( β ) touches the x-axis if = 1
|
|
- Brittany Douglas
- 5 years ago
- Views:
Transcription
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 using clculus nd show tht this point is minimum point. Find the minimum vlue of Hence, show tht the curve f ( β ) touches the -is if, where nd re constnts to e determined in terms of k. f in terms of k. α lies entirel ove the -is if < k <, k or k, γ cuts the -is in two distinct points if k < or k >. ii. Prove tht the stright line m f in two rel nd distinct points for intersects the curve ll rel nd finite vlues of m if nd onl if k <. () Let g 7 6. Using Reminder theorem repetedl show tht ( ) is fctor of Epress g in the form ( ) ( c) Deduce tht g for ll rel vlues of. g., where, nd c re constnts to e determined. nswer () f k k ( k ) k k ( k ) ( k k ) This is of the form f ( ), where k nd k k. When ( k,) Let k, f k k Is the turning point of f, where k k. k Δ nd k Δ, where Δ > f ( ) ( k Δ k) ( k ) f ( ) ( k Δ k) ( k ) k ( Δ ) ( Δ ) k f ( ) > f ( k) f ( ) > f ( k) k is the turning point nd the minimum point of f. - - Done B : Chndim Peiris (B.Sc - Specil)
2 Minimum vlue of f f ( k) k k If k k > then, the curve f lies entirel ove the -is. k k > k k < Sign of ( k )( k ) ( k )( k ) < < k < If < < k then, the curve f k then, the curve f If k k k k k k k k or k If k or If k < k then, the curve f k then, the curve f k k k < k > ( k )( k ) > k < or k > If k < or > k then, the curve f lies entirel ove the -is. touches the -is. touches the -is. cuts the -is in two distinct points. cuts the -is in two distinct points. Sign of ( k )( k ) Let m nd k k intersect ech other. m k k ( k m) ( k ) Here the discriminnt Δ ( k m) ( k ) If the line m intersects the curve f discriminnt ( Δ ) should e positive. Since ( k m) > for ll vlues of m, ( k ) ( k ) < ( k ) < k < in two rel nd distinct points for ll Rthen the should e less thn ero. Similrl, if k < then, the ove discriminnt will e positive. The stright line m finite vlues of m if nd onl if k < intersects the curve f in two rel nd distinct points for ll rel nd - - Done B : Chndim Peiris (B.Sc - Specil)
3 () g 7 6 g g (ccording to the reminder theorem) is fctor of g ( )( ) Let h h ( ) ( ) is fctor of h (ccording to the reminder theorem) g ( ) ( ) ( ) is fctor of g h ( )( ) g ( ) ( ) g of the form ( ) ( c) Here Let f. The discriminnt of for ll But R, >, Equlit holds when, where g for ll R, Equlit holds when, nd c. f (S Δ ) is. i.e. Δ < - - Done B : Chndim Peiris (B.Sc - Specil)
4 . B for ll R. () Find constnts nd B such tht Hence, determine () r Show tht n u r f for r Ζ, such tht u r f () r f ( r ), whereu r. r r ( n ) r r. Show tht the series r ur is convergent nd find the vlue of r () Sketch, in the sme figure, the grphs of nd. Hence, find the set of vlues of for which 6. B considering the grph of k, for n k R, in the sme figure find for wht vlue of l the eqution 6 l hs onl one rel solution. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur u. r nswer B When, When B ( ) 8 B () B B u r r ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) - - Done B : Chndim Peiris (B.Sc - Specil)
5 ( r ) ( r ) u r f () r f ( r ), where f () r. r Let S () r f () () () u r u f f u f f f r u f f f n f ( n ) f ( n ) f ( n) ( n) f u n u n u n n r n f n u u r f () f ( n ) ( ) ( n ) ( ) r r n S n n u lim S n ( ) r r n n lim n lim n ( n ) S This is finite vlue. Since S is finite vlue, the series ur is convergent. r The vlue of u r r - - Done B : Chndim Peiris (B.Sc - Specil)
6 () ; when ; when < ; when ; when < 8 6 Finding the vlue of ; Finding the vlue of ; The required rnge of is, When k, k ecomes. Then there is onl one non-trivil position where nd intersect ech other. 6 6 Here l. 6 6 When l, the eqution 6 l hs onl one non trivil solution Done B : Chndim Peiris (B.Sc - Specil)
