SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

Save this PDF as:

Size: px
Start display at page:

Transcription

1 SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting Formule nd Solving Equtions 7. Qudrtic Equtions A Reminder. Simultneous Equtions A Reminder Tutoril Eercises Dr Derek Hodson

2 Bsic Alger Opertions nd Epressions You will lred e fmilir with vrious forms of mthemticl epressions. It is importnt tht ou re confident in hndling nd mnipulting such epressions, so this section should refresh our memor regrding some sic lgeric concepts nd techniques. The Bsic Arithmetic Opertions In mthemticl epressions, numers nd / or vriles (i.e. unknowns re comined using the rithmetic opertions of ddition, sutrction, multipliction, division nd eponentition (i.e. squring, cuing, etc., long with rckets to group or seprte terms. If n epression contins onl sums nd differences, or contins onl multiplictions nd divisions, then the opertions cn e delt with from left to right. For emple, 0 ;. 0 For more involved epressions, we use the BODMAS principle to determine the order of evlution: Brckets ] first priorit Order (i.e. powers ] second priorit Division Multipliction Addition Sutrction third priorit fourth priorit. For emple: (B (O (M (A (

3 Another emple: (B (O (D (S ( Indices (Powers The simple use of powers or indices to represent repeted multipliction should e fmilir, for emple n... [ n terms], ut ou must lso e cler on the wider interprettion of indices. The lines elow collte the importnt spects of indices: 0 n n nd n n nd m m n m n m n m n m n m ( m ( m n m m m m m These results will e needed in oth the epnsion nd simplifiction of lgeric epressions.

4 Emple ( Epnd ( 7 (. 7 ( 7 ( ( 7 ( ( Epnd (. ( ( ( ( ( ( ( (c Simplif ( z z. ( z z z z z z See the Tutoril Eercises for prctice.

5 Common Mistkes Here re just few common mistkes to void when mnipulting lgeric epressions: ( : ( ( : ( d c d c : d c d c d c c c ( : c c ( n m n m : n m n m : Division of Algeric Epressions Providing the denomintor (i.e. the ottom it of division contins no s or s, the division itself is usull strightforwrd to crr out. Emple ( ( When the denomintor does contin s nd / or s, we m require polnomil division. Note:,, re emples of polnomils. 0 0 (

6 Eponentil Functions nd Logrithms An eponent is nother nme for power or inde. Eponents cn e used to crete eponentil functions:, ( > 0 : 0,. In this contet, is clled the se of the eponentil function. Commonl used ses re 0 nd the eponentil constnt e. 7 : 0 ; e. The function with the eponentil constnt e s its se is so importnt in mthemtics, science nd engineering tht it is referred to s the eponentil function, ll others eing suordinte. Relted to eponentil functions re logrithms. If we epress numer N s power of, i.e. N, then the power is defined to e the (se logrithm of N : log N ( log of N to the se Note tht ecuse of the importnce of the eponentil constnt, logrithms to se e re given the specil nme of nturl logrithms nd denoted ln (. Emple ( ( (c 00 0 log 0 ( log 0 (000 log ( The definition cn e etended to frctionl powers with logrithms to se 0 nd se e otinle from clcultors: Emple ( log 0 ( (to deciml plces ( ln (. 77 (to 7 deciml plces

7 Becuse logs re definition indices, we cn use the rules for comining indices to determine the so-clled lws of logrithms: log ( R S log R log S [L] R log log R log S S [L] n log ( R n log R. [L] We cn use these to epnd or contrct epressions involving logrithms: Emple ( Epnd log 0 z. log 0 z log 0 ( log 0 ( z [L] log 0 ( log 0 ( log 0 ( z [L] log 0 ( log 0 ( log 0 ( z [L] ( Write ln ( ln ( z ln ( s single logrithm. ln ( ln ( z ln ( ln ( ln[( z ] ln ( [L] ln[ ( z ] ln ( [L] ( ln z [L]

