TABLE OF CONTENTS 3 CHAPTER 1


 Scarlett Robbins
 11 months ago
 Views:
Transcription
1
2 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 7 Logrithmic & Exponentil Functions 5 CHAPTER 8 Stright Line Grphs 5 CHAPTER 9 Circulr Mesure 5 CHAPTER 10 Trigonometry
3 CHAPTER 11 Permuttions & Combintions CHAPTER 12 Binomil Expnsions CHAPTER 13 Vectors in 2 Dimensions CHAPTER 14 Mtrices CHAPTER 15 Differentition & Integrtion
4 1. SET LANGUAGE & NOTATION A welldefined collection of objects is clled set nd ech object is clled member or element of the set A set is denoted by cpitl letter nd is expressed by: o Listing its elements, e.g. V = {, e, i, o, u} o A set builder nottion R set of rel numbers R + set of positive rel numbers N set of nturl numbers Z set of integers Z + set of positive integers o e.g. {x: x is prime number nd x < 30} For ny finite set P, n(p) denotes the number of elements in P A null or empty set is denote by { } or For ny two sets P nd Q: o P = Q if they hve the sme elements o P Q if x P x Q o P Q = {x: x P nd x Q} o P Q = then P nd Q re disjoint sets o P Q = {x: x P or x Q} For ny set P nd universl set ξ o P ξ nd 0 n(p) n(ξ) o P = {x: x ξ nd x P} o P P = o P P = ξ 2. FUNCTIONS Onetoone functions: ech x vlue mps to one distinct y vlue e.g. f(x) = 3x 1 Mnytoone functions: there re some f(x) vlues which re generted by more thn one x vlue e.g. f(x) = x 2 2x + 3 Domin = x vlues Rnge = y vlues Nottion: f(x) cn lso be written s f: x To find rnge: o Complete the squre x 2 2x + 3 (x 1) o Work out min/mx point Minimum point = (1,2) ll y vlues re greter thn or equl to 2. f(x) 2 Onetomny functions do not exist Domin of g(x) = Rnge of g 1 (x) Solving functions: o f(2): substitute x = 2 nd solve for f(x) o fg(x): substitute x = g(x) o f 1 (x): let y = f(x) nd mke x the subject Trnsformtion of grphs: o f( x): reflection in the yxis o f(x): reflection in the xxis o f(x) + : trnsltion of units prllel to yxis o f(x + ): trnsltion of units prllel to xxis o f(x): stretch, scle fctor 1 prllel to xxis o f(x): stretch, scle fctor prllel to yxis Modulus function: o Denoted by f(x) o Modulus of number is its bsolute vlue o Never goes below xxis o Mkes negtive grph into positive by reflecting negtive prt into xxis Solving modulus function: o Sketch grphs nd find points of intersection o Squre the eqution nd solve qudrtic Reltionship of function nd its inverse: o The grph of the inverse of function is the reflection of grph of the function in y=x 3. QUADRATIC FUNCTIONS To sketch y = x 2 + bx + c 0 o Use the turning point: Express y = x 2 + bx + c s y = (x h) 2 + k by completing the squre x 2 + nx (x + n 2 2 ) ( n 2 2 ) (x + n) 2 + k Where the vertex is ( n, k) > 0 ushped minimum point < 0 nshped mximum point Find the xintercept: o Fctorize or use formul Type of root by clculting discriminnt b 2 4c o If b 2 4c = 0, rel nd equl roots o If b 2 4c > 0, rel nd distinct roots o If b 2 4c < 0, no rel roots PAGE 3 OF 7
5 Intersections of line nd curve: if the simultneous equtions of the line nd curve leds to simultneous eqution then: o If b 2 4c = 0, line is tngent to the curve o If b 2 4c > 0, line meets curve in two points o If b 2 4c < 0, line does not meet curve Qudrtic inequlity: o (x d)(x β) < 0 d < x < β o (x d)(x β) > 0 x < d or x > β 4. INDICES & SURDS Definitions: o for > 0 nd positive integers p nd q 0 = 1 1 p p = p = 1 p p q = ( Rules: o for > 0, b > 0 nd rtionl numbers m nd n m n = m+n n b n = (b) n m n n = m n n b n = ( b ) ( m ) n = mn 5. FACTORS OF POLYNOMIALS To find unknowns in given identity o Substitute suitble vlues of x OR o Equlize the given coefficients of like powers of x Fctor Theorem: If (x t) is fctor of the function p(x) then p(t) = 0 Reminder Theorem: If function f(x) is divided by (x t) then: Reminder = f(t) The formul for reminder theorem: Dividend = Divisor Quotient + Reminder 6. SIMULTANEOUS EQUATIONS Simultneous liner equtions cn be solved either by substitution or elimintion Simultneous liner nd nonliner equtions re generlly solved by substitution s follows: o Step 1: obtin n eqution in one unknown & solve it o Step 2: substitute the results from step 1 into the liner eqution to find the other unknown The points of