Simplifying Algebra. Simplifying Algebra. Solutions. Curriculum Ready.

Size: px
Start display at page:

Download "Simplifying Algebra. Simplifying Algebra. Solutions. Curriculum Ready."

Transcription

1 Silifig Alger Silifig Alger Curriulu Re

2

3 How oes it work? Silifig Alger Pge questios Multilig iviig e f r q r q 6 6qr r q q r g g g g g g g g g h i ^ h ^ h ^ h ^ h ^ h ^ h Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

4 How oes it work? Silifig Alger Pge questios Multilig iviig ' ' ^ h - ' 6 ^ gh 6 ^ h g g g g g e f l I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

5 How oes it work? Silifig Alger Pge questios Multilig iviig l Pge 6 questios Multilig iviig - 6 ] 6 - g ] - g - ^- h ] - g ] - g ] - g l - Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

6 How oes it work? Silifig Alger Pge questios Aig sutrtig Like ters Like ters Like ters Like ters Like ters Like ters Like ters Like ters e q -- q+ q -- q+ f Like ters Like ters q -q q q Like ters g h w -w-w- w w -w-w- w Like ters Like ters w -w -w-w -w - w I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

7 How oes it work? Silifig Alger Pge questios Aig sutrtig Like ters Like ters s - 6s+ s - + s s - 6s+ s - + s Like ters s + s - 6s+ s- s + s- Like ters Like ters Like ters Like ters Like ters Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

8 How oes it work? Silifig Alger Pge 0 questios Coiig the si oertios l 6 I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

9 How oes it work? Silifig Alger Pge 0 questios Coiig the si oertios ] - g - -q -q q - q q - - q q q q ] q - g q - Pge questios Coiig the si oertios q + q q - 6q 0q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0q Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

10 Where oes it work? Silifig Alger Pge questios Multilitio rule for eoets se eoet oeffiiet se eoet oeffiiet se eoet oeffiiet se eoet oeffiiet Write these i eoet for: h h h h h s s s s 6 q q q e f Pge questios Multilitio rule for eoets Short ut + Short ut + Short ut + + I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

11 Where oes it work? Silifig Alger Pge questios Multilitio rule for eoets Short ut Short ut h h h h h h h h h h h h Short ut 0h 0 h h h h h h h h h h h h h + 0h 0.. Short ut Short ut Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

12 Where oes it work? Silifig Alger Pge questios Divisio rule for eoets w w w w w w w w w w w w w w w w w w w ' Short ut - ' ' Short ut - ' 0 I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

13 Where oes it work? Silifig Alger Pge questios Divisio rule for eoets 6 Short ut 6 ' ' Short ut v v Short ut v v 6 6v ^ ' hv v v v v v v v v v v 6v 6 ' ^6 ' h 6 6 Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

14 Where oes it work? Silifig Alger Pge questios Divisio rule for eoets 0 0 ' 0 0 Short ut Short ut ` j 6 0 ' 0 ^0 ' 0h 6 0 Pge questios Coiig ultilitio ivisio rules ] g' ' ' Short ut w w w w w w w w w w ' 6+ 0 w w - I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

15 Where oes it work? Silifig Alger Pge questios Coiig ultilitio ivisio rules k k k k k + 0 ' 0 - k k 6 0 ^ ' hk 6 ^0 ' h 6k 0 Pge questios Coiig ultilitio ivisio rules ] g ' ] g ] ' g + 6 ] 6 ' g 6 ^ 6h ^ h 6 ] ' 6g - - Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

16 Where oes it work? Silifig Alger Pge 0 questios Eoet ogs uzzle I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

17 Where oes it work? Silifig Alger Pge 0 questios Eoet ogs uzzle ' 6 ^ ' h ' 6 6 ' ^ ' h' 6 6 ' ^ ' h' ^6 ' h 6 6 ' ' 6 ' ' ' ' 6 ' ' ' ^ h+ 6 ' 0 6 ' 6 ' 0 6 Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