8 Opertions nd their Inverses Ech sic rithmetic opertion tht we m encounter (with few eceptions hs ssocited with it corresponding inverse opertion. An opertion nd its inverse, when pplied in sequence, effectivel cncel out one nd other. For emple, suppose we strt off with. If we now dd nd then sutrct, we re ck to gin. Tht is,. We cn s tht the inverse of dding is sutrcting (nd lso vice vers!. Similrl, if we multipl, then divide, we re gin ck to :. So the inverse of multipling is dividing (nd vice vers. The tle elow shows opertions nd their inverses, including the eponentils nd logs from the previous section, nd some others ou m lred e fmilir with: This is specil cse of the entr ove. Multipling the sme s multipling nd dividing. Hence the inverse opertion is multipling. is or ( ( or ( or ( ( or ( n n n n or ( ( n n or n n ( log 0 0 log 0 (0 ln e ln ( e sin (sin sin (sin cos (cos cos(cos tn ( tn tn ( tn 7

9 Mnipulting Formule nd Solving Equtions The opertion / inverse opertion effect descried in the previous section provides the ke to mnipulting formule nd solving equtions. A formul is n eqution tht epresses reltionship etween quntities. In prticulr, it epresses how to determine the vlue of one quntit from the vlues of one or more other quntities. Below re some emples of formule tht ou m hve seen efore: C ( F V r π d u t t v u t v i R In ech of these formule, the quntit on the left is clled the suject of the formul. Often, when working with formul, we wnt to chnge the suject to one of the other quntities. For emple: v i R v i - i is now the suject. R This is ver simple emple. However, the mnipultion of formule is n spect of lger tht cn pose difficulties for students. Consider the formul in the ove list tht reltes temperture in degrees Celsius to temperture in degrees Fhrenheit: C ( F. It is quite es to mke F the suject of this formul. How ou would tckle this would prol depend on methods rought from school. If ou re thinking something like, move terms from one side of the eqution to the other to get F on its own, then plese think gin. Although the method of moving terms will prol get ou the correct nswer in this cse, it is not mthemticll correct w of thinking nd cn cuse prolems in more complicted formule. Let us now look t the correct w to mnipulte formul or eqution. This will let ou see wht is rell hppening when terms pper to move round n eqution. Wrning: Ignore the following t our peril!

10 First, let us look t the formul s given nd the mthemticl opertions within it: C ( F. Given vlue of F, (following BODMAS we would determine the corresponding vlue of C : sutrcting multipling. A formul (or, in fct, n eqution is like lnced set of scles: C ( F When working with n eqution, we must not upset the lnce. This mens tht if we do something to one side of the eqution, then we must do the ect sme thing to the other side. This is the one nd onl rule of lgeric mnipultion. [Note: We cn, of course, swp the sides round, ut tht is just like turning the scles round; it does not ffect the lnce.] To chnge the suject of formul, we ppl crefull chosen opertions to oth sides of the eqution whose net effect is to isolte the new suject. These opertions re the inverse opertions of the those within the originl formul. Tht is: sutrct multipl re reversed nd inverted to give multipl dd. These opertions re now pplied, in turn, to oth sides of the eqution.

11 The process in full is s follows: C ( F Multipl oth sides : C ( F C ( F C F Add to oth sides: C F C F Swp sides: F C... nd with pictures: Formul: C ( F Multipl oth sides : C ( F C F Add to oth sides: C F C F Swp sides: F C 0

12 In prctice, we needn t put in s much detil. For such simple formul we might simpl set down the following lines: C ( F Multipl oth sides : C F Add to oth sides: C F Swp sides: F C. This revited version m give the impression tht terms re moving round the eqution, ut the re not. You must lws keep in mind wht is trul hppening in the ckground. Alws think BALANCE. Further Emples Note: In the following emples, full detils re shown. In prctice, the level of detil ou show in ou own working will depend on our own confidence nd ilit; s these grow, ou will nturll displ less. Emple For the formul mke the suject. [Note: This is the sme s sing, solve for.] Anlse opertions. Given, how is clculted? squre multipl dd Reverse nd invert opertions: sutrct divide squre root

13 Appl inverse opertions to formul: Formul: Sutrct : Divide : At this point it is useful to swp sides Squre root: Shortened version (for the more confident: Formul: Sutrct : Swp sides nd divide : Squre root:

