Thomas Whitham Sixth Form


 Lynette Black
 9 months ago
 Views:
Transcription
1 Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors
2 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 Acos B sin Asin B A B cos Acos B sin Asin B tn A tn B tn( A B) tn Atn B tn A tn B tn( A B) tn Atn B Emple Solve the eqution sin cos cos sin for sin cos cos sin sin cos cos sin sin S T A C PV = Emple Given A B 5 nd tn A find without using clcultor the vlue of tn A B tn A tn B tn Atn B tn B t where t tn B. t
3 t t t t tn B Doule ngle formule sin sin cos cos cos sin sin cos Pge tn importnt rerrngements tn tn sin cos cos cos [see the rticle on integrtion] Emple Let Find the vlue of 5 tn without using clcultor. tn t t t t tn tn t t 8 t tn Emple Solve the eqution cos sin for cos sin sin sin sin sin sin sin sin S T A C
4 5 7 Pge R/ Methods where nd re positive constnts there re vlues of the positive R nd the cute ngle for which sin cos Rsin cos sin R cos sin cos Rsin cos sin R cos Emple Epress sin cos in the form R cos Identity method cos sin R cos cos Rcos sin Rsin R R cos R Rcos cos Rsin sin Rsin R cos cos sin Rcos cos Rsin sin sin R R tn cos sin cos sin cos in the form Rsin Emple Epress Alterntive method sin cos 5sin cos sin cos cos sin 5sin sin cos 5sin 5... tn Where 5
5 Pge Emple Solve the eqution sin cos for Answers to the nerest degree. LHS sin cos Rsin tn 8. Rsin cos R cos sin Rcos sin Rsin cos R 9 R Using the identity method you will find tht sin cos for sin 8. for sin S T A C PV=9. Geometry Prmetrics () Equtions of the form y f () or f ( y) re clled crtesin equtions. Emple y e y 5 () Equtions of the form f ( t) y g( t) where t is third vrile re clled prmetric equtions; t is the prmeter They define curve which hs points with coordintes of the form f ( t) g( t). As t vries the curve is defined.
6 Pge 5 () It is possile in some cses to otin the crtesin eqution of curve from prmetric equtions. This involves eliminting the prmeter etween the equtions. Emple A curve hs prmetric coordintes t t the crtesin eqution nd sketch the grph. The crtesin equtions re t y t From the first eqution t Sustitute into the second eqution y This is crtesin eqution of prol with verte ( ) Check point ( ) y. Find Since Emple A curve hs prmetric equtions sin sin nd to give y cos for Find the crtesin eqution of the curve nd sketch the grph. cos y use the identity sin cos y y ( ) Recognise this s the eqution of circle centre nd rdius
7 ecept tht for semi circle. Pge sin will e positive only. Hence the curve is y r = ( ) () In mny cses it isn t possile to otin coherent crtesin eqution Emple (5) Prmetric differentition Where t is prmeter t t y t t crtesin pproch. Applictions will e to tngents nd normls Emple doesn t respond well to Otin the eqution of the tngent to the curve dy d t t y t t t t y t t dy dt d dt t the point where t = d dt dy t t t dt putting t = grd t point ( ) Tngent y y dy d t t t
8 () Intersection prolems Pge 7 Emple Find the points t which the line y cuts the curve defined y prmetric coordintes t t The technique here is to sustitute prmetric coordintes y t into the eqution of the line. For intersections t t t t t t t Points re nd t Emple Refer to the emple in (5). The tngent y cuts the curve gin t point P. Find the coordintes of P For P t t t t t t t Since t = is doule root of this eqution (tngent t t = ) the cuic redily fctorises t At P 5 t P t y inspection Vector Geometry A collection of importnt results. k nd re prllel 8. ˆ where â is vector of unit mgnitude in the direction of. This cn e otherwise written s ˆ nd ˆ
9 Pge 8. P With n ssigned origin O the position of ny point P in p spce is specified uniquely y OP p nd is clled the O A position vector of P. O B AB O The point P dividing AB in the rtio m : n hs A m n m OP P m n n m B Proof: OP OA AP AB m n m m n m n m m n n m m n The midpoint of AB hs position vector m m For midpoint m n OP m