18 Where oes it work? Silifig Alger Pge questios Eoet rule for eoets ^ j h j j j j j ^ h j j 0 ^ j h j ^ h j 0 ^r h r r ^ h ^ h ^ h 6 6 ^ h ^ h ^ h ^ h ^r h r r r r Short ut + + Short ut r + r ^ h ^ h ^r h r r r 6 6 ^ k h k 0 0 ^z h z k z 0 k 6 z 6 I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

19 Where oes it work? Silifig Alger Pge questios Zero rule for eoets 0 0 ^h ^ h k ' ^k h 0 0 k k 6 6 e ^h Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

20 Where oes it work? Silifig Alger Pge questios Coiig ll the eoet rules g g g ^ h g g g g g g g g g g + g g - ^ h' ^ h 6 ^ h ] ' g ^ zh ^ zh z z z z z 0 z z I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

21 Where oes it work? Silifig Alger Pge 6 questios Alger teriolog uzzle glier fsili l g e r i s i l i f se etoe s e e o e t is elru s i u e r l viler elik sert v r i l e l i k e t e r s ee orf e e f o r Nuer Vrile Aswer o e f f i i e t Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

22 Wht else ou o? Silifig Alger Pge questios Perieter re roles (i) Perieter (ii) If, erieter + 0 uits uits (i) Perieter (ii) If, erieter + 0 uits + 0 uits Pge questios Perieter re roles (i) Perieter ^+ h+ ^+ h+ ^+ h+ ^+ h (ii) If, erieter + uits + uits uits (i) Perieter (ii) If, erieter + uits 0 + uits + uits 0 I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

23 Wht else ou o? Silifig Alger Pge 0 questios Perieter re roles (i) Are 6 (ii) If re, 6 uits 6 6 uits 6 uits (i) Are (ii) If re, 6 uits 6 6 uits uits Pge questios Perieter re roles 6 (i) Perieter (ii) If, erieter 6 uits uits (iii) Legth ^ + h (ultilig eh ter oe hlf will hge the oeffiiets ol) + With ^ h 0 ` Perieter Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

24 Silifig Alger Notes I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

25 Silifig Alger Notes Silifig Alger Mthletis Pssort P Lerig I SERIES TOPIC

26 Silifig Alger Notes I Silifig Alger SERIES TOPIC Mthletis Pssort P Lerig

27

28 Silifig Alger

Simplifying Algebra. Simplifying Algebra. Solutions. Curriculum Ready.

Simplifying Algebra. Simplifying Algebra. Solutions. Curriculum Ready. Siplifing Alger Siplifing Alger Curriulu Re www.thletis.o How oes it work? Siplifing Alger Pge questions Multipling n iviing 6 6 6 0 0 e f p r q p r q 6 6pqr p r q p q r g g g g g g g g g h n n n n n

More information

Pythagoras Theorem PYTHAGORAS THEOREM.

Pythagoras Theorem PYTHAGORAS THEOREM. Pthgors Theorem PYTHAGORAS THEOREM www.mthletis.om.u How oes it work? Solutions Pthgors Theorem Pge 3 questions Right-ngle tringles D E x z Hotenuse is sie: F Hotenuse is sie: DF Q k j l Hotenuse is sie:

More information

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready.

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready. Are n Perimeter Are n Perimeter Solutions Curriulum Rey www.mthletis.om How oes it work? Solutions Are n Perimeter Pge questions Are using unit squres Are = whole squres Are = 6 whole squres = units =

More information

H SERIES. Algebra Basics. Algebra Basics. Solutions. Curriculum Ready.