14 Emple 7 For the formul V, mke r the suject. r π Anlse opertions. Given r, how is V clculted? cue (or rise to the power π multipl Reverse nd invert opertions: multipl π cue root (or rise to the power Appl inverse opertions to formul: Formul: V r π V π r Multipl π : π π V π r V π r Swp sides: r V π Rise to the power r V π : ( ( r V ( π Shortened version: Formul: V π r Swp sides nd multipl π r V : π Rise to the power r V π : (

15 Emple For the eqution e, solve for. Note: For BODMAS purposes, ou hve to imgine rckets grouping the power ( terms together, i.e. e Anlse opertions. Given, how is clculted? multipl e to the power ( i.e. multipl ( e Reverse nd invert opertions: divide tke nturl log ( i.e. ln ( divide Appl inverse opertions to formul: Eqution: e Divide : e e Nturl log: ln ln ( e ln Swp sides: ln Divide : ln ln

16 Shortened version: Eqution: e Divide : e Swp sides nd tke nturl log: ln Divide : ln Emples (with less detil: Emple Determine t when 0. t. 0 This is not formul with suject, so we cnnot write down sequence of opertions. It is, however, n eqution (with n unknown nd the concept of lnce still pplies. In mnipulting this eqution, we wnt to end up with the form t (. Clerl we must mnipulte t down to the level of the equls sign. One of the lws of logrithms will help here: Eqution: 0. t 0 0.t Tke log (se 0 of oth sides: ( 0 log ( log t log0 ( Multipl oth sides : 0.t log ( 0 t t log0.0 (

17 Emple 0 t Determine t when e 0.. t Eqution: e 0. t Tke nturl log of oth sides: ln ( ln (0. e t ln (0. Divide oth sides : t ln (0. (sme s multipling t 0.7 Emple Determine when. Eqution: Tke log (se 0 of oth sides: ( log ( log 0 0 log0 ( log0 ( Divide oth sides log0 ( : log log 0 0 ( ( 0. Note: You could use nturl logs nd get the sme nswer.

18 7 Qudrtic Equtions A Reminder One tpe of eqution tht crops up quite frequentl is the qudrtic eqution: c 0. This tpe of eqution is solved either fctoristion (which isn t lws possile or use of the qudrtic formul ± c. Emple ( Solve 0 fctoristion. 0 ( ( 0 ( ( or 0 0 or ( Solve 0 the qudrtic formul. 0,, c ± c ± ( ± ± or or Note: Qudrtic equtions m hve two rel solutions, one rel solution or no rel solutions, depending on the vlue of the discriminnt c. 7

19 Simultneous Equtions A Reminder Equtions cn contin more thn one unknown. For emple, the eqution hs two unknowns. A solution of this eqution is mde up of n -vlue nd -vlue tht together stisf the eqution. We could hve ( 0, or (, or n one of n infinite numer of solutions. When plotted on Crtesin es sstem, ll possile solutions of this eqution lie on stright line (see the hnd-out Coordinte Geometr The Bsics. If we hve second eqution, s, this lso hs n infinite numer of solutions, which lso lie on stright line. However, there is one solution tht is common to oth equtions. Grphicll, this common solution is given the coordintes of the point of intersection of the two stright-line grphs. To find this solution, we could drw the grphs nd red off the vlues of nd. This would e fine for the ove equtions which, s it turns out, hve integer vlues in the solution. For more generl method of solving equtions simultneousl, we require n lgeric pproch. There re vrious ws of setting this out. Method Elimintion Sustitution Tke either eqution nd epress one unknown in terms of the other:. Sustitute this into the other eqution, there eliminting one of the unknowns nd leving n eqution with single unknown tht is esil solved: (. This vlue is then sustituted into to give. Solution:,

20 Method Elimintion the Addition or Sutrction of Equtions Line up the equtions one ove the other:. If necessr, multipl up one or other or oth equtions to otin common coefficients on one of the unknowns: or. Net, eliminte the unknown with the common coefficient dding or sutrcting corresponding sides of the equtions: Add : or Sutrct : Solve : Solve :. Solution:, Whichever method ou use, lws check our nswers sustituting the vlues into the originl equtions: st eqution : LHS RHS ( nd eqution : LHS ( RHS.