10 Pge 9. P The vector eqution of the line through A d O 5. In D spce the direction of d is given y r d A in position vector fied fied of ny point on point sclr direction the line i nd j re unit vectors in the directions nd y respectively. prmeter Any vector in this D spce cn e epressed in the form = component = y component Given v The direction rtios of v re :. v i j. In D spce i j nd k re unit vectors in the directions y nd z respectively. Any vector in this spce cn e written in the form v i j ck c r v y c j i
11 Pge The direction rtios re : : c NB A test for prllel vectors is tht their direction rtios should e the sme or reduce to the sme. 7. The sclr product of two vectors nd is written. nd is defined s:. cos where is the ngle etween nd Notice the directions of nd in the sclr product! Very importnt. nd re perpendiculr! Rememer tht c... c etc. where. my e seen ut is to e voided! Very importnt is the sclr product in component form. i.e. given i j k nd i j k.. 8. Pirs of lines in D spce r d nd r d my e (i) Prllel in which cse d kd (ii) Intersecting in which cse there re vlues of nd for which
12 (iii) Emple d d Pge Skew when neither prllel nor intersecting. Skew lines hve mutully perpendiculr line which will give closest distnce prt. Points A nd B hve position vectors nd reltive to the origin O. A further point Q hs position vector point P divides AB in the rtio : () Show on digrm the reltive positions of O A B P nd Q () Write down the position vector of P nd nother (c) Write down n epression for QP nd comment on its mgnitude nd A direction. O P Q B OP QP p q QP is prllel to OA nd is of OA Emple Find vector eqution for the line AB where A is the point ( ) nd B is the point (8 ). The perpendiculr from the origin O to AB meets it t N. Using the sclr product of AB nd ON find the coordintes of N. Deduce the coordintes of the point C (distinct from B) on AB such tht OC = OB.
13 Pge 8 8 r r or t r For N AB ON. 8. n n n 8 n n n 5 5 n n Hence N(5 ) which will e the midpoint of BC therefore for C( ) 5 8 Hence C( 8 ) Emple The lines L nd L re given y the equtions N B (8 ) P A ( ) r
14 Pge L : t r L : 8 s r (i) Write down the direction rtios of L (ii) Find unit vector in the direction of L (iii) clculte the cute ngle etween L nd L (iv) Show tht L nd L don t intersect (v) Verify tht is perpendiculr to oth lines (vi) Find points P nd Q on L nd L such tht PQ is prllel to (i) DR s of L re :  : (ii) Unit vector in direction of L = (iii) The required ngle is etween nd nd is given y the formul. cos
15 Pge cos (iv) For intersection the eqution s t t s t 8 hve to e consistent From nd eqution t = Su into st eqution s = Check in rd eqution LHS = + = RHS = 8 + = Equtions not consistent lines do not intersect. (v). is perpendiculr to L. is perpendiculr to L (vi) Tke p p p P nd q q Q 8 p q p p q p q PQ 5 nd to e prllel to k PQ
16 q p k p k 5 q p k Pge 5 q q p k p p p k p p Hence P( 5 ) nd Q( ) Alger Binomil epnsion n for n rtionl When n is not positive integer the epnsion will hve n infinite numer of terms.!! The epnsion is vlid for i.e. it is convergent for n n nn nn n nn n n... to The generl term is unlikely to e tested ut for the record it is: n! {Don t try n r! r! defined for positive integers.} Emple n n n... n r r r! s the coefficient ecuse it just won t work n! is only Otin the first four terms in the epnsions of (i) (ii)! nd write down in ech cse the rnge of vlues of for which the epnsion is convergent.
17 Pge (i) !! Epnsion convergent for i.e. (ii) y !! Epnsion convergent for i.e. Prtil Frctions For ech unrepeted liner fctor occurring in the denomintor there will rise prtil frction of the form p. The vlue of p cn e found using the cover up rule.
