Prelim Examination 2010 / 2011 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
|
|
- Bethany Norris
- 5 years ago
- Views:
Transcription
1 Prelim Examination 00 / 0 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Calculators may be used in this paper.. Candidates should answer all questions. Full credit will only be given where the solution contains appropriate working
2 All questions should be attempted. Use Gaussian elimination to solve the following system of equations x + y + z = x + y + 6z = 0 x + y - z = 0. Find the solution corresponding to y = -. sin xx ( ). Given that y =, 0 x <, show that dy dx = x + x x sin ( x) ( x).. Two complex numbers, z and z, are given by where k is a real number. z = i and z = 6 + ki, (a) Given that z z is a purely real number, find the value of k. (b) Express z z in the form x + iy, where x and y are real numbers.. The term independent of x in the expansion of 6 t x x is 60. Given that t > 0, find the value of t.. (a) The function f(x) is given by ( x) Express f(x) in the form ( x) x + x + 7 f =, 0 x < π. ( x + )( x + 9) p qx + r f = + where p, q and r are integers. x + ( x 9), + π (b) Hence show that f ( x) dx = + ln [Turn over Pegasys 00
3 6. For each of the following statements, decide whether it is true or false and prove your conclusion. A For all natural numbers n, if n is divisible by then n + 6n 9 is divisible by 9. B The sum of any prime number p and any even integer q is always odd. dy sec y 7. Find the solution of the differential equation = given that t dt e π y = when t = A curve is defined by the parametric equations x = t, y = + t t t for all t. (a) Find the coordinates of the stationary point of this curve. d y (b) Use dx to determine the nature of this stationary point. 9. The first two terms of a series are + and +. (a) (b) If the series is arithmetic, show that the common difference is, and that the sum of the first six terms is 6 6. If the series is geometric, show that it has a sum to infinity, and that this sum is ( + ). 0. Use the substitution ( ) x + = θ + to show that 0 dx x + 0 = ln. 6 [Turn over Pegasys 00
4 x + 8. Part of the graph of the function f(x) =, x k, x k is shown below. The graph has a vertical asymptote with equation x =, a non-vertical asymptote L and crosses the y-axis at the point P. The points (a, b) and (c, d) are the stationary points of the graph. y x = (c,d) L (a,b) O P x (a) Write down the value of k. (b) Determine algebraically the equation of the non-vertical asymptote, L. (c) (d) Write down the coordinates of the point P where the graph of f crosses the y-axis. The graph has stationary points with coordinates (a, b) and (c, d) as shown in the diagram. Use calculus to determine the values of a, b, c and d and justify the nature of the stationary points. (d) State the coordinates of the stationary points of the graph with equation g( x) = f ( x). [END OF QUESTION PAPER] Pegasys 00
5 Marking Scheme Advanced Higher Grade 00 / 0 Prelim (Assessing Units & ) ans: x = 7z, y = z +,' z = z' (or equivalent) marks 6 0 correct augmented matrix 0 correct first modified system 0 0 correct second modified system 0 0 correct third modified system x = 7z, y = z +,' z = z' (or equivalent) ans: x = 7, (y = -), z = - mark ans: Proof marks attempts to substitute into quotient rule differentiates correctly differentiates correctly and correct denominator simplifies correctly x = 7, (y = -), z = - (or equivalent) f g fg substitutes into " " g x x or ( x) ( x) ( x) ( ) sin sin ( ) x ( x) ( ) or x x and x ( ) ( x) dy x + x sin ( x) =. dx x ( x) Pegasys 00
6 (a) ans: k = marks starts correctly continues correctly i or 8 ki + i( k) (or equivalent) k = (b) ans: 0 + i (or equivalent) marks correct complex conjugate continues correctly ans: t = marks finds correct general term simplifies to find correct expression for power of x solves for r correctly finds correct term 6 + i 6 + i + i i + i 0 + i (or equivalent) 6 6 ( x ) r r x r = t = r t x r (a) ans: ( x) f = + marks x + x + 9 correct method one value correct other two values correct x + x + 7 p( x + 9) + ( qx + r)( x + ) p =, q = 0 or r = f ( x) = + x + x + 9 (b) ans: Proof marks integrates correctly integrates correctly substitutes correctly completes proof correctly ln x + tan x ln + tan ln+ tan 0 π ln Pegasys 00