intersection of two grphs re given by the solution of their simultneous equtions p ) q PAGE 4 OF 7 7. LOGARITHMIC & EXPONENTIAL FUNCTIONS Definition o for > 0 nd 1 y = x x = log y For log y to be defined y > 0 nd > 0, 1 When the logrithms re defined log 1 = 0 log b + log c log bc b log = 1 log b log c log c log b log b log b n n log b log When solving logrithmic equtions, check solution with originl eqution nd discrd ny solutions tht cuses logrithm to be undefined Solution of x = b where 1, 0, 1 If b cn be esily written s n, then x = n x = n Otherwise tke logrithms on both sides, i.e. log x = log b nd so x = log log 10 ln log e Logrithmic & Exponentil Grphs log b log 8. STRAIGHT LINE GRAPHS Eqution of stright line: y = mx + c y y 1 = m(x x 1 ) Grdient: m = y 2 y 1 x 2 x 1 Length of line segment: Length = (x 2 x 1 ) 2 + (y 2 y 1 ) 2
6 Midpoint of line segment: ( x 1 + x 2 2, y 1 + y 2 ) 2 Prllelogrm: o ABCD is prllelogrm digonls AC nd BD hve common midpoint o Specil prllelogrms = rhombuses, squres, rectngles Specil grdients: o Prllel lines: m 1 = m 2 o Perpendiculr lines: m 1 m 2 = 1 Perpendiculr bisector: line psses through midpoint To work out point of intersection of two lines/curves, solve equtions simultneously 9. CIRCULAR MEASURE Rdin mesure: π = 180 2π = 360 Degree to Rd = π 180 Arc length: Are of sector: s = rθ A = 1 2 r2 θ Rd to Degree = 180 π TANGENT CURVE CAST DIAGRAM Trigonometric rtios: sec θ = 1 cosec θ = 1 cot θ = 1 cos θ sin θ tn θ Trigonometric identities: sin θ tn θ = sin 2 θ + cos 2 θ = 1 cos θ cot 2 θ + 1 = cosec 2 θ tn 2 θ + 1 = sec 2 θ Sketching trigonometric grphs: 10. TRIGONOMETRY Trigonometric rtio of specil ngles: SINE CURVE COSINE CURVE 11. PERMUTATIONS & COMBINATIONS Bsic Counting Principle: to find the number of wys of performing severl tsks in succession, multiply the number of wys in which ech tsk cn be performed: e.g Fctoril: n! = n (n 1) (n 2) o NOTE: 0! = 1 Permuttions: o The number of ordered rrngements of r objects tken from n unlike objects is: o Order mtters Combintions: n P r = n! (n r)! o The number of wys of selecting r objects from n unlike objects is: n n! C r = r! (n r)! Order does not mtter PAGE 5 OF 7
7 12. BINOMIAL EXPANSIONS The binomil theorem llows expnsion of ny expression in the form ( + b) n (x + y) n n = C 0 x n n + C 1 x n 1 n y + C 2 x n 2 y 2 n + + C n y n e.g. Expnd (2x 1) 4 (2x 1) 4 4 = C 0 (2x) C 1 (2x) 3 ( 1) 4 + C 2 4 (2x) 2 ( 1) 2 + C 3 (2x) ( 1) 3 + C 4 ( 1) 4 = 1(2x) 4 + 4(2x) 3 ( 1) +6(2x) 2 ( 1) 2 + 4(2x) ( 1) 3 + 1( 1) 4 = 16x 4 32x x 2 8x + 1 The powers of x re in descending order 13. VECTORS IN 2 DIMENSIONS Position vector: position of point reltive to origin, OP Forms of vector: ( b ) AB p i bj Prllel vectors: sme direction but different mgnitude Generlly, AB = OB OA Mgnitude = i 2 + j 2 Unit vectors: vectors of mgnitude 1 o Exmples: consider vector AB AB = 2i + 3j AB = 13 Unit vector = 1 (2i + 3j) 13 Colliner vectors: vectors on the sme line Dot product: (i + bj). (ci + dj) = (ci + bdj) Angle between two diverging vectors:. b cos A = b Reltive Velocity Motion in the wter: V w = true velocity of wter V P/W = velocity of P reltive to W still wter Course tken by P is direction of V P/W Motion in the ir: V w = true velocity of wind or ir V P/W = velocity of P reltive to W still wind/ir Course tke by P is direction of V P/W V P/Q = V P V Q MATRICES Order of mtrix: mtrix with m rows nd n columns, Order = m n Adding/subtrcting mtrices: dd/subtrct ech corresponding element Sclr multipliction: to multiply mtrix by k, multiply ech element by k Multiplying mtrices: multiply row by column Identity mtric: I = ( 1 0 ) IA = A nd AI = I 0 1 Clculting the determinnt: A = ( b ) A = (d bc) c d Inverse of 2 by 2 mtrix: o Switch leding digonl, negte secondry digonl o Multiply by 1 A A = ( b c d ) A 1 = 1 ( d b d bc c ) A 1 A = AA 1 = I Solving simultneous liner equtions by mtrix method: x + by = h cx + = k Eqution cn be written s: b ( c d ) (x y ) = (h k ) Rerrnge it nd solve: ( x y ) = 1 d bc ( d b c ) (h k ) For mtrix to give unique solutions: b ( c d ) DIFFERENTIATION & INTEGRATION 15.1 Differentition FUNCTION 1ST DERIVATIVE 2 ND DERIVATIVE y = x n = d 2 y nxn 1 = n(n 1)xn 2 2 INCREASING FUNCTION DECREASING FUNCTION > 0 < 0 Sttionry point: equte first derivtive to zero = 0 PAGE 6 OF 7