H SERIES. Algebra Basics. Algebra Basics. Solutions. Curriculum Ready. Alger Bsis H SERIES Alger Bsis Curriulum Rey www.mthletis.om Copyright 009 P Lerning. All rights reserve. First eition printe 009 in Austrli. A tlogue reor for this ook is ville from P Lerning Lt. ISBN

More information

CH 45 INTRO TO FRACTIONS

CH 45 INTRO TO FRACTIONS CH INTRO TO FRACTIONS Itrotio W e re ot to erk o st of frtios. If o ve erstoo ritheti frtios efore, o ll fi tht lgeri frtios follo the se set of rles. If frtios re still ster, let s ke this the seester

More information

Simplifying Algebra. Simplifying Algebra. Curriculum Ready.

Simplifying Algebra. Simplifying Algebra. Curriculum Ready. Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give this

More information

ECE 102 Engineering Computation

ECE 102 Engineering Computation ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios Summtio Symol is the symol for summtio. Emple: N k N... 9 k k k k k the, If e e e f e f k Vetor Bsis A vetor is

More information

Cape Cod Community College

Cape Cod Community College Cpe Cod Couity College Deprtetl Syllus Prepred y the Deprtet of Mthetics Dte of Deprtetl Approvl: Noveer, 006 Dte pproved y Curriculu d Progrs: Jury 9, 007 Effective: Fll 007 1. Course Nuer: MAT110 Course

More information

ELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000]

ELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000] ELLIPSE. If the ltus rectum of ellipse e equl to hlf of its mior is, the its eccetricit is [Krtk CET 000] / / / d /. The legth of the ltus rectum of the ellipse is [MNR 7, 0, ] / / / d 0/. Eccetricit of

More information

Accuplacer Elementary Algebra Study Guide

Accuplacer Elementary Algebra Study Guide Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give

More information

Westchester Community College Elementary Algebra Study Guide for the ACCUPLACER

Westchester Community College Elementary Algebra Study Guide for the ACCUPLACER Westchester Commuity College Elemetry Alger Study Guide for the ACCUPLACER Courtesy of Aims Commuity College The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry

More information

The limit comparison test

The limit comparison test Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does

More information

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT ALGEBRA II (3 CREDIT HOURS)

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT ALGEBRA II (3 CREDIT HOURS) SINCLAIR COMMUNITY COLLEGE DAYTON OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT - ALGEBRA II (3 CREDIT HOURS) 1. COURSE DESCRIPTION: Ftorig; opertios with polyoils d rtiol expressios; solvig seod degree equtios

More information

Assessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)

Assessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT) Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give

More information

Section 2.2. Matrix Multiplication

Section 2.2. Matrix Multiplication Mtri Alger Mtri Multiplitio Setio.. Mtri Multiplitio Mtri multiplitio is little more omplite th mtri itio or slr multiplitio. If A is the prout A of A is the ompute s follow: m mtri, the is k mtri, 9 m

More information

CH 20 SOLVING FORMULAS

CH 20 SOLVING FORMULAS CH 20 SOLVING FORMULAS 179 Itrodutio S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them.

More information

CH 19 SOLVING FORMULAS

CH 19 SOLVING FORMULAS 1 CH 19 SOLVING FORMULAS INTRODUCTION S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them.

More information

Mu Alpha Theta National Convention: Denver, 2001 Sequences & Series Topic Test Alpha Division

Mu Alpha Theta National Convention: Denver, 2001 Sequences & Series Topic Test Alpha Division Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio. Wht is the commo rtio of the geometric sequece, 7, 9,? 7 (C) 5. The commo differece of the rithmetic sequece,, 0, is 5 (C)

More information

SIMPLE NONLINEAR GRAPHS

SIMPLE NONLINEAR GRAPHS S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle

More information

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum 0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.

