UNIVERSITY of LIMERICK

Size: px
Start display at page:

Download "UNIVERSITY of LIMERICK"

Transcription

1 UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4701 SEMESTER: Autumn 2011/12 MODULE TITLE: Technological Mathematics 1 DURATION OF EXAMINATION: hours LECTURER: J. O Shea PERCENTAGE OF TOTAL MARKS: 85 % EXTERNAL EXAMINER: Prof. T. Myers INSTRUCTIONS TO CANDIDATES: Questions One (Q1) is compulsory and carries 40 marks. Answer any other four questions worth 15 marks each. N.B. There are some useful formulae at the end of the paper.

2 1. (a) Find the equation of the line passing through the point (1, 3) and which is perpendicular to the line 2x y + 1 = 0. 4 (b) Solve the equation ln (4x + 1) = 3. (ln = log e ) 4 (c) Find the first derivative of the following functions (i) y = 3x 4 + ln 2x (ii) f(x) = (x 2 4x + 1) 4 (iii) g(x) = e 4x sin x (iv) f(t) = 3t2 +2 2t+1. 8 (d) Find the equation of the tangent to the curve y = 3x 3 6x 2 5x + 11 at the point (2, 1). 4 (e) Use De Moivre s theorem to evaluate (1 + 2i) 8. 4 (f) (i) If z = 3 + 5i evaluate z z. (ii) Express e 3i in the form a + bi. 4 (g) ā = 3i + j 2k, find the unit vector in the direction of ā. 4 (h) Evaluate the matrix product ( ) ( ) (i) ( ) ( ) (ii) (i) From the following augmented matrix, state the solution to the linear system:

3 2. (a) (i) Find f 1 (x) the inverse of the function f(x) = 2x + 1. (ii) f(x) = 3x 2 and g(x) = sin 4x, find the composite function f g(x). (iii) Prove that f(x) = (iv) Evaluate 3x is an odd function. x 2 +1 lim x 2x 5 x + 1. (v) Find the local minimum point of y = x 2 4x+7 and hence sketch the function. 10 (b) (i) Find the value of x R and y R given that ( ) ( ) ( ) x 4x 1 9 =. y 2y (ii) A = Find det(a). 5 2

4 3. (a) The function y = 4 e 0.002t denotes the process known as radioactive decay where 4 grams is the initial level of radium and t is the time in years. How long will it take for the radium to reduce to 2 grams? 5 (b) Determine the amplitude A, angular velocity ω, period T and frequency f of the function y = 4 sin 2t. Sketch the function. 5 (c) For ABC, AB = 3cm, AC = 4cm and BAC = 60. A C B Find (i) BC correct to 1 decimal place. (ii) ABC to the nearest degree (a) Let f(x) = x 3 12x + 4. Determine the stationary points of f(x) and classify them as local maxima or minima. Find also the inflection point. Sketch the curve. 10 (b) A ball is thrown vertically upwards. The height h metres of the ball, t seconds after it is thrown is given by the formula h = 3t(10 t). Find (i) the speed of the ball after 1 second. (ii) the time when the ball reaches its maximum height. (iii) the maximum height reached. 5 3

5 5. (a) Given z 1 = i and z 2 = 3 - i. Find (i) 4z 1 - z 2. (ii) 3 z 1 in the form a + bi. (iii) the real number k so that 5k = 1 + z 1. (iv) the polar form of z 2. 8 (b) Express -16 in general polar form. Find the four fourth roots of ( 16) 1/4. Plot the roots in the complex plane (a) Given a = 2i j + k b = 3i + 2j + 4k Find (i) 3a b. (ii) the scalar product a.b (iii) the vector product a b. 11 (b) An object is experiencing two perpendicular forces F 1 of 20N and F 2 of 15N as shown in the diagram. Calculate the magnitude of the resultant force F and determine the angle θ that F makes with the vertical. θ F 1 F 2 F 4 4

6 7. (a) Solve the following system of linear equations using Gaussian elimination, or otherwise x + 2y + 3z = 9 3x + y + 2z = 11 2x y + z = 8 8 (b) Verify by drawing a suitable graph that the values of Q and P given by the table below satisfy a law of the form Q = a + bp 2. Determine the best values for a and b using the following b = nσxy ΣxΣy and a = Σy nσx 2 (Σx) 2 n bσx n where a and b are the intercept and slope of the least squares line y = a+bx. P Q

