Stage 11 Prompt Sheet

Size: px
Start display at page:

Download "Stage 11 Prompt Sheet"

Transcription

1 Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions in surds dd & sutrct m ± n = (m±n) Exmple 1 + = (+) = Multiply & divide x = Exmple x 1 = 4 = Exmple (4 + )( - ) = = - = Exmple = = 11/ Rtionlise surd denomintors To remove surd from the denomintor multiply the numertor & the denomintor y tht surd Exmple 1 1 = x = 1 1 = = = (Multiply oth top & ottom y 1) (ncel y ) = 11/ lculte with upper & lower ounds If is rounded to nerest x Upper ound = + ½x Lower ound = ½x Exmple: if 1.8 is rounded to 1dp Upper ound = ½(.1) = 1.8 Lower ound = 1.8 ½(.1) = 1.7 lculting using ounds dding ounds Mximum = Upper + upper Minimum = Lower + lower Sutrcting ounds Mximum = Upper - lower Minimum = Lower upper Multiplying Mximum = Upper x upper Minimum = Lower x lower Dividing Mximum = Upper lower Minimum = Lower upper 11/4 lgeric frctions dding & sutrcting lgeric frctions Exmple 1 x + + x (common denomintor is 1) 4 = (x + ) + 4(x ) 1 1 = x + + 4x 1 = 7x 11 1 Exmple - (common denomintor is (x+1)(x+) (x + 1) (x + ) = (x + ) (x + 1) (x+1)(x+) = x + 1 x (x+1)(x+) = x + 7 (x+1)(x+)

2 Simplifying lgeric frctions Exmple: x + x + 1 (fctorise) x -x 4 = (x + 1)(x + 1) (cncel) (x 4)(x + 1) = (x + 1) (x 4) 11/ Solve equtions with frctions x + 4 = 1 ommon denomintor (x-)(x+1) x x + 1 x(x+1)+ 4(x-) =1 (x-)(x+1) x + x + 8x - 1 =1 (x-)(x+1) x + x 1 =1(x-)(x+1) x + x -1 = x x - (-x from oth sides) x -1 = x x - (-x from ech side) -1 = x 1x - (+1 to ech side) = x 1x + (fctorise) (x + )(x + 1) = x = - or x = -1 11/ Solve qudrtic eqution y fctoring ut eqution in form x + x + c = x =x + x x = Fctorise the left hnd side (x )(x + 1) = Equte ech fctor to zero x = or x + 1 = x =. or x = -1 11/7 Interpret expressions s functions function is rule tht tkes numers s inputs nd ssigns to ech input exctly one numer s output. The output is function of the input. Simple expressions s functions Exmple: y = x + f(x) = x + (Replce y with f of x ) f(4) = (4) + = 17 Inverse function This is the reverse process tht tkes you ck to the originl vlues We write the inverse of f(x) s f -1 (x) Exmple: if f(x) = x + We sy y = x + x + = y x = y Rerrnge in terms of x f -1 (x) = x hnge y ck to x x Inverse function using flow digrm x x f -1 (x) = x + x- omposite function pplying one function to the results of nother Exmple 1: To comine these two functions f(x) = x nd g(x) = x - 1 gf(x) mens g(x) = (x) 1 = x - 1 Replce x in the function g(x) with x x+ fg(x) mens f(x - 1) = (x - 1) = x Replce x in the function f(x) with x-1 Exmple : To evlute the composition of functions If f(x) = x - 1 nd g(x) = x +, work out fg() Find g() = ()+ = Then f() = -1 = -1 - x This is the input into the function This is the output of the function