7 - 7 - Done B : Chndim Peiris (B.Sc - Specil). () Let e mtri. Show tht O I, where I is the identit mtri nd O is the ero mtri. Hence, find. Let 6 B e mtri. Show tht B B. Hence or otherwise find non-ero mtri C such tht O BC. () Let e comple numer. Prove tht nd Re. Hence, show tht for n two comple numers nd. Deduce tht. If < i then, show tht < <. Shde the region R consisting the set of points in the rgnd digrm which represent the comple numer for which i nd rg. nswer () I ) ( O I (); O I O I I 6 B 6 B
8 6 B B Let C c d c d BC 6 c d 6c 6d Since BC O, c nd 6c c d nd 6d d t Let t, where t is prmeter. c k Let k, where k is prmeter. d C () Let i, where, R i () ( i)( i) i () ( Q i ) From () nd (); Since >, > Re, when. When, Re Re, Equlit holds when. Let i nd i, where,,, R. i Wnt: ( )( ) [( ) i( )]( [ ) i( )] ( ) ( ) ( ) Done B : Chndim Peiris (B.Sc - Specil)
9 - 9 - Done B : Chndim Peiris (B.Sc - Specil) Since,. i.e. Let i i i < < i This is circle whose centre, nd rdius. ccording to the ove rgnd digrm, O is the minimum distnce from O to the nd OB is the mimum distnce from O to the. Min nd M < < Where rg i R O,,, B Re Im O, Re Im, B R
10 . () B considering onl first derivtive find the minimum nd mimum vlue of Sketch the grph of. 7 Hence, find for wht vlues of k, the eqution k 7k, where k is rel, hs (i) two coincident rel roots, (ii) three coincident rel roots, (iii) two distinct rel roots, (iv) no rel roots.. 7 () Consider rectngle BCD with B nd BC ( < ). Let P e movle point on CD. The length of PB L, where DP. P is L. Show tht Find the minimum length of L nd the position of P on CD corresponding to this minimum length. lso, find the mimum length of L. nswer () Let 7 Differentiting with respect to. d ( 7) d 7 d d ( 8 ) ( 7) ( ) ( 7) ( )( ) ( 7) ( )( )( ) ( 7) For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur d Criticl points of re given d, nd re the criticl points of. - - Done B : Chndim Peiris (B.Sc - Specil)
11 When,. When, When, < < < < < > ( ) - ( ) - d d (, ) is inflection point. - -, is reltive minimum point., is reltive mimum point. lim nd lim. - - Done B : Chndim Peiris (B.Sc - Specil)
12 Therefore the following figure illustrtes the grph of 7... Let k. i.e. k 7 Let f ( k ) k 7 k 7k. (i) (ii) (iii) (iv) When k or k, f hs two coincident rel roots. When k, re two coincident rel roots of f nd when k, coincident rel roots of f. When k, f hs three coincident rel roots. Here re three coincident rel roots. When < k < or < k <, f hs two distinct rel roots. When k > or k <, f hs no rel roots. re two - - Done B : Chndim Peiris (B.Sc - Specil)
13 () D ( ) P C B PB ( ) P Let L P PB L Differentiting L with respect to dl d dl d Criticl points of ( ) ( ) ( ) ( ) dl L re given d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( )( ) ( ) Since nd,. - - Done B : Chndim Peiris (B.Sc - Specil)
14 - - Done B : Chndim Peiris (B.Sc - Specil) d dl d L d d L d > d L d L hs reltive minimum point when, ccording to the nd derivtive test. L Min f Min f When, L hs mimum vlue. L M f M f
15 . () Show tht ( sin cos ) d () Using integrtion prts, or otherwise find tn d. 8. (c) Using prtil frctions find ( ) nswer () Let I ( sin cos )d sin cos I sin d d. ( sin cos )( sin cos sin cos ) ( sin cos )( sin cos ) sin cos sin cos d sin cos cos cos d cos sin sin d [ cos ] [ sin ] sin cos d cos ( sin )d [ cos cos] [ sin sin ] sin d ( sin ) cos d( cos ) sin cos [ ] sin sin cos cos [ ] 6 I I [ ] [ ] () Let I tn d I tn d d tn d tn d d I d - - Done B : Chndim Peiris (B.Sc - Specil)