7 6 ans: A True & Proof; B False & Proof marks 7 A - correct conclusion A starts proof A completes proof B correct conclusion B completes proof ans: y 7 = sin + t e marks rearrange correctly correct strategy; i.e. rewrite and integrate integrate correctly substitute correctly True ( k ) + 6( k) 9 9( + k ) k which is divisible by 9 False p = and (e.g.) q = gives p + q = 6 which is even dy dt = t sec y e t cos ydy = e dt t e sin y = + C ( 0) sin π = e + C 6 7 y = sin + t e 7 (accept sin y = + t e ) Pegasys 00
8 8(a) ans:, marks attempts to substitute into correct formula differentiates correctly (twice) equates with zero and solves for t correctly correct coordinates ( ) ( ) " substitutes into " dy y t = dx x t 8t + t + t t =, 8(b) ans: Maximum (turning point) marks finds second derivative correctly substitutes correctly correct sign and conclusion 8 6 t + + t t t t t t + t + ( + ) d y <0 MaximumT.P. dx Pegasys 00
9 9(a) ans: Proof marks correct method + ( + ) correct value = correct method 6 ( ) + + correct value =... = 6 6 9(b) ans: Proof marks correct method correct conclusion correct method correct value < < S exists + ( + )( + ) ( + ) ( + ) =... = ( + ) conjugate surd used to form answer 0 ans: Proof 6 marks starts substitution correctly continues substitution correctly changes limits correctly integrates correctly substitutes correctly completes proof correctly ( θ + ) dθ θ x = θ =, x = 0 θ = [ θ + lnθ ] { + ln + ln } 0 + ln ln =... = ln ( ) ( ) ( ( ) Pegasys 00
10 (a) ans: k = mark correct value k = (b) ans: y = x + marks correct method continues correctly x x y = x + + x y = x + (c) ans: (, 8) 0 mark correct coordinates (, 8) 0 (accept -8) (d) a =, b =, c = & d = 8 ans: (, ) (,8) MinimumT. P. MaximumT. P. & differentiates correctly solves f ( x) = 0 correctly correct y-coordinates correct formula for second derivative correct nature of both points marks f ( x) = 9 ( x ) x =, a =, b =, c = & d = 8and/or (, ) & (,8) 8 f ( x) = ( x ) f ( ) < 0 MaximumT.P. & f () > 0 MinimumT.P. (d) ans: (, )(,, 6) marks correct point correct point (, ) (, 6) TOTAL MARKS = 70 Pegasys 00
Prelim Examination 2015/2016 (Assessing all 3 Units) MATHEMATICS. CFE Advanced Higher Grade. Time allowed - 3 hours
Prelim Eamination /6 (Assessing all Units) MATHEMATICS CFE Advanced Higher Grade Time allowed - hours Total marks Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions
More informationG H. Extended Unit Tests A L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests
M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Extended Unit Tests A (more demanding tests covering all levels) Contents Extended Unit Tests Detailed marking schemes
More informationAdvanced Higher Grade
Practice Eamination A (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours 0 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working..
More informationG H. Extended Unit Tests B L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests
M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Etended Unit Tests B (more demanding tests covering all levels) Contents 3 Etended Unit Tests Detailed marking schemes
More informationAdvanced Higher Grade
Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators
More informationX100/701 MATHEMATICS ADVANCED HIGHER. Read carefully
X/7 N A T I O N A L Q U A L I F I C A T I O N S 9 T H U R S D A Y, M A Y. P M. P M MATHEMATICS ADVANCED HIGHER Read carefully. Calculators may be used in this paper.. Candidates should answer all questions.
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More information2018 Mathematics. Advanced Higher. Finalised Marking Instructions
National Qualifications 08 08 Mathematics Advanced Higher Finalised Marking Instructions Scottish Qualifications Authority 08 The information in this publication may be reproduced to support SQA qualifications
More informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationCreated by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.