18 Pge 7 Emple Epress in prtil frctions For ny repeted liner fctor occurring in the denomintor there will rise two prtil frctions Q P. The vlue of Q only cn e found y using the cover up rule. The vlue of P is found y setting up denomintor free identity Emple Epress in prtil frctions P P P P P Clculus Tngents nd norml to curve t specific point Find the grdient through differentition nd use m y y Emple Find the equtions of the tngent nd norml to the grph of y ln t the point where e
19 y ln When e dy ln. d ln Pge 8 dy d y e ln( e ) e nd Tngent y e e y e Norml Grd = y e e y e Integrtion techniques Integrtion y sustitution the sustitution will e given Emple Find (i) sin d y sustituting u cos (ii) d (i) sin d sin sin (ii) cos du d y sustituting u du sin d du sin d u du u u du u A cos cos A u cos In definite integrl the limits re chnged ccording to the sustitution
20 d u u u du du u Pge 9 u du d du d u u Integrtion y prts This method is used for some products such s for emple: sin cos e ln It cn lso e used to integrte for emple: ln sin tn dv d Formul u d uv v du d d e Emple sin d sin d cos cos cos cos d cos sin C.d u du d dv sin d v cos d In words for the evlution of first second integrl Integrl derivtive First times minus Integrl of Times of second lredy found of first
21 Emple ln d Pge Whether you use the formul or words the order here hs to e chnged ln d ln Emple tn.d tn d ln d ln A ln tn d Here introduce s the second tn tn tn (Use sme method for ln d d d ln d A A sin d etc) Do them! For definite integrl Emple e d e e e e e e d. d
22 Pge e e e e e Using prtil frctions Emple Evlute (i) d (ii) d (i) C d d ln ln (ii) A A A A A d d C d d d ln ln ln ln
23 Pge d ln ln ln ln 5 ln ln 5 5 ln ln Integrtion of sin nd cos etc Here we need the rerrnged doule ngle formule for cos i.e. cos cos sin cos Emple d cos d cos sin d sin C sin C {of this ooklet see pge 7 for cos d } Using the tles of stndrd derivtives nd integrls in the formul ooklet Relevnt to Core re derivtives of inverse trig functions nd of nd cosec (Pge 5) The results cn e reversed. Emple Emple d sin d tn C C On pge the integrls of tn cot Core sec cot Emple cosec d cot C cosec nd sec re relevnt to
24 Volumes of revolution y Pge When the shded region is rotted through c out the volume of revolution will e given y V y d y.out y V dy Emple Find the volume of revolution of the grph of y cos from to out through c y V cos d cos d sin sin sin
25 Differentil equtions Pge () Equtions of the form dy d y f () or g ( ) integrte t once d d Emple dy d e y e C * Emple d y d dy sin cos A d y sin A B * * These re the generl solutions of the differentil equtions where A B C re ritrry constnts. Prticulr solutions to differentil equtions cn e found if oundry conditions re given d Emple Solve the eqution t dt d dt t t t A t A A t t given tht when {generl solution} t () Equtions which reduce to f ( ) d g( y) dy re clled vriles () seprle. Integrte oth sides ut include just one ritrry constnt. Emple dy d y dy d y ln y C
26 y e Pge 5 C y Ae e e c Ae () An importnt ppliction is to rte of growth nd rte of decy. It is importnt to recll tht rte of chnge (ROC) (with respect to time unless otherwise specified) is derivtive with respect to time). ROC positive there is growth (increse) ROC negtive there is decy (decrese) Emple The rte of decy of certin rdioctive element t ny time is proportionl to the mss of the element remining t tht instnt. After dys one third of given mss m hs disintegrted. How much is left fter further dys? Let m e the mss remining t time t. The initil mss is m (when t ) Decy implies loss of mss nd rte of decy is given y derivtive dm dt Seprte vriles m km where k dm m kdt dm kdt m ln m kt A m when t ln m A ln m kt ln m
27 ln m ln m Pge m ln m m m kt kt e kt m m e kt m when t m m m e k e k e k m m t When m m t m 9 m
28 Pge 7 Notes
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 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 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 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 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 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 informationMathematics. 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 informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
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 informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationCHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee
ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Eercise
More informationHigher 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 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 information( x )( x) dx. Year 12 Extension 2 Term Question 1 (15 Marks) (a) Sketch the curve (x + 1)(y 2) = 1 2
Yer Etension Term 7 Question (5 Mrks) Mrks () Sketch the curve ( + )(y ) (b) Write the function in prt () in the form y f(). Hence, or otherwise, sketch the curve (i) y f( ) (ii) y f () (c) Evlute (i)
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 informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section  4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More information( β ) touches the xaxis 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( ) 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 informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
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 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 informationA. Limits  L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. 1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
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 informationFP3 past questions  conics
Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions  conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. 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..). Fig.. Suppose we rotte the line m round