16 tn tn I tn tn C, where C is constnt C D (c) ( ) ( ) ( ) ( )( ) B( ) ( C D)( ) B When When When 8 B B D B C D B D () C D () When 6 B 9C 9D 6 9C 9D C D () C D () () (); C D D C () From () nd (); ( D) C D D C D D C () () ; C C 8 From () ; D D 6 From () ; ( ) ( ) d d d ( ) ( ) ( ) d ln ln( ) tn C, where C is constnt Done B : Chndim Peiris (B.Sc - Specil)
17 6. () Find the equtions of the isectors of the ngle etween two non prllel stright lines l c nd l c. Show tht the isector of the cute ngle etween two stright lines given nd is the isector of the otuse ngle etween two stright lines given 7 8 nd 8. () Show tht, for ll vlues of g nd f the circle g f r isects the circumference of the circle r., touching the stright line nd isecting the circumference of the circle. Find the equtions of these two circles. Show tht two circles cn e drwn through the point (,) nswer () l c d d (, ) l l c d d c c l is isector of the ngle etween two stright lines l ndl., is n point on the linel. l is isector of the ngle etween two stright lines l ndl, d d. Let Let ( ) Since c c c c ± B considering nd - of the right hnd side of the ove eqution, we hve two equtions of the stright lines. These re the equtions of the isectors of the ngles etween two non prllel stright lines l c nd l c. One of these equtions is the isector of the cute ngle etween l nd l nd other one is the eqution of the isector of the otuse ngle Done B : Chndim Peiris (B.Sc - Specil)
18 l θ θ l The equtions of the isectors of the ngle etween l nd l re given, ± ± B considering ; B considering -; () () If the eqution () is the eqution of the isector of the cute ngle etween l ndl then, θ <. i.e. tnθ should e less thn one. m, m m m tnθ m m < Here tn θ < is the eqution of the isector of the cute ngle etween l 7 8 φ φ l 8 Done B : Chndim Peiris (B.Sc - Specil)
19 The equtions of the isectors of the ngle etween l ndl re given, ± ± ( 8 ) B considering ; B considering -; () () Let us ssume tht the eqution () is the eqution of the isector of the cute ngle etween l nd l. m, m 8 m m tnφ m m 8 8 > Here tnφ > φ > Therefore is not the eqution of the isector of the cute ngle etween l nd l. i.e. the eqution of the isector of the otuse ngle etween l nd l. is the eqution of the isector of the cute ngle etween nd nd lso it is the eqution of the isector of the otuse ngle etween 7 8 nd 8. () g f r () r () () ; g f g f g (*) f This is of the form m Done B : Chndim Peiris (B.Sc - Specil)
20 (*) is the eqution of the line joining the points of intersection of () nd (). For ll vlues of g nd f, stright line (*) isects the circumference of the eqution circle. For ll vlues of g nd f, the circle g f r isects the circumference of the circle r. (Q(*) is the eqution of dimeter of the circle r ) Let S g f r, where g, f R, is the eqution of the required circle. Centre C ( g, f ) Since the circle S psses the point (,), g f r g f r () Let R is the rdius of the circle S. R g f r Since the circle S touches the line, f R f R ( f ) ( f ) g f r () Since the circle S isects the circumference of the circle, r. Sustituting r in (); g f g f () Sustituting f g f From () nd (); f f r in (); () f f f f f f f 8 f This is qudrtic eqution of f. There re two circles cn e drwn ccording to the ove given conditions. f 8 f f f f or f When f, g When f, g Their equtions: 7. () For tringle BC, prove in the usul nottion, tht - - Done B : Chndim Peiris (B.Sc - Specil)