IYGB GCE Mathematics MP Advanced Level Practice Paper M Difficulty Rating:.8750/1.176 Time: hours Candidates may use any calculator allowed by the regulations of this examination. Information for Candidates
More informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationFurther Mathematics SAMPLE. Marking Scheme
Further Mathematics SAMPLE Marking Scheme This marking scheme has been prepared as a guide only to markers. This is not a set of model answers, or the exclusive answers to the questions, and there will
More informationMathematics 136 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 19 and 21, 2016
Mathematics 36 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 9 and 2, 206 Every rational function (quotient of polynomials) can be written as a polynomial
More information2010 Maths. Advanced Higher. Finalised Marking Instructions
00 Maths Advanced Higher Finalised Marking Instructions Scottish Qualifications Authority 00 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationTopics and Concepts. 1. Limits
Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationTHE INVERSE TRIGONOMETRIC FUNCTIONS
THE INVERSE TRIGONOMETRIC FUNCTIONS Question 1 (**+) Solve the following trigonometric equation ( x ) π + 3arccos + 1 = 0. 1 x = Question (***) It is given that arcsin x = arccos y. Show, by a clear method,
More informationFinal practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90
Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationBook 4. June 2013 June 2014 June Name :
Book 4 June 2013 June 2014 June 2015 Name : June 2013 1. Given that 4 3 2 2 ax bx c 2 2 3x 2x 5x 4 dxe x 4 x 4, x 2 find the values of the constants a, b, c, d and e. 2. Given that f(x) = ln x, x > 0 sketch
More information" $ CALCULUS 2 WORKSHEET #21. t, y = t + 1. are A) x = 0, y = 0 B) x = 0 only C) x = 1, y = 0 D) x = 1 only E) x= 0, y = 1
CALCULUS 2 WORKSHEET #2. The asymptotes of the graph of the parametric equations x = t t, y = t + are A) x = 0, y = 0 B) x = 0 only C) x =, y = 0 D) x = only E) x= 0, y = 2. What are the coordinates of
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
National Quali cations AHEXEMPLAR PAPER ONLY EP/AH/0 Mathematics Date Not applicable Duration hours Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions
More informationMATHEMATICS. Higher 2 (Syllabus 9740)
MATHEMATICS Higher (Syllabus 9740) CONTENTS Page AIMS ASSESSMENT OBJECTIVES (AO) USE OF GRAPHING CALCULATOR (GC) 3 LIST OF FORMULAE 3 INTEGRATION AND APPLICATION 3 SCHEME OF EXAMINATION PAPERS 3 CONTENT
More information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationCore Mathematics C4. You must have: Mathematical Formulae and Statistical Tables (Pink)
Write your name here Surname Other names Pearson Edexcel GCE Centre Number Core Mathematics C4 Advanced Candidate Number Friday 23 June 2017 Morning Time: 1 hour 30 minutes Paper Reference 6666/01 You
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationPURE MATHEMATICS AM 27
AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationAB Calculus Diagnostic Test
AB Calculus Diagnostic Test The Exam AP Calculus AB Exam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour and 5 minutes Number of Questions
More informationNational Quali cations
National Quali cations AH08 X747/77/ Mathematics THURSDAY, MAY 9:00 AM :00 NOON Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationand lim lim 6. The Squeeze Theorem
Limits (day 3) Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationsin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationHSC Marking Feedback 2017
HSC Marking Feedback 017 Mathematics Extension 1 Written Examination Question 11 Part (a) The large majority of responses showed correct substitution into the formula x = kx +lx 1 k+l given on the Reference
More information2016 Mathematics. Advanced Higher. Finalised Marking Instructions
National Qualifications 06 06 Mathematics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 06 The information in this publication may be reproduced to support SQA qualifications
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More informationH I G H E R S T I L L. Extended Unit Tests Higher Still Higher Mathematics. (more demanding tests covering all levels)
M A T H E M A T I C S H I G H E R S T I L L Higher Still Higher Mathematics Extended Unit Tests 00-0 (more demanding tests covering all levels) Contents Unit Tests (at levels A, B and C) Detailed marking
More informationReview session Midterm 1
AS.110.109: Calculus II (Eng) Review session Midterm 1 Yi Wang, Johns Hopkins University Fall 2018 7.1: Integration by parts Basic integration method: u-sub, integration table Integration By Parts formula
More informationPossible C2 questions from past papers P1 P3
Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationx n cos 2x dx. dx = nx n 1 and v = 1 2 sin(2x). Andreas Fring (City University London) AS1051 Lecture Autumn / 36
We saw in Example 5.4. that we sometimes need to apply integration by parts several times in the course of a single calculation. Example 5.4.4: For n let S n = x n cos x dx. Find an expression for S n
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationHorizontal and Vertical Asymptotes from section 2.6
Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:
More information2017 HSC Mathematics Extension 1 Marking Guidelines
07 HSC Mathematics Extension Marking Guidelines Section I Multiple-choice Answer Key Question Answer A B 3 B 4 C 5 D 6 D 7 A 8 C 9 C 0 B NESA 07 HSC Mathematics Extension Marking Guidelines Section II
More informationMath 122 Test 3. April 15, 2014
SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not
More informationCore Mathematics C34
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C34 Advanced Tuesday 20 June 2017 Afternoon Time: 2 hours 30 minutes
More informationTime: 1 hour 30 minutes
www.londonnews47.com Paper Reference(s) 6665/0 Edexcel GCE Core Mathematics C Bronze Level B4 Time: hour 0 minutes Materials required for examination papers Mathematical Formulae (Green) Items included
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationMATHEMATICS Higher Grade - Paper I (Non~calculator)
Prelim Eamination 006 / 007 (Assessing Units & ) MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefully. Calculators may not be used in this paper.. Full credit
More information0606 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maximum raw mark 80
More informationH2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
More informationMath 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
More informationIYGB Mathematical Methods 1
IYGB Mathematical Methods Practice Paper A Time: 3 hours Candidates may use any non programmable, non graphical calculator which does not have the capability of storing data or manipulating algebraic expressions
More informationFirst Midterm Examination
Çankaya University Department of Mathematics 016-017 Fall Semester MATH 155 - Calculus for Engineering I First Midterm Eamination 1) Find the domain and range of the following functions. Eplain your solution.