3 11/8 Deduce turning point of qudrtic functions y completing the squre To complete the squre x + 4x + 4 = (x+) (perfect squre) x + 4x + = (x+) 1 (completed squre form) Rules to complete the squre Exmple 1: x +4x+ (x+) x in rcket with ½ of +4 (x+) put squred sign on rcket (x+) - 4 sut the squre of the new end numer (x+) -4+ dd /sutrct the originl end numer (x+) -1 simplify Exmple : x + x + divide ll terms y (x + x +.) ((x + 1.) -.) + (x + 1.) (x+1.) +. Deduce turning point of qudrtic Exmple: y = x +4x+ x + 4x + = (x+) 1 complete the squre Turning point is (-,-1) identify the coordintes / Solve qudrtic eqution y completing the squre 11/1 Solve qudrtic equtions y formul x + x + c = Formul (to lern): x = - ± 4c Exmple To solve: x + 4x = x = - ± 4c x = -4 ± (-4) 4()(-) () = -4± 1+4 = -4± 4 x = OR -4-4 x =.(dp) OR -1.7 (dp) 11/11 Solve qudrtic inequlities Exmple: x + x > 1 x + x 1 = Replce inequlity symol with = (x - )(x + ) = Fctorise x = nd x = - Solve repre the numer line Test x=- in inequlity (-) +(-) =18 18>1 TRUE Test x= in inequlity () +() = >1 FLSE - Solution set on numer line = = 4 c = - These re EXT vlues Test x= in inequlity () +() =18 18>1 TRUE Mke the coefficient of x squre x + 1x + = (mult y ) 4x + x + 1 = dd numer to oth sides to mke perfect squre 4x + x + 1 = (dd 1) 4x + x + = 1 (x + ) = 1 Squre root oth sides x + = ± 1 (- from oth sides) x =- + 1 OR x = -. OR (dp) - Set nottion: x<- nd x> Solution on grph

4 11/1 Rerrnge more complex formule (inc where suject ppers twice) ollect ll the terms with the new suject Fctorise to isolte the new suject 11/14 Grphs of trigonometric functions LERN THE SHES OF THE GRHS Grph of y=sin x Exmple: to mke the new suject = 7 (multiply oth sides y ( ) - ( ) = 7 (Expnd the rcket) = 7 (ollect terms in new suject) 7 + = (+ to oth sides) 7 + = + (fctorise to isolte ) (7 + ) = + ( (7 + ) oth sides) (7 + ) (7 + ) = + (7 + ) Grph y = cos x -1 sin x 1 11/1 Exponentil grphs The grph of the exponentil function is: y = k x Exmple: y = x It hs no mximum or minimum point It crosses the y-xis t (,1) It never crosses the x-xis Grph y = tn x -1 cos x 1 tn x is undefined t, 7... Solutions to trigonometric equtions cn e found on the clcultor nd y using the symmetry of these grphs Exmple: If sin x =. x =, 1, (See the solutions on sin grph ove or from clcultor)

5 11/1 Trnsformtion of functions For ny grph y = f(x) LERN the trnsformtions y=f(x) ± Trnsltion ( ) moves up(+)/down(-) ± y=f(x± ) y=-f(x) y=f(-x) Trnsltion ( ± ) moves right(-)/left(+) Reflection in the x-xis (horizontlly) Reflection in the y-xis (verticlly) 11/17 Grph of the circle The grph of circle is of the form: x + y = r where r is the rdius nd the centre is (,) 11/1 Grdient of curve Exmple: To find grdient t point x=4 Drw tngent t x=4 to the curve ick points on the tngent (x 1,y 1) & (x,y ) Work out rise & run or use y y 1 x x 1 heck if positive or negtive run Rise = 4 Run =.8 Grdient = 4.8 = 4 8 = Its slope is positive 11/1 re under curve Split into trpeziums Find the sum of their res 14 rise 1 tngent 1 4 re = ½ x 1 x( +( ) + 1) = ½ x 1 x ( ) = 4 units The curve is concve, so it will e slight over-estimte onvex curves give n over-estimte This circle of rdius nd centre (,) The grph of this circle is x + y = x + y = 11/17 Eqution of tngent to circle Eqution of tngent: y y 1 = m(x x 1) m= grdient of tngent t the point It is perpendiculr to the rdius so m rdius x m tngent = -1 (x 1, y 1) = point on circle where tngent meets Exmple entre (,) (,1) Grdient of rdius = ½ Grdient of tngent = - (m rdius x m tngent = -1) y y 1 = m(x x 1) y 1 = -(x - ) y 1 = -x + 4 y =-x + (eqution of tngent)