21 c sin sin B sin C B C Deduce tht ( c) cos cosec () Show tht, for n rel vlue ofθ, the epression tnθ tnθ cnnot tke n vlue etween -7 nd. (c) Epress cos θ 8cosθ sinθ 9sin θ in the form of the cos( θ α ), where nd re constnts nd α is n ngle independent ofθ. Hence or otherwise find the generl solution of the eqution. 8 ( cos sin ) ( cos sin ) 9. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur nswer () Sine rule c sin sin B c sin C B C Proof: Cse : For n cute ngled tringle Drw the line D perpendiculr from to the side BC. c Then BD is right ngled tringle. B D sin B B D B sin B c sin B () C D Similrl, in the right ngle tringle CD, D sin C C D C sin C sin C () c From () nd (); csin B sin C sin B sin C - - Done B : Chndim Peiris (B.Sc - Specil)
22 c In the sme w, it cn e proved tht drwing perpendiculr line from B to the side BC. sin sin C c For n cute ngled tringle, sin sin B sin C Cse : For right ngled tringle Let B is the right ngle of the tringle BC. Drw the line BD perpendiculr from B to the side C. D c Then BD is right ngled tringle. BD sin B BD B sin c sin () B C Similrl, in the right ngled tringle BCD, BD sin C BC BD BC sin C sin C () c From () nd (); sin C csin sin sin C B Since BC is right ngled tringle, sin C B C sin C sin C () C But sin B sin9 We cn write B c sin B (6) c From () nd (6); sin C csin B sin B sin C c For right ngled tringle, sin sin B sin C Cse : For n otuse ngled tringle c B Let the ngle C of the tringle BC is n otuse ngle. Drw the line BD perpendiculr from B to produced C. BCD ˆ C D In the right ngled tringle BCD, BD sin ( C ) BC BD BC sin C sin C (7) Similrl, in the right ngled tringle BD, BD sin B BD B sin c sin (8) c From (7) nd (8); sin C csin sin sin C In the sme w, it cn e proved tht drwing perpendiculr line from C to the side B. sin B sin C - - Done B : Chndim Peiris (B.Sc - Specil)
23 For n otuse ngled tringle, sin sin B ccording to the ove three cses, for n tringle c sin C sin c. sin B sin C c Let k, where k is constnt. sin sin B sin C k sin, k sin B, c k sinc Let us consider ( c) k sin ( c) k( sin B sin C). sin sin B sin C sin B C B C cos sin sin cos Q B C B C cos sin sin cos Q cos sin B C sin sin cos B C sin B C cos cosec ( c) B C B C cos cosec c () Let tnθ tnθ tnθ tn tnθ tnθ tn ( tnθ ) tnθ tnθ - - Done B : Chndim Peiris (B.Sc - Specil)
24 ( tnθ ) tnθ ( tnθ ) tnθ tnθ tnθ tn θ tnθ ( ) tnθ ( ) tn θ This is qudrtic eqution of tn θ. Since tn θ R, the discriminnt of the ove eqution should e. ( ) ( ) 8 Sign curve of ( 7)( ) 6 7 ( 7)( ) 7 [, ) (, 7] The vlue of cnnot tke n vlue etween -7 nd. i.e. The epression tnθ tnθ cnnot tke n vlue etween -7 nd. (c) Let E cos θ 8cosθ sinθ 9sin θ E cos θ sin θ 8cosθ sinθ sin θ 9sin θ cosθ ( Q cos θ sin θ sin θ cos θ ) 7 9sin θ cosθ 9 7 sin θ cos θ E 7 cos θ sin θ If cosα then sin α E 7 ( cosθ cosα sin θ sinα ) E 7 cos θ α, where 7, ndsin α, 8 cos sin cos sin This is of the form cos( θ α ) 9 cos α. ( cos sin cos ) ( cos sin sin cos sin ) 9 ( cos ) 6sin cos ( sin cos ) sin cos 8sin 9 8 sin 8 sin cos 6sin cos 8sin sin 9 cos 6sin cos 8sin 9 ( cos 8sin cos 9sin ) 9 ( 7 cos( α )) 9, wheresin α, cos α. ( α ) 9 ( α ) cos cos cos ( α ) cos - - Done B : Chndim Peiris (B.Sc - Specil)
25 α n ± n ± α α n ±, n Ζ. Where 6 sin α, cos α. For /L Comined Mths (Group/Individul) Clsses / Contct :778 P.C.P.Peiris B.Sc (Mths Specil) Universit of Sri Jewrdenepur - - Done B : Chndim Peiris (B.Sc - Specil)
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 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 information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
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 Bord-pproved clcultors my be used A tble of stndrd
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 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 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 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.
