clculus, we were interested in the derivtive of given rel-vlued function, whether it ws lgeric, eponentil or logrithmic.

More information

For example, the system. 22 may be represented by the augmented matrix

For example, the system. 22 may be represented by the augmented matrix Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be

More information

Introduction to Integration

Introduction to Integration Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition.

More information

The Reciprocal Function Family. Objectives To graph reciprocal functions To graph translations of reciprocal functions

The Reciprocal Function Family. Objectives To graph reciprocal functions To graph translations of reciprocal functions - The Reciprocl Function Fmil Objectives To grph reciprocl functions To grph trnsltions of reciprocl functions Content Stndrds F.BF.3 Identif the effect on the grph of replcing f () b f() k, kf(), f(k),

More information

such that the S i cover S, or equivalently S

such that the S i cover S, or equivalently S MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i

More information

x )Scales are the reciprocal of each other. e

x )Scales are the reciprocal of each other. e 9. Reciprocls A Complete Slide Rule Mnul - eville W Young Chpter 9 Further Applictions of the LL scles The LL (e x ) scles nd the corresponding LL 0 (e -x or Exmple : 0.244 4.. Set the hir line over 4.

More information

MATH 2530: WORKSHEET 7. x 2 y dz dy dx =

MATH 2530: WORKSHEET 7. x 2 y dz dy dx = MATH 253: WORKSHT 7 () Wrm-up: () Review: polr coordintes, integrls involving polr coordintes, triple Riemnn sums, triple integrls, the pplictions of triple integrls (especilly to volume), nd cylindricl

More information

Tree Structured Symmetrical Systems of Linear Equations and their Graphical Solution

Tree Structured Symmetrical Systems of Linear Equations and their Graphical Solution Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA Tree Structured Symmetricl Systems of Liner Equtions nd their Grphicl Solution Jime

More information

Math/CS 467/667 Programming Assignment 01. Adaptive Gauss Quadrature. q(x)p 4 (x) = 0

Math/CS 467/667 Programming Assignment 01. Adaptive Gauss Quadrature. q(x)p 4 (x) = 0 Adptive Guss Qudrture 1. Find n orthogonl polynomil p 4 of degree 4 such tht 1 1 q(x)p 4 (x) = 0 for every polynomil q(x) of degree 3 or less. You my use Mple nd the Grm Schmidt process s done in clss.

More information

SIMPLIFYING ALGEBRA PASSPORT.

SIMPLIFYING ALGEBRA PASSPORT. SIMPLIFYING ALGEBRA PASSPORT www.mthletics.com.u This booklet is ll bout turning complex problems into something simple. You will be ble to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give

More information

called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.

called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. Review of conic sections Conic sections re grphs of the form REVIEW OF CONIC SECTIONS prols ellipses hperols P(, ) F(, p) O p =_p REVIEW OF CONIC SECTIONS In this section we give geometric definitions

More information

INTRODUCTION TO SIMPLICIAL COMPLEXES

INTRODUCTION TO SIMPLICIAL COMPLEXES INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min

More information

Math 142, Exam 1 Information.

Math 142, Exam 1 Information. Mth 14, Exm 1 Informtion. 9/14/10, LC 41, 9:30-10:45. Exm 1 will be bsed on: Sections 7.1-7.5. The corresponding ssigned homework problems (see http://www.mth.sc.edu/ boyln/sccourses/14f10/14.html) At

More information

1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers?

1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers? 1.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Prepring for 2A.6.K, 2A.7.I Intervl Nottion nd Set Nottion Essentil Question When is it convenient to use set-uilder nottion to represent set of numers? A collection

More information

EXPONENTIAL & POWER GRAPHS

EXPONENTIAL & POWER GRAPHS Eponentil & Power Grphs EXPONENTIAL & POWER GRAPHS www.mthletics.com.u Eponentil EXPONENTIAL & Power & Grphs POWER GRAPHS These re grphs which result from equtions tht re not liner or qudrtic. The eponentil

More information

ECE 468/573 Midterm 1 September 28, 2012

ECE 468/573 Midterm 1 September 28, 2012 ECE 468/573 Midterm 1 September 28, 2012 Nme:! Purdue emil:! Plese sign the following: I ffirm tht the nswers given on this test re mine nd mine lone. I did not receive help from ny person or mteril (other

More information

Chapter Spline Method of Interpolation More Examples Electrical Engineering

Chapter Spline Method of Interpolation More Examples Electrical Engineering Chpter. Spline Method of Interpoltion More Exmples Electricl Engineering Exmple Thermistors re used to mesure the temperture of bodies. Thermistors re bsed on mterils chnge in resistnce with temperture.

More information

Lesson 11 MA Nick Egbert

Lesson 11 MA Nick Egbert Lesson MA 62 Nick Eert Overview In this lesson we return to stndrd Clculus II mteril with res etween curves. Recll rom irst semester clculus tht the deinite interl hd eometric menin, nmel the re under

More information

Functor (1A) Young Won Lim 10/5/17

Functor (1A) Young Won Lim 10/5/17 Copyright (c) 2016-2017 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version published

More information

this grammar generates the following language: Because this symbol will also be used in a later step, it receives the

this grammar generates the following language: Because this symbol will also be used in a later step, it receives the LR() nlysis Drwcks of LR(). Look-hed symols s eplined efore, concerning LR(), it is possile to consult the net set to determine, in the reduction sttes, for which symols it would e possile to perform reductions.