6 Wter level in cm Wter level in cm 11/1 Solve simultneous equtions~one liner, one qudrtic lgericlly Rewrite the liner with one letter in terms of the other Sustitute the liner into the qudrtic Solve the qudrtic y fctorising Exmple: To solve y=x- nd y=x -x- Sustitute y=x- into y=x -x- x- = x -x- x -x 4 = (fctorise) (x 4)(x + 1) = x = 4 or x = -1 when x= 4, y = (4)- = when x= -1, y = (-1)- = -4 See points of intersection of grphs for solutions y=x- 7 Grph of y=x -x Solutions re: (4, ) nd (-1, -4) 11/ Interpret grdient of tngent & chord The instntneous rte of chnge of quntity t given time is the grdient of the tngent to the grph t tht time e.g. min Time in minutes Instntneous rte of rise of wter level t min 4cm. min.7cm/min The verge rte of chnge is the grdient of the chord etween the two given times Time in minutes verge rte of chnge etween & 4min 1cm min = cm/min

7 11/ Use itertion to solve equtions Itertion mens repeting process. Ech repetition is clled itertion. The result of n itertion is used s the strting point of the next itertion e.g. x x + 1 = Write x in terms of x (here is one wy) x = x + 1 x = x + 1 Then write s the itertion formul x n+1 = x n + 1 (n=previous term; n+1=next term) hoose vlue for x 1 (it my e given/found from grph e.g. x 1 =. Find x y sustituting x1 into the itertion formul x = (.) + 1 =. Find x y sustituting x into the itertion formul X = (.) + 1 =.4... ontinue until nswer converges to given numer of d.p. (in this cse.(dp)) Quick method with clcultor.= (NS +1) = = = etc till it converges 11/ re of tringle height not known Exmple re = ½ sin re = ½ c sin re = ½ c sin Formul NOT provided 11/ ircle Theorem proofs ngle in semicircle = Drw in rdius s shown ngle t centre = x ngle t circumference Drw in rdius s shown ngles in the sme segment re equl Drw ngle t centre from the chord s shown x y z c In the old tringle: + = 18 ( ) => + = x=18 - y=18 - z= - (x + y) =-(--) = + =( + ) Opposite ngles of cyclic qudrilterl = 18 Drw in two rdii s shown x y c= (lredy proved) c= (lredy proved) = = x= (lredy proved) y= (lredy proved) x + y = + = + = re = ½ sin = ½ x 8 x x sin8 = 14.8 cm (1dp) =8cm =cm 8 ngle etween tngent nd its chord is equl to the ngle in the 'lternte segment' Drw tngent nd ngle in semicircle s shown + = x + = = x x