7 Formulae 1. m = slope PQ = y 2 y 1 x 2 x 1 equation PQ : y y 1 = m(x x 1 ) P (x 1,y 1 ) Q (x 2,y 2 ) 2. Logarithms a x = y log a y = x 3. De Moivre s theorem 4. Vector operations [r(cos θ + i sin θ)] n = r n (cos nθ + i sin nθ) = r n e inθ v 1 = x 1 i + y 1 j and v 2 = x 2 ī + y 2 j v 1 = x y1 2 v 1 v 2 = x 1 x 2 + y 1 y 2 = v 1 v 2 cos θ (norm) (scalar product) ( a b 5. Inverse of a matrix A = c d = ad bc 0 6. Calculus ) ( is 1 d b det (A) c a ) where det(a) f(x) x n f (x) nx n 1 ln x 1 e x e ax cos x sin x e x ae ax sin x cos x 6

8 Product Rule: Quotient Rule: y = uv dy dx = udv dx + v du dx 7. Trigonometry Tables Page 9 and π Rad. = 180 y = u v dy dx = v du u dv dx dx v 2 7

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH FACULTY OF SCIENCE AND ENGINEERING DEPARTMENT OF MATHEMATICS & STATISTICS END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA440 SEMESTER: Autumn 04 MODULE TITLE:

More information

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics and Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4006 SEMESTER: Spring 2011 MODULE TITLE:

More information

Review for the Final Exam

Review 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 information

C3 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) 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

b) The trend is for the average slope at x = 1 to decrease. The slope at x = 1 is 1.

b) The trend is for the average slope at x = 1 to decrease. The slope at x = 1 is 1. Chapters 1 to 8 Course Review Chapters 1 to 8 Course Review Question 1 Page 509 a) i) ii) [2(16) 12 + 4][2 3+ 4] 4 1 [2(2.25) 4.5+ 4][2 3+ 4] 1.51 = 21 3 = 7 = 1 0.5 = 2 [2(1.21) 3.3+ 4][2 3+ 4] iii) =

More information

UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test

UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following

More information

College Algebra and College Algebra with Review Final Review

College Algebra and College Algebra with Review Final Review The final exam comprises 30 questions. Each of the 20 multiple choice questions is worth 3 points and each of the 10 open-ended questions is worth 4 points. Instructions for the Actual Final Exam: Work

More information

MTH Calculus with Analytic Geom I TEST 1

MTH Calculus with Analytic Geom I TEST 1 MTH 229-105 Calculus with Analytic Geom I TEST 1 Name Please write your solutions in a clear and precise manner. SHOW your work entirely. (1) Find the equation of a straight line perpendicular to the line

More information

Core 3 (A2) Practice Examination Questions

Core 3 (A2) Practice Examination Questions Core 3 (A) Practice Examination Questions Trigonometry Mr A Slack Trigonometric Identities and Equations I know what secant; cosecant and cotangent graphs look like and can identify appropriate restricted

More information

MAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists

MAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists Data provided: Formula sheet MAS53/MAS59 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics (Materials Mathematics For Chemists Spring Semester 203 204 3 hours All questions are compulsory. The marks awarded

More information

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.

Candidates 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 information

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph. Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

More information

Math 1120 Calculus Final Exam

Math 1120 Calculus Final Exam May 4, 2001 Name The first five problems count 7 points each (total 35 points) and rest count as marked. There are 195 points available. Good luck. 1. Consider the function f defined by: { 2x 2 3 if x

More information

SECTION 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. 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 information

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON UNIVERSITY OF SOUTHAMPTON MATH03W SEMESTER EXAMINATION 0/ MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min This paper has two parts, part A and part B. Answer all questions from part

More information

MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 2015

MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 2015 MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 05 Copyright School Curriculum and Standards Authority, 05 This document apart from any third party copyright material contained in it may be freely

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS ADDITIONAL MATHEMATICS GCE NORMAL ACADEMIC LEVEL (016) (Syllabus 4044) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE

More information

Math 180, Final Exam, Fall 2007 Problem 1 Solution

Math 180, Final Exam, Fall 2007 Problem 1 Solution Problem Solution. Differentiate with respect to x. Write your answers showing the use of the appropriate techniques. Do not simplify. (a) x 27 x 2/3 (b) (x 2 2x + 2)e x (c) ln(x 2 + 4) (a) Use the Power

More information

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS4613 SEMESTER: Autumn 2002/03 MODULE TITLE: Vector Analysis DURATION OF EXAMINATION:

More information

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS425 SEMESTER: Autumn 25/6 MODULE TITLE: Applied Analysis DURATION OF EXAMINATION:

More information

a 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).

a 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 information

4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity 4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

More information

OR MSc Maths Revision Course

OR MSc Maths Revision Course OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision

More information

Integration by Parts

Integration by Parts Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u

More information

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0)

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 2017 First certification from June 2018 2

More information

Week 12: Optimisation and Course Review.

Week 12: Optimisation and Course Review. Week 12: Optimisation and Course Review. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway November 21-22, 2016 Assignments. Problem

More information

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook

More information

C3 papers June 2007 to 2008

C3 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 information

March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work.

March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work. March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work. 1. (12 points) Consider the cubic curve f(x) = 2x 3 + 3x + 2. (a) What

More information

Summer Math Packet for AP Calculus BC

Summer Math Packet for AP Calculus BC Class: Date: Summer Math Packet for AP Calculus BC 018-19 1. Find the smallest value in the range of the function f (x) = x + 4x + 40. a. 4 b. 5 c. 6 d. 7 e. 8 f. 16 g. 4 h. 40. Find the smallest value

More information

Core Mathematics C34

Core 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 information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.9 Antiderivatives In this section, we will learn about: Antiderivatives and how they are useful in solving certain scientific problems.

More information

MLC Practice Final Exam. Recitation Instructor: Page Points Score Total: 200.

MLC Practice Final Exam. Recitation Instructor: Page Points Score Total: 200. Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 20 4 30 5 20 6 20 7 20 8 20 9 25 10 25 11 20 Total: 200 Page 1 of 11 Name: Section:

More information

Semester 1 Exam Review - Precalculus Test ID:

Semester 1 Exam Review - Precalculus Test ID: 203-4 Semester Exam Review - Precalculus Test ID: Use interval notation to describe the interval of real numbers. ) x is greater than or equal to 0 and less than or equal to 4. ) A) [0, 4) B) (0, 4] C)

More information

m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule)

m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule) Chapter 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) = c f (x) = 0 g(x) = x g (x) = 1 h(x) = x n h (x) = n x n-1 (The Power Rule) k(x)

More information

Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours

Paper 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 information

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS Surname Centre Number Candidate Number Other Names 0 WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS A.M. MONDAY, 22 June 2015 2 hours 30 minutes S15-9550-01 For s use ADDITIONAL MATERIALS A calculator

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

More information

Math. 151, WebCalc Sections December Final Examination Solutions

Math. 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 information

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA GRADE 12 EXAMINATION NOVEMBER 2016 ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA Time: 2 hours 200 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper

More information

Announcements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 5.2 5.7, 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems

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?

= 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 information

Review for Cumulative Test 2

Review for Cumulative Test 2 Review for Cumulative Test We will have our second course-wide cumulative test on Tuesday February 9 th or Wednesday February 10 th, covering from the beginning of the course up to section 4.3 in our textbook.

More information

We first review various rules for easy differentiation of common functions: The same procedure works for a larger number of terms.