More informationMathematics Specialist Units 3 & 4 Program 2018
Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
More informationUNIVERSITY OF SOUTHAMPTON. A foreign language dictionary (paper version) is permitted provided it contains no notes, additions or annotations.
UNIVERSITY OF SOUTHAMPTON MATH055W SEMESTER EXAMINATION 03/4 MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min Solutions Only University approved calculators may be used. A foreign language
More information1969 AP Calculus BC: Section I
969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationIYGB Mathematical Methods 1
IYGB Mathematical Methods Practice Paper B Time: 3 hours Candidates may use any non programmable, non graphical calculator which does not have the capability of storing data or manipulating algebraic expressions
More informationWorkbook for Calculus I
Workbook for Calculus I By Hüseyin Yüce New York 2007 1 Functions 1.1 Four Ways to Represent a Function 1. Find the domain and range of the function f(x) = 1 + x + 1 and sketch its graph. y 3 2 1-3 -2-1
More information(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,
Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),
More informationTopics from Algebra and Pre-Calculus. (Key contains solved problems)
Topics from Algebra and Pre-Calculus (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the calculator, except on p. (8) and
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
More informationParametric Equations and Polar Coordinates
Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Exam 4 Review 1. Trig substitution
More information9709 MATHEMATICS 8719 HIGHER MATHEMATICS
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary and Advanced Level MARK SCHEME for the June 005 question paper 9709 MATHEMATICS 8719 HIGHER MATHEMATICS 9709/03, 8719/03 Paper
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More information8.3 Partial Fraction Decomposition
8.3 partial fraction decomposition 575 8.3 Partial Fraction Decomposition Rational functions (polynomials divided by polynomials) and their integrals play important roles in mathematics and applications,
More informationC3 papers June 2007 to 2008
physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+
More informationMULTIVARIABLE CALCULUS
MULTIVARIABLE CALCULUS Summer Assignment Welcome to Multivariable Calculus, Multivariable Calculus is a course commonly taken by second and third year college students. The general concept is to take the
More informationab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationTechniques of Integration
Chapter 8 Techniques of Integration 8. Trigonometric Integrals Summary (a) Integrals of the form sin m x cos n x. () sin k+ x cos n x = ( cos x) k cos n x (sin x ), then apply the substitution u = cos
More informationMath 122 Test 3. April 17, 2018
SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator
More informationSTEP Support Programme. Pure STEP 1 Questions
STEP Support Programme Pure STEP 1 Questions 2012 S1 Q4 1 Preparation Find the equation of the tangent to the curve y = x at the point where x = 4. Recall that x means the positive square root. Solve the
More informationUniversity of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 2012 First Year - CHE, CIV, IND, LME, MEC, MSE
University of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 212 First Year - CHE, CIV, IND, LME, MEC, MSE MAT187H1F - CALCULUS II Exam Type: A Examiner: D. Burbulla INSTRUCTIONS:
More informationMathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
More informationCore Mathematics C4. You must have: Mathematical Formulae and Statistical Tables (Pink)
Write your name here Surname Other names Pearson Edexcel GCE Centre Number Core Mathematics C4 Advanced Candidate Number Friday 24 June 2016 Morning Time: 1 hour 30 minutes Paper Reference 6666/01 You
More informationCore Mathematics C34
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C34 Advanced Monday 16 June 2014 Morning Time: 2 hours 30 minutes You
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationSolutions to Math 41 First Exam October 15, 2013
Solutions to Math 41 First Exam October 15, 2013 1. (16 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether
More informationPhysicsAndMathsTutor.com. Core Mathematics C4. You must have: Mathematical Formulae and Statistical Tables (Pink)
Write your name here Surname Other names Pearson Edexcel GCE Centre Number Core Mathematics C4 Advanced Candidate Number Friday 24 June 2016 Morning Time: 1 hour 30 minutes Paper Reference 6666/01 You
More informationPaper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours
1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mark scheme Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question
More information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationFinal Exam Review Problems
Final Exam Review Problems Name: Date: June 23, 2013 P 1.4. 33. Determine whether the line x = 4 represens y as a function of x. P 1.5. 37. Graph f(x) = 3x 1 x 6. Find the x and y-intercepts and asymptotes
More informationMath. 151, WebCalc Sections December Final Examination Solutions
Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for
More informationTest one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
More informationC3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009)
C3 Revision Questions (using questions from January 2006, January 2007, January 2008 and January 2009) 1 2 1. f(x) = 1 3 x 2 + 3, x 2. 2 ( x 2) (a) 2 x x 1 Show that f(x) =, x 2. 2 ( x 2) (4) (b) Show
More information