More information

K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:

K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c: Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to

More information

Intermediate Arithmetic

Intermediate Arithmetic Git Lerig Guides Iteredite Arithetic Nuer Syste, Surds d Idices Author: Rghu M.D. NUMBER SYSTEM Nuer syste: Nuer systes re clssified s Nturl, Whole, Itegers, Rtiol d Irrtiol uers. The syste hs ee digrticlly

More information

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled

More information

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +

More information

Topic 4 Fourier Series. Today

Topic 4 Fourier Series. Today Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will

More information

Polynomials. Polynomials. Curriculum Ready ACMNA:

Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

More information

Advanced Higher Grade

Advanced Higher Grade Prelim Emitio / (Assessig Uits & ) MATHEMATICS Avce Higher Gre Time llowe - hors Re Crefll. Fll creit will be give ol where the soltio cotis pproprite workig.. Clcltors m be se i this pper.. Aswers obtie

More information

General properties of definite integrals

General properties of definite integrals Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

More information

LAWS OF INDICES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

LAWS OF INDICES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski M.K. HOME TUITION Mthetics Revisio Guides Level: GCSE Higher Tier LAWS OF INDICES Versio:. Dte: 0--0 Mthetics Revisio Guides Lws of

More information

( ) 2 3 ( ) I. Order of operations II. Scientific Notation. Simplify. Write answers in scientific notation. III.

( ) 2 3 ( ) I. Order of operations II. Scientific Notation. Simplify. Write answers in scientific notation. III. Assessmet Ceter Elemetry Alger Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give

More information

AT100 - Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a

AT100 - Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7

More information

Formal Languages The Pumping Lemma for CFLs

Formal Languages The Pumping Lemma for CFLs Forl Lguges The Pupig Le for CFLs Review: pupig le for regulr lguges Tke ifiite cotext-free lguge Geertes ifiite uer of differet strigs Exple: 3 I derivtio of log strig, vriles re repeted derivtio: 4 Derivtio

More information

Applications of Regular Closure

Applications of Regular Closure Applictios of Regulr Closure 1 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 L2 cotext free regulr Regulr Closure L1 L 2 cotext-free 2 Liz 6 th, sectio 8.2, exple 8.7, pge

More information

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following.

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following. Reteching Lines nd Angles Not ll lines nd plnes intersect. prllel plnes. prllel. } shows tht lines or plnes re prllel: < > < > A } ens Line A is prllel to line. skew. A } plne EFHG A plne FH } plne AEG

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

Pre-Calculus - Chapter 3 Sections Notes

Pre-Calculus - Chapter 3 Sections Notes Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied

More information

Objective Mathematics

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

More information

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () =... 0.00 b) 0 c). l d =... c

More information

Review of Linear Algebra

Review of Linear Algebra PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr

More information

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233, Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :

More information

Course 121, , Test III (JF Hilary Term)

Course 121, , Test III (JF Hilary Term) Course 2, 989 9, Test III (JF Hilry Term) Fridy 2d Februry 99, 3. 4.3pm Aswer y THREE questios. Let f: R R d g: R R be differetible fuctios o R. Stte the Product Rule d the Quotiet Rule for differetitig

More information

Introduction of Fourier Series to First Year Undergraduate Engineering Students

Introduction of Fourier Series to First Year Undergraduate Engineering Students Itertiol Jourl of Adved Reserh i Computer Egieerig & Tehology (IJARCET) Volume 3 Issue 4, April 4 Itrodutio of Fourier Series to First Yer Udergrdute Egieerig Studets Pwr Tejkumr Dtttry, Hiremth Suresh

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

«A first lesson on Mathematical Induction»

«A first lesson on Mathematical Induction» Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,

More information

Algebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by

Algebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio

More information

NURTURE COURSE TARGET : JEE (MAIN) Test Type : ALL INDIA OPEN TEST TEST DATE : ANSWER KEY HINT SHEET. 1. Ans.