We first review various rules for easy differentiation of common functions: The same procedure works for a larger number of terms. 1 Math 182 Lecture Notes 1. Review of Differentiation To differentiate a function y = f(x) is to find its derivative f '(x). Another standard notation for the derivative is Dx(f(x)). Recall the meanings

More information

Sydney Grammar School

Sydney Grammar School 1. (a) Evaluate 3 0 Sydney Grammar School 4 unit mathematics Trial HSC Examination 2000 x x 2 +1 dx. (b) The integral I n is defined by I n = 1 0 xn e x dx. (i) Show that I n = ni n 1 e 1. (ii) Hence show

More information

Exam 3 MATH Calculus I

Exam 3 MATH Calculus I Trinity College December 03, 2015 MATH 131-01 Calculus I By signing below, you attest that you have neither given nor received help of any kind on this exam. Signature: Printed Name: Instructions: Show

More information

UNIVERSITY OF REGINA Department of Mathematics and Statistics. Calculus I Mathematics 110. Final Exam, Winter 2013 (April 25 th )

UNIVERSITY OF REGINA Department of Mathematics and Statistics. Calculus I Mathematics 110. Final Exam, Winter 2013 (April 25 th ) UNIVERSITY OF REGINA Department of Mathematics and Statistics Calculus I Mathematics 110 Final Exam, Winter 2013 (April 25 th ) Time: 3 hours Pages: 11 Full Name: Student Number: Instructor: (check one)

More information

Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4

Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4 Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x

More information

x 1 2 i 1 5 2i 11 9x 9 3x 3 1 y 2 3y 4 y 2 1 Poudre School District s College Algebra Course Review

x 1 2 i 1 5 2i 11 9x 9 3x 3 1 y 2 3y 4 y 2 1 Poudre School District s College Algebra Course Review . Perform the indicated operations and simplify the result. Leave the answer in factored form. 9x 9 x a. b. 9x 9 x x. Solve: x 7x x 0. a. x., b. x 0,., x,0,. x.,0,. Find the quotient and the remainder

More information

Part I: Multiple Choice Questions

Part I: Multiple Choice Questions Name: Part I: Multiple Choice Questions. What is the slope of the line y=3 A) 0 B) -3 ) C) 3 D) Undefined. What is the slope of the line perpendicular to the line x + 4y = 3 A) -/ B) / ) C) - D) 3. Find

More information

Suppose that f is continuous on [a, b] and differentiable on (a, b). Then

Suppose that f is continuous on [a, b] and differentiable on (a, b). Then Lectures 1/18 Derivatives and Graphs When we have a picture of the graph of a function f(x), we can make a picture of the derivative f (x) using the slopes of the tangents to the graph of f. In this section

More information

*P46958A0244* IAL PAPER JANUARY 2016 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA. 1. f(x) = (3 2x) 4, x 3 2

*P46958A0244* IAL PAPER JANUARY 2016 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA. 1. f(x) = (3 2x) 4, x 3 2 Edexcel "International A level" "C3/4" papers from 016 and 015 IAL PAPER JANUARY 016 Please use extra loose-leaf sheets of paper where you run out of space in this booklet. 1. f(x) = (3 x) 4, x 3 Find

More information

2017 LCHL Paper 1 Table of Contents

2017 LCHL Paper 1 Table of Contents 3 7 10 2 2017 PAPER 1 INSTRUCTIONS There are two sections in this examination paper. Section A Concepts and Skills 150 marks 6 questions Section B Contexts and Applications 150 marks 3 questions Answer

More information

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question

More information

APPM 1350 Final Exam Fall 2017

APPM 1350 Final Exam Fall 2017 APPM 350 Final Exam Fall 207. (26 pts) Evaluate the following. (a) Let g(x) cos 3 (π 2x). Find g (π/3). (b) Let y ( x) x. Find y (4). (c) lim r 0 e /r ln(r) + (a) (9 pt) g (x) 3 cos 2 (π 2x)( sin(π 2x))(

More information

Mathematics (JAN12MPC201) General Certificate of Education Advanced Subsidiary Examination January Unit Pure Core TOTAL

Mathematics (JAN12MPC201) General Certificate of Education Advanced Subsidiary Examination January Unit Pure Core TOTAL Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Pure Core 2 Friday 13 January 2012 General Certificate of Education Advanced

More information

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics. MATH C Test #1. October 23, 2013.

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics. MATH C Test #1. October 23, 2013. YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 1505 6.00 C Test #1 October 3, 013 Solutions Family Name (print): Student No: Given Name: Signature: INSTRUCTIONS: 1. Please

More information

AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period:

AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: WORKSHEET: Series, Taylor Series AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: 1 Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The

More information

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 611b Assignment #6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the function graphed. 1) 1) A) f(x) = 5 + x, x < -

More information

Core 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 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 information

MATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010)

MATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010) Course Prerequisites MATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010) As a prerequisite to this course, students are required to have a reasonable mastery of precalculus mathematics

More information

ABSOLUTE VALUE INEQUALITIES, LINES, AND FUNCTIONS MODULE 1. Exercise 1. Solve for x. Write your answer in interval notation. (a) 2.