21 Tutoril Eercises ( Epnsion of Algeric Epressions Contining Brckets (Revision (. Epnd the following epressions involving products nd powers: (i ( ( (ii ( ( (iii ( (7 (iv ( ( 7 (v ( ( 7 (vi ( ( (vii ( ( (viii ( ( (i ( ( ( ( ( ( (. (. Open out the following rcketed epressions: (i ( ( (ii ( ( 7 (iii ( ( (iv ( ( (v ( (vi ( (vii ( (viii ( ( (i ( ( 7 ( ( ( (. (. Fctorise the following qudrtic epressions: (i (ii (iii (iv (v (vi (vii (viii (i (. (. Show tht ( ( nd use the result to (i epnd: ( ( ; ( ( (ii fctorise: ;. 0

22 ( Division of Algeric Epressions (. Perform the following divisions: (i (ii (iii 7c d (iv c d (v (vi π r h π r π r h h. ( Eponentils nd Logrithms The Bsics (. Use our clcultor functions, nmel log (which is ctull log 0 nd 0, to evlute the following epressions. Comment on the results for prts (v (viii. (i log 0 (000 (ii log 0 (. (iii. 0 (iv 0... (v log0 (0 (vi log0 (0 (vii log 0 (. 0 (viii log 0 ( (. Use our clcultor functions, nmel ln (which is ctull log e nd e, to evlute the following epressions. Comment on the results for prts (v (viii. (i ln (. (ii ln (.7 (iii 0. e (iv. e 0.. (v ln ( e (vi ln ( e (vii ln (. e (viii ln ( 0.7 e.

23 (. Write ech of the following epressions s sums nd differences of logrithms (where possile, without using powers: (i log 0 (ii ln (iii log 0 00 (iv ln e (v log (vi 0 ln ( ( (. (. Write ech of the following s single logrithm: (i (ii (iii (iv log0 log0 log0 z log0 ( log0 z ln ln ln ( ln z.

24 ( Chnging the Suject of Formul (. For ech of the following formule, chnge the suject to the quntit indicted in rckets: (i (ii (iii (iv (v v u t ( t s ( t t ( u s u t s u t t ( t v u s ( s (vi (vii v u s ( u ( (viii t i e ( t (i i t e ( t ( (i ( 0 ( ( 0 ( (ii ( t. k t c0 0 e (. For ech of the following formule, chnge the suject to the quntit indicted in rckets: (i ( (ii ( (iii z ( (iv C C C C C ( C.

25 ( Solution of Equtions (. Solve the following equtions: (i (ii 0 (iii (iv (v (vi 0 0 (vii e 0. 7 (viii e 0. (i e 7 ( e (i ln ( (ii ln ( 0 (iii ln ( 0 (iv ln ( 0. (v 0. (vi 0. 7 (vii log 0 ( 0. (viii log 0 ( 0. (i ( (. Solve the following qudrtic equtions fctoristion, then repet using the qudrtic formul: (i 0 (ii 0 0 (iii 0 0 (iv 7 0.

26 (. The following qudrtic equtions do not hve n rel solutions. Tr solving them the qudrtic formul nd see wht hppens: (i 0 (ii 0. (. Solve the following sets of simultneous equtions: (i (ii (iii 7 0 (iv 0...

27 Answers (. (i (iii (v (vii (i (ii (iv (vi 0 (viii ( 7 (. (i (ii (iii (iv 00 0 (v (vi (vii (viii 0 (i ( 7 0 (. (i ( ( (ii ( ( (iii (iv ( ( ( (v ( ( (vi ( ( (vii ( ( (viii ( ( (i ( ( ( ( ( (. (i ( ( ; ( ( (ii ( ( ; ( ( (. (i (ii (iii (v 7 (iv c d (vi h r

28 (. (i (ii (iii.77 (iv (v. (vi. (vii. (viii 0. (. (i (ii.000 (iii.00 (iv 0.00 (v 0. (vi. (vii. (viii 0.7 (. (i log0 log0 log0 (ii ln ln ln (iii log0 ( (iv ln ( (v log ( 0 (vi [ ln ( ln ( ln ( ] (. (i log 0 z (ii log 0 ( z (iii ( ln (iv ln z 7

29 (. (i t v u (ii t s (iii u s t ( s (iv t t u t (v s v u (vi u v s (vii (i (viii t ln i t ln ( [ log0 ( ] i (i log ( ] (ii [ 0 t k ln 0 c 0 (. (i (ii (iii z (iv z C C C C C (. (i (ii (iii (iv (v, (vi (vii ln ( (viii ln ( (i ln (. 0 ( ln ( (i e 0. 0 (ii e (iii e 7. 0 (iv ( e 0. (v log 0 ( (vi log (.7 ] [ (vii (viii [0 ] (i / Over the pge...