8 11/ ircle Theorem proofs (continued) Equl tngents from point to the circumference rdius, perpendiculr to chord isects the chord 11/4 Use circle theorems See Stge 1 rompt Sheet 11/ ythgors Theorem in D 1 cm 4 cm cm Q Exmple: Identify the tringle in the D shpe contining the unknown side Q R Use ythgors in ΔRQ to find RQ RQ = ( + 4 ) = cm 4cm R O O Q M In ΔO & ΔO O =O (rdii) O is common ngles etween tngent & rdius = ΔO & ΔO re congruent (RHS) = In ΔOM & ΔQM O =OQ (rdii) OM is common ngles = ΔOM & ΔQM re congruent (RHS) QM = M Leve in surd form when needing the EXT vlue 11/ Trigonometry in D Identify the tringle in the D shpe contining the unknown ngle QR Use ythgors in ΔRQ to find RQ RQ = ( + 4 ) = cm Use Trigonometry in ΔRQ to find QR Tn QR = 1 =.4 Tn -1 QR = /7 Sine Rule (non-right ngled tringles) Use SINE RULE when given: two sides nd non-included ngle ny two ngles nd one side To find n ngle, use: sin = sin = sin c Exmple: To find ngle =7.1cm 1cm sin = sin c sin = sin 7.1 sin = sin x 7.1 sin = = sin -1 (.44...) = 8. (1dp) R =cm cm Q Formul NOT provided Use ythgors in ΔRQ to find Q Q= (1 + ) = 1cm OR = 1 = 1cm 1cm R Q cm Q cm

9 11/7 Sine Rule (continued) To find side use: = = c sin sin sin Exmple: To find side Formul NOT provided To find side given sides & included ngle use: = + c c cos = + c c cos c = + cos Exmple: To find side Formul NOT provided =cm c=4.cm 11 =.7cm 4 = sin sin = sin sin 4 = x sin sin 4 =.8 cm (1dp) = + c c cos = x.7x4. cos11 = 41. =.48(dp) 11/ Vectors =? 11/8 osine Rule (non-right ngled tringles) Use OSINE RULE when given: sides sides nd the included ngle To find n ngle, given sides use: Formul NOT provided Vector nottion This vector cn e written ( ) or or Exmple: To find ngle c=cm cos = + c c cos = + c c cos = + c =8cm =cm vector hs mgnitude(length) & direction(shown y n rrow) Mgnitude cn e found y ythgors Theorem = + = 1 =. prllel vector with sme mgnitude ut opposite direction cos = + c cos = 8 + x8x cos = = cos -1 ( ) = 8. (1dp) Q is equl in length to Vector Q direction so we sy: Q = - ut opposite in

10 11/ Vectors (continued) prllel vector with sme direction ut different mgnitude Vector sutrction = ( ) = ( ) D - Vector D is twice (sclr ) the mgnitude ut sme direction so we sy: D = negtive sclr would reverse the direction Vector ddition dding grphiclly, the vectors go nose to til = ( ) = ( ) + - The comintion of these two vectors: - = = - =( ) - ( ) = ( ) is clled the RESULTNT vector The sum of vectors M N = + M + M The comintion of these two vectors: = + + = = ( ) + ( ) = ( 1 4 ) The vector is equl to the sum of these vectors or it could e different route: = N + N go vi N Strt point End point olliner points (in sme stright line) To prove points re colliner: hoose two line segments, e.g. nd. rove tht they hve: (i) ommon direction (equl grdients) nd (ii) common point (e.g. )

11 frequency density (no. ooks per ) Frequency density 11/ Histogrms lss intervls re not equl Verticl xis is the frequency density Frequency is re of r not the height Frequency = clss width x frequency density Frequency density = frequency clss width To drw histogrm lculte the frequency density Exmple rice () in ( ) f = width x height(fd) < x 8 = 4 < 1 x 1 = 1 < 1 x = < 4 x = 4 ge (x yers) lss width f Frequency density < x 8 8 = 1.4 < x 1 1 =.4 < x = 4 < x = 1. Scle the frequency density xis up to.4 Histogrm of ge groups ge in yers To interpret histogrm Histogrm to show the numer of ooks sold rice () in

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Maths. Self Check Booklet. visit   for a wealth of free online maths resources at all levels from S1 to S6 Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

More information

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+ .1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

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

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

TABLE OF CONTENTS 3 CHAPTER 1

TABLE OF CONTENTS 3 CHAPTER 1 TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER

More information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the

More information

Loudoun Valley High School Calculus Summertime Fun Packet

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

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the

More information

Summer Work Packet for MPH Math Classes

Summer Work Packet for MPH Math Classes Summer Work Pcket for MPH Mth Clsses Students going into Pre-clculus AC Sept. 018 Nme: This pcket is designed to help students sty current with their mth skills. Ech mth clss expects certin level of number

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors: Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

More information

Preparation for A Level Wadebridge School

Preparation for A Level Wadebridge School Preprtion for A Level Mths @ Wdebridge School Bridging the gp between GCSE nd A Level Nme: CONTENTS Chpter Removing brckets pge Chpter Liner equtions Chpter Simultneous equtions 6 Chpter Fctorising 7 Chpter

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

IMPOSSIBLE NAVIGATION

IMPOSSIBLE NAVIGATION Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

THE DISCRIMINANT & ITS APPLICATIONS

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

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Summary Information and Formulae MTH109 College Algebra

Summary Information and Formulae MTH109 College Algebra Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

More information

MATHEMATICS AND STATISTICS 1.2

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

More information

Number systems: the Real Number System

Number systems: the Real Number System Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset

More information

AP Calculus AB Summer Packet

AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (

More information

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

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

More information

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0) Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.

A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence. Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12. Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k

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

TO: Next Year s AP Calculus Students

TO: Next Year s AP Calculus Students TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

More information

Chapter 6 Techniques of Integration

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

More information

SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day!

SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day! SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesdy After Lor Dy! This summer ssignment is designed to prepre you for Functions/Trigonometry. Nothing on the summer ssignment is new. Everything

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

Background knowledge

Background knowledge Bckground knowledge Contents: A B C D E F G H I J K L Surds nd rdicls Scientific nottion (stndrd form) Numer systems nd set nottion Algeric simplifiction Liner equtions nd inequlities Modulus or solute

More information

Precalculus Spring 2017

Precalculus Spring 2017 Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

More information

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

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

More information

I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics

I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics AAT-A Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #17-45 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

More information

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

( β ) touches the x-axis if = 1

( β ) touches the x-axis if = 1 Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without

More information

Trigonometric Functions

Trigonometric Functions Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

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

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

More information

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically. Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

More information

Mathematics. toughest areas of the 2017 exam papers. Edexcel GCSE (9-1) Higher. guided exam support on the top 10 toughest

Mathematics. toughest areas of the 2017 exam papers. Edexcel GCSE (9-1) Higher. guided exam support on the top 10 toughest toughest res of the 07 em ppers Edecel GSE (9-) Mthemtics Higher guided em support on the top 0 toughest res of the 07 Higher tier ppers from Top 0 Edecel GSE Mths Higher tier 07 Help our students ctch

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

Section 7.1 Area of a Region Between Two Curves

Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

More information

5.1 How do we Measure Distance Traveled given Velocity? Student Notes

5.1 How do we Measure Distance Traveled given Velocity? Student Notes . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

More information

I look forward to seeing you in August. Have a wonderful rest of your summer!

I look forward to seeing you in August. Have a wonderful rest of your summer! PHYSICS Summer Homework 016 Sister Dominic, OP M First & Lst Nme: Due Dte: Der Physics Students, Welcome to Physics where we get to study how our universe works!! In order to do this, we need to effectively

More information

Section 3.1: Exponent Properties

Section 3.1: Exponent Properties Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted

More information

Integration Techniques

Integration Techniques Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

PHYS 1114, Lecture 1, January 18 Contents:

PHYS 1114, Lecture 1, January 18 Contents: PHYS 1114, Lecture 1, Jnury 18 Contents: 1 Discussed Syllus (four pges). The syllus is the most importnt document. You should purchse the ExpertTA Access Code nd the L Mnul soon! 2 Reviewed Alger nd Strted

More information

Coordinate geometry and vectors

Coordinate geometry and vectors MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2

More information

2008 Mathematical Methods (CAS) GA 3: Examination 2

2008 Mathematical Methods (CAS) GA 3: Examination 2 Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information