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 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 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 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 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 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 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 informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationEllipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION
FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied
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 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 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 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
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/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More informationLesson-5 ELLIPSE 2 1 = 0
Lesson-5 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 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. Bord-pproved clcultors my be used A tble of
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 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 informationTriangles 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 informationMTH 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 informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationSet 1 Paper 2. 1 Pearson Education Asia Limited 2017
. A. A. C. B. C 6. A 7. A 8. B 9. C. D. A. B. A. B. C 6. D 7. C 8. B 9. C. D. C. A. B. A. A 6. A 7. A 8. D 9. B. C. B. D. D. D. D 6. D 7. B 8. C 9. C. D. B. B. A. D. C Section A. A (68 ) [ ( ) n ( n 6n
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationGEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS:
GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0
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 rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 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 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 informationJEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse
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 informationCalculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
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 information10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm
HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then
More informationTime : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
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 information15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )
- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
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 information1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D
. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )
More informationDrill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte
More informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p
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/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION
More informationMathematics Extension 2
S Y D N E Y B O Y S H I G H S C H O O L M O O R E P A R K, S U R R Y H I L L S 005 HIGHER SCHOOL CERTIFICATE TRIAL PAPER Mthemtics Extension Generl Instructions Totl Mrks 0 Reding Time 5 Minutes Attempt
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 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 information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationTO: 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 informationREVIEW SHEET FOR PRE-CALCULUS MIDTERM
. If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,
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 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 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 informationSet 6 Paper 2. Set 6 Paper 2. 1 Pearson Education Asia Limited 2017
Set 6 Pper Set 6 Pper. C. C. A. D. B 6. D 7. D 8. A 9. D 0. A. B. B. A. B. B 6. B 7. D 8. C 9. D 0. D. A. A. B. B. C 6. C 7. A 8. B 9. A 0. A. C. D. B. B. B 6. A 7. D 8. A 9. C 0. C. C. D. C. C. D Section
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 informationBelievethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
More informationForm 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6
Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms
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 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 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 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 informationAnti-derivatives/Indefinite Integrals of Basic Functions
Anti-derivtives/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 informationFINALTERM 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 informationMEP 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 informationAP 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 information1 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 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 informationPARABOLA EXERCISE 3(B)
PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationNat 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 informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
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 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 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 informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
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 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 informationSUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 2012
MOCK TEST 0 SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 0 ANSWERS. () π π Tke cos - (- ) then sin [ cos - (- )]sin [ ]/. () Since sin - + sin - y + sin - z π, -; y -, z - 50 + y 50 + z 50 - + +
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
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 informationMath Sequences and Series RETest Worksheet. Short Answer
Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning
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 informationMDPT Practice Test 1 (Math Analysis)
MDPT Prctice Test (Mth Anlysis). Wht is the rdin mesure of n ngle whose degree mesure is 7? ) 5 π π 5 c) π 5 d) 5 5. In the figure to the right, AB is the dimeter of the circle with center O. If the length
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 informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice
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 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 right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in
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 informationGEOMETRY 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 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 informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
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 Bord-pproved clcultors m be used A tble of stndrd
More informationAP 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