More information

The Basic Properties of the Integral

The Basic Properties of the Integral The Bsic Properties of the Integrl When we compute the derivtive of complicted function, like + sin, we usull use differentition rules, like d [f()+g()] d f()+ d g(), to reduce the computtion d d d to

More information

EECS 281: Homework #4 Due: Thursday, October 7, 2004

EECS 281: Homework #4 Due: Thursday, October 7, 2004 EECS 28: Homework #4 Due: Thursdy, October 7, 24 Nme: Emil:. Convert the 24-bit number x44243 to mime bse64: QUJD First, set is to brek 8-bit blocks into 6-bit blocks, nd then convert: x44243 b b 6 2 9

More information

Angle properties of lines and polygons

Angle properties of lines and polygons chievement Stndrd 91031 pply geometric resoning in solving problems Copy correctly Up to 3% of workbook Copying or scnning from ES workbooks is subject to the NZ Copyright ct which limits copying to 3%

More information

CS 241 Week 4 Tutorial Solutions

CS 241 Week 4 Tutorial Solutions CS 4 Week 4 Tutoril Solutions Writing n Assemler, Prt & Regulr Lnguges Prt Winter 8 Assemling instrutions utomtilly. slt $d, $s, $t. Solution: $d, $s, nd $t ll fit in -it signed integers sine they re 5-it

More information

Applications of the Definite Integral ( Areas and Volumes)

Applications of the Definite Integral ( Areas and Volumes) Mth1242 Project II Nme: Applictions of the Definite Integrl ( Ares nd Volumes) In this project, we explore some pplictions of the definite integrl. We use integrls to find the re etween the grphs of two

More information

George Boole. IT 3123 Hardware and Software Concepts. Switching Algebra. Boolean Functions. Boolean Functions. Truth Tables

George Boole. IT 3123 Hardware and Software Concepts. Switching Algebra. Boolean Functions. Boolean Functions. Truth Tables George Boole IT 3123 Hrdwre nd Softwre Concepts My 28 Digitl Logic The Little Mn Computer 1815 1864 British mthemticin nd philosopher Mny contriutions to mthemtics. Boolen lger: n lger over finite sets

More information

Compilers Spring 2013 PRACTICE Midterm Exam

Compilers Spring 2013 PRACTICE Midterm Exam Compilers Spring 2013 PRACTICE Midterm Exm This is full length prctice midterm exm. If you wnt to tke it t exm pce, give yourself 7 minutes to tke the entire test. Just like the rel exm, ech question hs

More information

Pointers and Arrays. More Pointer Examples. Pointers CS 217

Pointers and Arrays. More Pointer Examples. Pointers CS 217 Pointers nd Arrs CS 21 1 2 Pointers More Pointer Emples Wht is pointer A vrile whose vlue is the ddress of nother vrile p is pointer to vrile v Opertions &: ddress of (reference) *: indirection (dereference)

More information

Questions About Numbers. Number Systems and Arithmetic. Introduction to Binary Numbers. Negative Numbers?

Questions About Numbers. Number Systems and Arithmetic. Introduction to Binary Numbers. Negative Numbers? Questions About Numbers Number Systems nd Arithmetic or Computers go to elementry school How do you represent negtive numbers? frctions? relly lrge numbers? relly smll numbers? How do you do rithmetic?

More information

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών. Lecture 3b Lexical Analysis Elias Athanasopoulos

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών. Lecture 3b Lexical Analysis Elias Athanasopoulos ΕΠΛ323 - Θωρία και Πρακτική Μταγλωττιστών Lecture 3 Lexicl Anlysis Elis Athnsopoulos elisthn@cs.ucy.c.cy RecogniNon of Tokens if expressions nd relnonl opertors if è if then è then else è else relop è

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

Assignment 4. Due 09/18/17

Assignment 4. Due 09/18/17 Assignment 4. ue 09/18/17 1. ). Write regulr expressions tht define the strings recognized by the following finite utomt: b d b b b c c b) Write FA tht recognizes the tokens defined by the following regulr

More information

Dr. D.M. Akbar Hussain

Dr. D.M. Akbar Hussain Dr. D.M. Akr Hussin Lexicl Anlysis. Bsic Ide: Red the source code nd generte tokens, it is similr wht humns will do to red in; just tking on the input nd reking it down in pieces. Ech token is sequence

More information

Fall 2018 Midterm 1 October 11, ˆ You may not ask questions about the exam except for language clarifications.

