Year 2009 VCE Mathematical Methods CAS Solutions Trial Examination 2
|
|
- Dwain Snow
- 5 years ago
- Views:
Transcription
1 Yer 9 VCE Mthemticl Methods CAS Solutions Tril Emintion KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC AUSTRALIA TEL: () FAX: () 987 kilbh@gmil.com KILBAHA PTY LTD 9
2 IMPORTANT COPYRIGHT NOTICE This mteril is copyright. Subject to sttutory eception nd to the provisions of the relevnt collective licensing greements, no reproduction of ny prt my tke plce without the written permission of Kilbh Pty Ltd. The contents of this work re copyrighted. Unuthorised copying of ny prt of this work is illegl nd detrimentl to the interests of the uthor. For uthorised copying within Austrli plese check tht your institution hs licence from Copyright Agency Limited. This permits the copying of smll prts of the mteril, in limited quntities, within the conditions set out in the licence. Techers nd students re reminded tht for the purposes of school requirements nd eternl ssessments, students must submit work tht is clerly their own. Schools which purchse licence to use this mteril my distribute this electronic file to the students t the school for their eclusive use. This distribution cn be done either on n Intrnet Server or on medi for the use on stnd-lone computers. Schools which purchse licence to use this mteril my distribute this printed file to the students t the school for their eclusive use. The Word file (if supplied) is for use ONLY within the school. It my be modified to suit the school syllbus nd for teching purposes. All modified versions of the file must crry this copyright notice. Commercil use of this mteril is epressly prohibited. KILBAHA PTY LTD 9
3 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge SECTION ANSWERS A B C D E A B C D E A B C D E A B C D E 5 A B C D E A B C D E 7 A B C D E 8 A B C D E 9 A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E 5 A B C D E A B C D E 7 A B C D E 8 A B C D E 9 A B C D E A B C D E A B C D E A B C D E KILBAHA PTY LTD 9
4 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge SECTION Question Answer D log ( ) f = g differentiting using the product rule e g f ( ) = g ( ) loge ( ) + e e e f = g loge ( e) + g e e e f = + = e Question Answer A p+ + ( p+ ) y = nd p+ y = q, in mtri form s p = y q The determinnt p + = p( p + ) = p p = ( p + p ) = ( p )( p + ) p so when p= ndp= there is no unique solution. If p = the equtions become () nd () + y = + y = q + y = nd + y = q when q = the two equtions re the sme eqution, so when p= ndq= there is n infinite number of solutions. Question Answer B f ( ) e Question y 7+ e + e d= = + verge vlue is Answer E = into y = ( + ) or y+ = ( + ) y = y' + nd = ' + become ' = nd y' = y in mtri form T y = + y KILBAHA PTY LTD 9
5 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge 5 Question 5 Answer C Let y = m + c nd y = +, the tngent to the grph t dy dy the point P, where =. = + = = m so B. is true. d d = = y = 9+ 9 = P, is on the tngent, At y = m+ c = 9+ c c= so A. is true, lso D. is true. b The re A= ( y y) d = b=, so tht = + ( ) + ( + ) E. is true, C. is flse. Question Answer A f ( + h) f ( ) + hf ( ) with f ( ) = = h=. so tht f.f Question 7.99 Answer B A m c d The required re is below the -is, so tking the bsolute vlue, mkes the re positive. A. is true ( ) d this is lso equl to D. which is true ( ) by symmetry C. is true ( ) t, now the inverse function is d. The grph of re bounded by the curve nd the y-is is Question 8 Answer C d, y =, this crosses the y-is = y y = + y = +, the + d, so tht E. is true, B. is flse. f : y = + dom f = R rn f =, f = y + trnsposing y = y =± but rn f = R dom f =, so we must tke the negtive, :(, ), f R f = KILBAHA PTY LTD 9
6 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge Question 9 Answer A ( f ( ) ) d f ( ) d [ ] f = f, f is n even function, nd f d=, then f d= 5 = = 5 = Question Answer B The function is not defined when =, ll of A, C, D. nd E. re flse, The function is n even function, symmetricl bout the y-is. y = log e ( ) dy loge( ) ( loge( ) ) d = + = + dy for turning points, d =, since loge ( ) = = e =, e the grph hs minimums t =± e y Question Answer E f : y = b + b f = b+ b= y b= y b y b b f ( ) = y = b+ so f = f b The domin nd rnge of both f nd f re R \{ b }. Since nd b, the grph of y = f ( ) psses through, b b nd the grph of y = f ( ) psses through b, b. All of A. B. C. D. re true, however E. is flse y = f nd y = f lwys intersects on the line y = t the points The grph of ( b, b ) ± ± only if >. KILBAHA PTY LTD 9
7 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge 7 Question Answer C dy cos cos y d sin = = = + c to find c, use d 5π = sin + c= + c= c= y = sin now when = y = sin = 5π y = Question Answer D b b b + b b y = = = b+ hs y = b s horizontl symptote nd = s verticl symptote. Question Let f [ π ] R f ( ) Answer D π :,, = cos. The period is T = = π The grph of f is trnsformed by reflection in the -is, the rule is g( ) = cos, we only hve one-qurter of cycle now diltion of fctor from the y-is, replce with g:, [ π ] R, g( ) = cos since we must hve one-qurter of cycle, the new domin is [ ], π then diltion by fctor of from the -is, multiply y by :,, = cos Answer E the eqution becomes g [ π ] R g( ) Question 5 ( A B ) + b p= ( A B ) + p= b ( A B ) = + p ( + b) Pr or Pr Pr A A B p b p b B p? b KILBAHA PTY LTD 9
8 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge 8 Question f = f = 8 f = turning points t =, =± for the function to be one-one, we require < Answer D y Question 7 Answer B d (?,.) n ( X = ) = e e loge (.) log (.) X = Bi n= p= Betty winning gme. Pr.. n log. log. n = 5. so n= e Question 8 Answer B d X = N μx = μ, σ = 9 X μ μ μ Pr ( X > μ ) = Pr Z > Pr Z.5 = > = μ Question 9 Answer C Let g ( ) = f ( t) dt then g ( ) = f ( ) g = g = f = 9 g = f = KILBAHA PTY LTD 9