NURTURE COURSE TARGET : JEE (MAIN) Test Type : ALL INDIA OPEN TEST TEST DATE : ANSWER KEY HINT SHEET. 1. Ans. Test Type : LL INDI OPEN TEST Paper Code : 0000CT005 00 CLSSROOM CONTCT PROGRMME (cadeic Sessio : 05-06) NURTURE COURSE TRGET : JEE (MIN) 07 TEST DTE : - 0-06 NSWER KEY HINT SHEET Corporate Office : CREER

More information

GRADE 12 SEPTEMBER 2012 MATHEMATICS P2

GRADE 12 SEPTEMBER 2012 MATHEMATICS P2 Provice of the EASTERN CAPE EDUCATION NATIONAL SENIOR CERTIFICATE GRADE SEPTEMBER 0 MATHEMATICS P MARKS: 50 TIME: 3 hours *MATHE* This questio paper cosists of 4 pages, icludig a formula sheet ad 4 diagram

More information

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Name: Date: Part I Questios UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS. For the quadratic fuctio show, the coordiates. of its verte are 0, (3) 6, (), 7 (4) 3, 6. A quadratic fuctio

More information

Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral

Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ

More information

Factorising FACTORISING.

Factorising FACTORISING. Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will

More information

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questios. For the quadratic fuctio show below, the coordiates of its verte are () 0, (), 7 (3) 6, (4) 3, 6. A quadratic

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

EE 380. Linear Control Systems. Lecture 10

EE 380. Linear Control Systems. Lecture 10 EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.

More information

Test One (Answer Key)

Test One (Answer Key) CS395/Ma395 (Sprig 2005) Test Oe Name: Page 1 Test Oe (Aswer Key) CS395/Ma395: Aalysis of Algorithms This is a closed book, closed otes, 70 miute examiatio. It is worth 100 poits. There are twelve (12)

More information

Nonlocal Boundary Value Problem for Nonlinear Impulsive q k Symmetric Integrodifference Equation

Nonlocal Boundary Value Problem for Nonlinear Impulsive q k Symmetric Integrodifference Equation OSR ol o Mec OSR-M e-ssn: 78-578 -SSN: 9-765X Vole e Ve M - A 7 PP 95- wwwojolog Nolocl Bo Vle Poble o Nole lve - Sec egoeece Eo Log Ceg Ceg Ho * Yeg He ee o Mec Yb Uve Yj PR C Abc: A oe ole lve egoeece

More information

Unit 2 Exponents Study Guide

Unit 2 Exponents Study Guide Unit Eponents Stud Guide 7. Integer Eponents Prt : Zero Eponents Algeric Definition: 0 where cn e n non-zero vlue 0 ecuse 0 rised to n power less thn or equl to zero is n undefined vlue. Eple: 0 If ou

More information

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1 MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]

More information

CH 39 USING THE GCF TO REDUCE FRACTIONS

CH 39 USING THE GCF TO REDUCE FRACTIONS 359 CH 39 USING THE GCF TO EDUCE FACTIONS educig Algeric Frctios M ost of us lered to reduce rithmetic frctio dividig the top d the ottom of the frctio the sme (o-zero) umer. For exmple, 30 30 5 75 75

More information

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 4 Solutions [Numerical Methods]

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 4 Solutions [Numerical Methods] ENGI 3 Advaced Calculus or Egieerig Facult o Egieerig ad Applied Sciece Problem Set Solutios [Numerical Methods]. Use Simpso s rule with our itervals to estimate I si d a, b, h a si si.889 si 3 si.889

More information

Objective Mathematics

Objective Mathematics . If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic

More information

G8-11 Congruence Rules

G8-11 Congruence Rules G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th

More information

M098 Carson Elementary and Intermediate Algebra 3e Section 10.2

M098 Carson Elementary and Intermediate Algebra 3e Section 10.2 M09 Crso Eleetry d Iteredite Alger e Sectio 0. Ojectives. Evlute rtiol epoets.. Write rdicls s epressios rised to rtiol epoets.. Siplify epressios with rtiol uer epoets usig the rules of epoets.. Use rtiol