Fall 2018 Midterm 1 October 11, ˆ You may not ask questions about the exam except for language clarifications. 15-112 Fll 2018 Midterm 1 October 11, 2018 Nme: Andrew ID: Recittion Section: ˆ You my not use ny books, notes, extr pper, or electronic devices during this exm. There should be nothing on your desk or

More information

Lists in Lisp and Scheme

Lists in Lisp and Scheme Lists in Lisp nd Scheme Lists in Lisp nd Scheme Lists re Lisp s fundmentl dt structures, ut there re others Arrys, chrcters, strings, etc. Common Lisp hs moved on from eing merely LISt Processor However,

More information

CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE

CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE 3.1 Scheimpflug Configurtion nd Perspective Distortion Scheimpflug criterion were found out to be the best lyout configurtion for Stereoscopic PIV, becuse

More information

Digital Design. Chapter 6: Optimizations and Tradeoffs

Digital Design. Chapter 6: Optimizations and Tradeoffs Digitl Design Chpter 6: Optimiztions nd Trdeoffs Slides to ccompny the tetbook Digitl Design, with RTL Design, VHDL, nd Verilog, 2nd Edition, by Frnk Vhid, John Wiley nd Sons Publishers, 2. http://www.ddvhid.com

More information

Answer Key Lesson 6: Workshop: Angles and Lines

Answer Key Lesson 6: Workshop: Angles and Lines nswer Key esson 6: tudent Guide ngles nd ines Questions 1 3 (G p. 406) 1. 120 ; 360 2. hey re the sme. 3. 360 Here re four different ptterns tht re used to mke quilts. Work with your group. se your Power

More information

Example: 2:1 Multiplexer

Example: 2:1 Multiplexer Exmple: 2:1 Multiplexer Exmple #1 reg ; lwys @( or or s) egin if (s == 1') egin = ; else egin = ; 1 s B. Bs 114 Exmple: 2:1 Multiplexer Exmple #2 Normlly lwys include egin nd sttements even though they

More information

What do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers

What do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers Wht do ll those bits men now? bits (...) Number Systems nd Arithmetic or Computers go to elementry school instruction R-formt I-formt... integer dt number text chrs... floting point signed unsigned single

More information

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula:

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula: 5 AMC LECTURES Lecture Anlytic Geometry Distnce nd Lines BASIC KNOWLEDGE. Distnce formul The distnce (d) between two points P ( x, y) nd P ( x, y) cn be clculted by the following formul: d ( x y () x )

More information

Functor (1A) Young Won Lim 8/2/17

Functor (1A) Young Won Lim 8/2/17 Copyright (c) 2016-2017 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version published

More information

EXPONENT RULES Add Multiply Subtraction Flip

EXPONENT RULES Add Multiply Subtraction Flip Algebr II Finl Em Review Nme Chpter 7 REVIEW: EXPONENT RULES Add Multiply Subtrction Flip Simplify the epression using the properties of eponents. Assume ll vribles re positive. 4 4. 8 8.. 4. 5. 9 9 5

More information

Yoplait with Areas and Volumes

Yoplait with Areas and Volumes Yoplit with Ares nd Volumes Yoplit yogurt comes in two differently shped continers. One is truncted cone nd the other is n ellipticl cylinder (see photos below). In this exercise, you will determine the

More information

Fig.25: the Role of LEX

Fig.25: the Role of LEX The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing

More information

Before We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):

Before We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1): Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters

More information

Control-Flow Analysis and Loop Detection

Control-Flow Analysis and Loop Detection ! Control-Flow Anlysis nd Loop Detection!Lst time! PRE!Tody! Control-flow nlysis! Loops! Identifying loops using domintors! Reducibility! Using loop identifiction to identify induction vribles CS553 Lecture

More information

If you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs.

If you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs. Lecture 5 Wlks, Trils, Pths nd Connectedness Reding: Some of the mteril in this lecture comes from Section 1.2 of Dieter Jungnickel (2008), Grphs, Networks nd Algorithms, 3rd edition, which is ville online

More information

MATH 25 CLASS 5 NOTES, SEP

MATH 25 CLASS 5 NOTES, SEP MATH 25 CLASS 5 NOTES, SEP 30 2011 Contents 1. A brief diversion: reltively prime numbers 1 2. Lest common multiples 3 3. Finding ll solutions to x + by = c 4 Quick links to definitions/theorems Euclid

More information

Stack. A list whose end points are pointed by top and bottom

Stack. A list whose end points are pointed by top and bottom 4. Stck Stck A list whose end points re pointed by top nd bottom Insertion nd deletion tke plce t the top (cf: Wht is the difference between Stck nd Arry?) Bottom is constnt, but top grows nd shrinks!

More information

CSCI 3130: Formal Languages and Automata Theory Lecture 12 The Chinese University of Hong Kong, Fall 2011

CSCI 3130: Formal Languages and Automata Theory Lecture 12 The Chinese University of Hong Kong, Fall 2011 CSCI 3130: Forml Lnguges nd utomt Theory Lecture 12 The Chinese University of Hong Kong, Fll 2011 ndrej Bogdnov In progrmming lnguges, uilding prse trees is significnt tsk ecuse prse trees tell us the

More information

Date: 9.1. Conics: Parabolas

Date: 9.1. Conics: Parabolas Dte: 9. Conics: Prols Preclculus H. Notes: Unit 9 Conics Conic Sections: curves tht re formed y the intersection of plne nd doulenpped cone Syllus Ojectives:. The student will grph reltions or functions,

More information