8 2 nd Derivtive: finds nture of the sttionry point o If vlue +ve, min. point negtive sttionry point o If vlue ve, mx. point positive sttionry point Chin rule: = du du Product rule: Quotient rule: = v Specil Differentils dv du = u + v du dv u v 2 of sin x = cos x of cos x = sin x of tn x = sec2 x of ex+b = e x+b of ln x = 1 x of ln(f(x)) = f (x) f(x) Relted rtes of chnge: o If x nd y re relted by the eqution y = f(x), then the rtes of chnge nd re relted by: dt dt dt = dt Smll chnges: o If y = f(x) nd smll chnge δx in x cuses smll chnge δy in y, then δy ( ) δx x=k Logs Algebr Trig e To find re under the grph (curve nd xxis): o Integrte curve o Substitute boundries of x o Subtrct one from nother (ignore c) y c To find volume under the grph (curve nd xxis): o Squre the function o Integrte nd substitute o Multiply by π d πy 2 c To find re/volume between curve nd yxis: o Mke x subject of the formul o Follow bove method using yvlues insted of xvlues Specil Integrls d sin(x + b) = 1 cos(x + b) + c cos(x + b) = 1 sin(x + b) + c sec 2 (x + b) = 1 tn(x + b) + c 1 x + b = 1 ln x + b + c 15.3 Kinemtics e x+b = 1 ex+b + c 15.2 Integrtion xn+1 x n = (n + 1) + c (x + b) n (x + b)n+1 = + c (n + 1) Definite integrl: substitute coordintes/vlues & find c Integrting by prts: u dv du = uv v o Wht to mke u: LATE Prticle t instntneous rest, v = 0 Mximum displcement from origin, v = 0 Mximum velocity, = 0 PAGE 7 OF 7
9
Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationPolynomials 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 informationEdexcel 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 information03 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
APDF Wtermrk DEMO: Purchse from www.apdf.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 information03 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
APDF Wtermrk DEMO: Purchse from www.apdf.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 informationChapter 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 informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More informationThomas 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 informationA 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 informationTABLE OF CONTENTS 2 CHAPTER 1
TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationSOLUTIONS 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 informationLecture 0. MATH REVIEW for ECE : LINEAR CIRCUIT ANALYSIS II
Lecture 0 MATH REVIEW for ECE 000 : LINEAR CIRCUIT ANALYSIS II Aung Kyi Sn Grdute Lecturer School of Electricl nd Computer Engineering Purdue University Summer 014 Lecture 0 : Mth Review Lecture 0 is intended
More informationx 2 + n(n 1)(n 2) x 3 +
Core 4 Module Revision Sheet The C4 exm is hour 30 minutes long nd is in two sections Section A 36 mrks 5 7 short questions worth t most 8 mrks ech Section B 36 mrks questions worth bout 8 mrks ech You
More informationMATH 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 informationPrerequisite 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 informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationSummary 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 informationTHE 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 informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationSCHEME OF WORK FOR IB MATHS STANDARD LEVEL
Snnrpsgymnsiet Lott Hydén Mthemtics, Stndrd Level Curriculum SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Min resource: Mthemtics for the interntionl student, Mthemtics SL, Hese PART 1 Sequences nd Series
More informationHigher 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 informationAB 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 informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationStage 11 Prompt Sheet
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
More information( ) 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).
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 informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationAlg. 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 information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
More informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationMath 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 informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationLesson 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 informationA 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 informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry and basic calculus
ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems
More informationSUMMER 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 informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationChapter 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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationMASTER 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/MCUPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationSummer Work Packet for MPH Math Classes
Summer Work Pcket for MPH Mth Clsses Students going into Preclculus 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 informationS56 (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 informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More informationBRIEF NOTES ADDITIONAL MATHEMATICS FORM
BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht
More information( ) Same as above but m = f x = f x  symmetric to yaxis. 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 informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel 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 informationChapter 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 informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationMORE 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 informationFinal Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed
Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y)
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationThe 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 informationR(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 informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationLoudoun 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 informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationIntroduction and Review
Chpter 6A Notes Pge of Introuction n Review Derivtives y = f(x) y x = f (x) Evlute erivtive t x = : y = x x= f f(+h) f() () = lim h h Geometric Interprettion: see figure slope of the line tngent to f t
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationTHE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES
THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationBridging 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 informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationHAND BOOK OF MATHEMATICS (Definitions and Formulae) CLASS 12 SUBJECT: MATHEMATICS
HAND BOOK OF MATHEMATICS (Definitions nd Formule) CLASS 12 SUBJECT: MATHEMATICS D.SREENIVASULU PGT(Mthemtics) KENDRIYA VIDYALAYA D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA.
More informationThe Product Rule state that if f and g are differentiable functions, then
Chpter 6 Techniques of Integrtion 6. Integrtion by Prts Every differentition rule hs corresponding integrtion rule. For instnce, the Substitution Rule for integrtion corresponds to the Chin Rule for differentition.
More informationJEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS
JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODED. Let, b nd c be three nonzero vectors such tht no two of them re colliner nd, b c b c. If is the ngle
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More informationLinear 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. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos(  1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin(  1 ) = π 2 6 2 6 Cn you do similr problems?
More informationMA 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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More information( ) 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 informationPrecalculus 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 informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More information