More information

Name: Period: Date: 2.1 Rules of Exponents

Name: Period: Date: 2.1 Rules of Exponents SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,

More information

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and Sec. 7., Boyce & DiPrim, p. Sectio 7., Systems of Lier Algeric Equtios; Lier Idepedece, Eigevlues, Eigevectors I. Systems of Lier Algeric Equtios.. We c represet the system...... usig mtrices d vectors

More information

MODEL SSLC EXAMINATION KEY FOR MATHEMATICS

MODEL SSLC EXAMINATION KEY FOR MATHEMATICS MODEL SSLC EXAMINATION 018 KEY FOR MATHEMATICS SECTION I Q. Key Aswer Q. Key Aswer No No 1. () A \ B = A B 9. (4) 60 m. (4) {,4,5}. (1) a A.P 10. (4) 4.5 cm 4. (4) a k+5 11. () ta θ 5. () cx + bx + a =

More information

Solutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors.

Solutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors. Quiz November 4th, 23 Sigals & Systems (5-575-) P. Reist & Prof. R. D Adrea Solutios Exam Duratio: 4 miutes Number of Problems: 4 Permitted aids: Noe. Use oly the prepared sheets for your solutios. Additioal

More information

Honors Algebra 2 Summer Assignment

Honors Algebra 2 Summer Assignment Hoors Algera Summer Assigmet Dear Future Hoors Algera Studet, Cogratulatios o your erollmet i Hoors Algera! Below you will fid the summer assigmet questios. It is assumed that these cocepts, alog with

More information

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k. . Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric

More information

Name: MATH 65 LAB INTEGER EXPONENTS and SCIENTIFIC NOTATION. Instructor: T. Henson

Name: MATH 65 LAB INTEGER EXPONENTS and SCIENTIFIC NOTATION. Instructor: T. Henson MATH 6 LAB INTEGER EXPONENTS ad SCIENTIFIC NOTATION Name: Istructor: T. Heso Purpose: Epoets are used i may formulas, especially i the scieces where epoets are used to write very small umbers ad very large

More information

f ( x) ( ) dx =

f ( x) ( ) dx = Defiite Itegrls & Numeric Itegrtio Show ll work. Clcultor permitted o, 6,, d Multiple Choice. (Clcultor Permitted) If the midpoits of equl-width rectgles is used to pproximte the re eclosed etwee the x-xis

More information

* power rule: * fraction raised to negative exponent: * expanded power rule:

* power rule: * fraction raised to negative exponent: * expanded power rule: Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures

More information

Project 3: Using Identities to Rewrite Expressions

Project 3: Using Identities to Rewrite Expressions MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht

More information

Student Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)

Student Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT) Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples

More information

Circular Functions (Trigonometry)

Circular Functions (Trigonometry) Circular Fuctios (Trigoometry) Circular fuctios Revisio Where do si cos ad ta come from? Uit circle (of radius ) cos is the coordiate si is the y coordiate si ta cos all are measures of legth. Remember

More information

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

); 5 units 5. x = 3 6. r = 5 7. n = 2 8. t =

); 5 units 5. x = 3 6. r = 5 7. n = 2 8. t = . Sample answer: dilation with center at the origin and a scale factor of 1 followed b a translation units right and 1 unit down 5. Sample answer: reflection in the -axis followed b a dilation with center

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios

More information

GRADE 12 SEPTEMBER 2016 MATHEMATICS P1

GRADE 12 SEPTEMBER 2016 MATHEMATICS P1 NATIONAL SENIOR CERTIFICATE GRADE SEPTEMBER 06 MATHEMATICS P MARKS: 50 TIME: 3 hours *MATHE* This questio pper cosists of pges icludig iformtio sheet MATHEMATICS P (EC/SEPTEMBER 06 INSTRUCTIONS AND INFORMATION

More information

Northwest High School s Algebra 2

Northwest High School s Algebra 2 Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:

More information

Dynamics of Marine Biological Resources * * * REVIEW OF SOME MATHEMATICS * * *

Dynamics of Marine Biological Resources * * * REVIEW OF SOME MATHEMATICS * * * Dmis o Mrie Biologil Resores A FUNCTION * * * REVIEW OF SOME MATHEMATICS * * * z () z g(,) A tio is rle or orml whih estlishes reltioshi etwee deedet vrile (z) d oe or more ideedet vriles (,) sh tht there

More information

PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES

PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with

More information

CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION EXAMINATION ADDITIONAL MATHEMATICS. Paper 02 - General Proficiency

CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION EXAMINATION ADDITIONAL MATHEMATICS. Paper 02 - General Proficiency TEST CODE 01254020 FORM TP 2015037 MAY/JUNE 2015 CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION CERTIFICATE@ EXAMINATION ADDITIONAL MATHEMATICS Paper 02 - Geeral Proficiecy 2 hours 40 miutes

More information

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

INDEPENDENT COMPONENT ANALYSIS

INDEPENDENT COMPONENT ANALYSIS www.seechehcemet.org 공사중 SeechEhcemet.tistor.com INDEPENDENT COMPONENT ANALYSIS -7- Seugil Kim E-mil : goodksi@gmil.com, Twitter : @DrSoicwve Itroductio ICA is owerful. ot difficult. ot differet from other

More information

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd

More information

, we would have a series, designated as + j 1

, we would have a series, designated as + j 1 Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite

More information

COMP 2804 Solutions Assignment 1

COMP 2804 Solutions Assignment 1 COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

d dx where k is a spring constant

d dx where k is a spring constant Vorlesug IX Harmoic Oscillator 1 Basic efiitios a properties a classical mechaics Oscillator is efie as a particle subject to a liear force fiel The force F ca be epresse i terms of potetial fuctio V F

More information

3. Supppose the amount of information available on the web is multiplied by 27 every year. How much information will be available a.

3. Supppose the amount of information available on the web is multiplied by 27 every year. How much information will be available a. Lesso -A The Root of the Prole H ow would epoets work if the were fractios? To fid out, we we will use the iteret cocept, where the ase uer represets what the aout of iforatio gets ultiplied ever ear.

More information

EXPONENTS AND LOGARITHMS

EXPONENTS AND LOGARITHMS 978--07-6- Mthemtis Stdrd Level for IB Diplom Eerpt EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW The rules of epoets: m = m+ m = m ( m ) = m m m = = () = The reltioship etwee epoets d rithms: = g where

More information

FREE Download Study Package from website: &

FREE Download Study Package from website:  & FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om SHORT REVISION. Defiitio : Retgulr rry of m umers. Ulike determits it hs

More information

MATH2007* Partial Answers to Review Exercises Fall 2004

MATH2007* Partial Answers to Review Exercises Fall 2004 MATH27* Partial Aswers to Review Eercises Fall 24 Evaluate each of the followig itegrals:. Let u cos. The du si ad Hece si ( cos 2 )(si ) (u 2 ) du. si u 2 cos 7 u 7 du Please fiish this. 2. We use itegratio

More information

ENGR 3861 Digital Logic Boolean Algebra. Fall 2007

ENGR 3861 Digital Logic Boolean Algebra. Fall 2007 ENGR 386 Digitl Logi Boole Alger Fll 007 Boole Alger A two vlued lgeri system Iveted y George Boole i 854 Very similr to the lger tht you lredy kow Sme opertios ivolved dditio sutrtio multiplitio Repled

More information

Calculus II - Problem Drill 21: Power Series, Taylor and Maclaurin Polynomial Series

Calculus II - Problem Drill 21: Power Series, Taylor and Maclaurin Polynomial Series Calculus II - Problem Drill : Power Series, Taylor ad Maclauri Polyomial Series Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 3 4 3 4. Fill

More information

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced

More information