30 (i log0 log [could lso use nturl logs] ( log0 log0. [could lso use nturl logs] (. (i, (ii, (iii [repeted root] (iv, (. Negtive vlues under the squre root sign indicte no rel solutions. Solutions onl possile moving into comple numers. (. (i, (ii, (iii, (iv.,.

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

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

Chapter 6 Solving equations

Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

Exponential and Logarithmic Functions

Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

4.0 5-Minute Review: Rational Functions

mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

CONIC SECTIONS. Chapter 11

CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

Mathematics Higher Level

Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

Section A-4 Rational Expressions: Basic Operations

A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

Pure C4. Revision Notes

Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

Math 135 Circles and Completing the Square Examples

Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

Solving Linear Equations - Formulas

1. Solving Liner Equtions - Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem

Section 7-4 Translation of Axes

62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

Section 5-4 Trigonometric Functions

5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

Sequences and Series

Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

Solutions to Section 1

Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

PhysicsAndMathsTutor.com

C Integrtion Volumes PhsicsAndMthsTutor.com. Using the sustitution cos u, or otherwise, find the ect vlue of d 7 The digrm ove shows sketch of prt of the curve with eqution, <

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

Introduction to Integration Part 2: The Definite Integral

Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the

Math I EB127. Arab Academy For Science & Technology. [Basic and Applied Science Dept.]

Ar Acdem For Science & Technolog [Bsic nd Applied Science Dept] Mth [EB] Anltic Geometr Determinnts Mtrices Sstem of Liner Equtions Curve Fitting Liner Progrmming Mth I EB Sllus for Mthemtics I Course

Integration by Substitution

Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

0.1 Basic Set Theory and Interval Notation

0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b.

5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the -is = nd two verticl lines = nd =. =

A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method

Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN - Vol. - A Note on Complement of Trpezoidl Fuzzy Numers Using the α-cut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

A new algorithm for generating Pythagorean triples

A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

Factoring Polynomials

Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

5.6 POSITIVE INTEGRAL EXPONENTS

54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

Quadratic Equations. Math 99 N1 Chapter 8

Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

Experiment 6: Friction

Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

Double Integrals over General Regions

Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

Vectors 2. 1. Recap of vectors

Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

Ae2 Mathematics : Fourier Series

Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl word-for-word with my lectures which will

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

Two special Right-triangles 1. The

Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The

Simple Nonlinear Graphs

Simple Nonliner Grphs Curriulum Re www.mthletis.om Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle Liner euse their grphs re stright

4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

1.2 The Integers and Rational Numbers

.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

The Quadratic Formula and the Discriminant

9-9 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt

Brief review of prerequisites for ECON4140/4145

1 ECON4140/4145, August 2010 K.S., A.S. Brief review of prerequisites for ECON4140/4145 References: EMEA: K. Sdsæter nd P. Hmmond: Essentil Mthemtics for Economic Anlsis, 3rd ed., FT Prentice Hll, 2008.

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

Variable Dry Run (for Python)

Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

Teaching and Learning Guide 10: Matrices

Teching nd Lerning Guide : Mtrices Teching nd Lerning Guide : Mtrices Tle of Contents Section : Introduction to the guide. Section : Definitions nd Opertions.. The concept of definitions nd opertions.

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

THE RATIONAL NUMBERS CHAPTER

CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

Lesson 10. Parametric Curves

Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +

Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our

Strong acids and bases

Monoprotic Acid-Bse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple

1 Numerical Solution to Quadratic Equations

cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

Sect 8.3 Triangles and Hexagons

13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS)

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470 - COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010

/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed

Net Change and Displacement

mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the