9 Mthemticl Methods CAS Tril Emintion 9 Solutions Section Pge 9 Question Answer D Option D. hs nd ( ) Which is the grph required. f = e g = f g = e Question Answer A da dl A = L = L given cm/s dl dt = da da dl =. = L = 9L dt dl dt da = 9 cm /s dt L= Question Answer E b b Pr = = = + = + = A. is true Since ( X ) b ( b) b E ( X) = Pr( X = ) = + b+ = + b= ( b ) B. is true b E( X ) = Pr( X = ) = ( ) + ( ) + b+ = + + b+ b= ( + b) = C. is true, since A. is true. ( ) vr X = E X E X = b = b + 8b D. is true b 5 = = = + + = X E. is flse, E Pr( X ) b ( b ) END OF SECTION SUGGESTED ANSWERS KILBAHA PTY LTD 9
10 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge SECTION Question.i f = + c+ d f = + c = is turning point so f ( ) = ( + )( 9) = ( + )( ) but Epnding gives c = 9, lso u =, f = 5= c+ d = + 9+ d so tht d = nd f = v= 7 7 7= 7 v = 7 ii. The grph of y = 9 hs mimum vlue of 5, nd minimum vlue of 7, nd crosses the -is t three distinct points. The grph of y = 9+ d will therefore cross the -is t three distinct points, provided tht d 5, 7 or 5 < d < 7 A b. f = + c+ d f = + c, for two distinct turning points, we require Δ= c > M c< nd d R c. f + p = + p + p + c + p + d f + p = + p + p p+ c + p p + cp+ d = therefore p = p= nd p p+ c= since p= c= = c = nd p p + cp+ d = so d = lterntive method, if so tht p= c= nd d = y = = + y = KILBAHA PTY LTD 9
11 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge b d. A = f ( ) d = b= h= n= f = + c+ d L h f f f f = = () () M R= = h f + f () + f + f f f f f = () () = f + f () + f + f 8= f f = d 8 + c+ d = c c = c = Now subtrcting gives + d = d 8= 8 d = 8 M Question. the mplitude is.5, so tht =.5 π π one-hlf cycle is 8, so tht T = = n= n 8 b. y ( k e ) ( k e ) = psses through the origin O(,) nd B (,8) 8= = e k k e = e = k = log e k k = log e ( ) KILBAHA PTY LTD 9
12 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge c.i reflect in the y-is trnslte 8 units, to the right, wy from the y-is or trnslte 8 units, to the right prllel to the -is. ii. [ ] k( 8) f :,8 R, f = e A must give domin. k d.i π A = e sin d 8 ii. k π A= + e + cos k π 8 ech term k π k A= + e + cos + + cos k π but e = M k π 8 A = + k k π A = 8 k π p = 8 q= nd r = A Question. the function is continuous the totl re under the curve is one. 8 b t dt + c 8 t dt = f = b= c c= b b + 8c= nd c= b solving gives b= nd c= KILBAHA PTY LTD 9
13 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge b. must show point t y.5,. nd zero for t 8ndt G t c. Pr ( T > ) = ( 8 ) 8 t dt or the re of tringle s M Pr ( T > ) = c= Pr ( T > ) = 8 E T t dt t t dt d. = + ( 8 ) M E( T ) =. +. =. minutes KILBAHA PTY LTD 9
14 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge e. Since m t dt =. the medin time m is given by ( 8 t) dt =. M ( m m ) + 8 =. 8 solving for m with < m< 8 m =.5 minutes f. X is the running time of the movie in minutes ( 9, ) d X N μ σ = = = Pr ( X > 9) 9 9 = Pr Z > = Pr Z >.5 =.8 d g. Y = Bi( n=, p=.8) ( Y ) = Pr( Y = ) + Pr( Y = ) ( Y ) = + C Pr Pr Pr ( Y ) =. h. Pr ( comedies) = ACC + CAC + CCA M = =.9 M i =.89 or lterntively A.5.5 C =.. A C in the long run, the percentge of movies which re ctions re 8.9% KILBAHA PTY LTD 9
15 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge 5 Question. P, O(,) s = d( OP) = ( ) + + s = + = since > s = + M b.i. ds = = for minimum distnce d + 5 = = 5 ds if > consider =.8 =.9> d 5 ds nd if < consider =.7 =.5< d by the sign test it is minimum. M ii. S min =. 8 P, f 8 m = T = c.i t the point m N = 8 = or 8 norml y ( ) y = ii. norml psses through origin (, ) then + = 8 = 5 = = KILBAHA PTY LTD 9
16 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge Question 5. ( ) ( ) ( k ) π sin + cos = = where k Z b. since the period of both re π, it follows tht g ( + π ) = g ( ) T = π c. f :, [ π ] R, f ( ) = sin ( ) + cos ( ) sin( ) + cos( ) = sin( ) = cos( ) tn ( ) = 5π π 7π π =,,, = cos sin = d.i. f ( ) ( ) ( ) cos( ) = sin( ) tn ( ) = π π 7π 5π =,,, ii. m π 7π, nd,, min π 5π, nd, e. grph on correct domin, correct -intercepts G nd correct m nd min. G KILBAHA PTY LTD 9
17 Mthemtics Methods CAS Tril Emintion 9 Solutions Section Pge 7 π π f = sin + cos = sin + = sin + π trnslte sin( ), to the left prllel to the -is A = π α = f. END OF SECTION SUGGESTED ANSWERS KILBAHA PTY LTD 9
2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More 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 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 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 co-ordintes of the point R which
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 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 informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationYear 2011 VCE. Mathematical Methods CAS. Trial Examination 1
Year 0 VCE Mathematical Methods CAS Trial Examination KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC 30 AUSTRALIA TEL: (03) 908 5376 FAX: (03) 987 4334 kilbaha@gmail.com http://kilbaha.com.au IMPORTANT
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 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 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 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 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 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 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 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 informationYear 2013 VCE Mathematical Methods CAS Trial Examination 2 Suggested Solutions
Year VCE Mathematical Methods CAS Trial Examination Suggested Solutions KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC AUSTRALIA TEL: () 98 576 FAX: () 987 44 kilbaha@gmail.com IMPORTANT COPYRIGHT NOTICE
More informationPre-Calculus TMTA Test 2018
. For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to
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 Bord-pproved clcultors my be used Write your Student
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationWarm-up for Honors Calculus
Summer Work Assignment Wrm-up for Honors Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Honors Clculus in the fll of 018. Due Dte: The