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 informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bordpproved clcultors my be used A tble of
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 informationYear 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks
THGS Mthemtics etension Tril 00 Yer Tril Emintion Mthemtics Etension Question One mrks (Strt on new pge) Mrks ) If P is the point (, 5) nd Q is the point (, ), find the coordintes of the point R which
More informationSession Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN
School of Science & Sport Pisley Cmpus Session 056 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x)
ntegrtion (p) Integrtion by Inspection When differentiting using function of function or the chin rule: If y f(u), where in turn u f( y y So, to differentite u where u +, we write ( + ) nd get ( + ) (.
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
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 information4 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 information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
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 informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationChapter 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 informationTABLE 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 informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
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 informationMat 210 Updated on April 28, 2013
Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationFormulae For. Standard Formulae Of Integrals: x dx k, n 1. log. a dx a k. cosec x.cot xdx cosec. e dx e k. sec. ax dx ax k. 1 1 a x.
Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin
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 informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
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.
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 informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationExploring parametric representation with the TI84 Plus CE graphing calculator
Exploring prmetric representtion with the TI84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology
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 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 information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bordpproved clcultors my be used A tble of stndrd
More informationThings 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 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 informationOn 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 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 information7.1 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 informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bordpproved clcultors m be used A tble of stndrd
More informationCHAPTER 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 twicedifferentile function of x, then t
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 informationLesson5 ELLIPSE 2 1 = 0
Lesson5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
More informationSAINT IGNATIUS COLLEGE
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
More informationFINALTERM EXAMINATION 9 (Session  ) Clculus & Anlyticl GeometryI Question No: ( Mrs: )  Plese choose one f ( x) x According to PowerRule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
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 informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
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 informationIntegration 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 informationCET MATHEMATICS 2013
CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m  therefore ngle is o A sin o (). The
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 informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester  I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT 1 Curvture Let f : I R be sufficiently
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 informationFurther applications of integration UNCORRECTED PAGE PROOFS
. Kick off with CAS. Integrtion recognition. Solids of revolution. Volumes Further pplictions of integrtion. Arc length, numericl integrtion nd grphs of ntiderivtives.6 Wter flow.7 Review . Kick off with
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
More informationTime in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.
4. Are under Curve A cr is trveling so tht its speed is never decresing during 1second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
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 informationMAT137 Calculus! Lecture 28
officil wesite http://uoft.me/mat137 MAT137 Clculus! Lecture 28 Tody: Antiderivtives Fundmentl Theorem of Clculus Net: More FTC (review v. 8.58.7) 5.7 Sustitution (v. 9.19.4) Properties of the Definite
More information7.1 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 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 informationSection  2 MORE PROPERTIES
LOCUS Section  MORE PROPERTES n section , we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
More informationMathematics Extension Two
Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time  hours Write using blck or blue pen Bordpproved clcultors my be used Write your Student
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
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 information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationEigen Values and Eigen Vectors of a given matrix
Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
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 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 informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More information