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 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 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 informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show
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 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 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 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 informationFirst Semester Review Calculus BC
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.
More 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 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 informationEach term is formed by adding a constant to the previous term. Geometric progression
Chpter 4 Mthemticl Progressions PROGRESSION AND SEQUENCE Sequence A sequence is succession of numbers ech of which is formed ccording to definite lw tht is the sme throughout the sequence. Arithmetic Progression
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
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 informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne 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 information( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationLevel I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38
Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score
More informationWritten examination 1 (Facts, skills and applications)
MATHEMATICAL METHDS (CAS) PILT STUDY Written emintion (Fcts, skills nd pplictions) Frid 5 November 004 Reding time: 9.00 m to 9.5 m (5 minutes) Writing time: 9.5 m to 0.45 m ( hour 0 minutes) PART I MULTIPLE-CHICE
More informationMinnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017
Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.
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 informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
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 informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
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 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 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 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 informationSection The Precise Definition Of A Limit
Section 2.4 - The Precise Definition Of A imit Introduction So fr we hve tken n intuitive pproch to the concept of limit. In this section we will stte the forml definition nd use this definition to prove
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
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 informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More 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 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 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 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 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 informationAP * Calculus Review
AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril.
More informationCBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
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 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 informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
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 informationAPPM 1360 Exam 2 Spring 2016
APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the
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 informationCBSE 2013 ALL INDIA EXAMINATION [Set 1 With Solutions]
M Mrks : Q Write the principl vlue of CBSE ALL INDIA EXAMINATION [Set With Solutions] SECTION A tn ( ) cot ( ) Time Allowed : Hours Sol tn ( ) cot ( ) tn tn cot cot cot cot [Rnge of tn :,, cot : ], [ 5
More informationMATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2
MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points
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 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 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 informationthan 1. It means in particular that the function is decreasing and approaching the x-
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
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 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 informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
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 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 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 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 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 informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
More informationIf C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a
Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Events I P I P I P I P 5 7 0 0 S S S S Group Events G G G G 80 00 0 c 8 c c c d d 6 d 5 d 85 Individul Event I.,
More informationLesson 5.3 Graph General Rational Functions
Copright Houghton Mifflin Hrcourt Publishing Compn. All rights reserved. Averge cost ($) C 8 6 4 O 4 6 8 Number of people number of hits.. number of times t bt.5 n n 4 b. 4.5 4.5.5; No, btting verge of.5
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 informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More 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 x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
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 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 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 informationSULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING
SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
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 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 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 informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More information