ABSOLUTE VALUE INEQUALITIES, LINES, AND FUNCTIONS MODULE 1. Exercise 1. Solve for x. Write your answer in interval notation. (a) 2. MODULE ABSOLUTE VALUE INEQUALITIES, LINES, AND FUNCTIONS Name: Points: Exercise. Solve for x. Write your answer in interval notation. (a) 2 4x 2 < 8 (b) ( 2) 4x 2 8 2 MODULE : ABSOLUTE VALUE INEQUALITIES,

More information

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra

More information

MATHEMATICS Unit Pure Core 2

MATHEMATICS Unit Pure Core 2 General Certificate of Education June 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Thursday 15 May 2008 9.00 am to 10.30 am For this paper you must have: an 8-page answer book

More information

McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS A CALCULUS I EXAMINER: Professor K. K. Tam DATE: December 11, 1998 ASSOCIATE

McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS A CALCULUS I EXAMINER: Professor K. K. Tam DATE: December 11, 1998 ASSOCIATE NOTE TO PRINTER (These instructions are for the printer. They should not be duplicated.) This examination should be printed on 8 1 2 14 paper, and stapled with 3 side staples, so that it opens like a long

More information

AS PURE MATHS REVISION NOTES

AS 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 information

Solutionbank Edexcel AS and A Level Modular Mathematics

Solutionbank Edexcel AS and A Level Modular Mathematics Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x

More information

PreCalculus Second Semester Review Ch. P to Ch. 3 (1st Semester) ~ No Calculator

PreCalculus Second Semester Review Ch. P to Ch. 3 (1st Semester) ~ No Calculator PreCalculus Second Semester Review Ch. P to Ch. 3 (1st Semester) ~ No Calculator Solve. Express answer using interval notation where appropriate. Check for extraneous solutions. P3 1. x x+ 5 1 3x = P5.

More information

UNIVERSITY OF BOLTON WESTERN INTERNATIONAL COLLEGE FZE BENG (HONS) CIVIL ENGINEERING SEMESTER TWO EXAMINATION 2015/2016

UNIVERSITY OF BOLTON WESTERN INTERNATIONAL COLLEGE FZE BENG (HONS) CIVIL ENGINEERING SEMESTER TWO EXAMINATION 2015/2016 OCD74 UNIVERSITY OF BOLTON WESTERN INTERNATIONAL COLLEGE FZE BENG (HONS) CIVIL ENGINEERING SEMESTER TWO EXAMINATION 015/016 MATHEMATICS AND STRUCTURAL DESIGN MODULE NO: CIE401 Date: Saturday 8 May 016

More information

Calculus first semester exam information and practice problems

Calculus first semester exam information and practice problems Calculus first semester exam information and practice problems As I ve been promising for the past year, the first semester exam in this course encompasses all three semesters of Math SL thus far. It is

More information

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA GRADE 1 EXAMINATION NOVEMBER 017 ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA Time: hours 00 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists

More information

CSSA Trial HSC Examination

CSSA Trial HSC Examination CSSA Trial HSC Examination. (a) Mathematics Extension 2 2002 The diagram shows the graph of y = f(x) where f(x) = x2 x 2 +. (i) Find the equation of the asymptote L. (ii) On separate diagrams sketch the

More information

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS ADDITIONAL MATHEMATICS GCE Ordinary Level (06) (Syllabus 4047) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE

More information

Possible C4 questions from past papers P1 P3

Possible C4 questions from past papers P1 P3 Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.

More information

Time: 1 hour 30 minutes

Time: 1 hour 30 minutes Paper Reference (complete below) Centre No. Surname Initial(s) Candidate No. Signature Paper Reference(s) 6663 Edexcel GCE Pure Mathematics C Advanced Subsidiary Specimen Paper Time: hour 30 minutes Examiner

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS ADDITIONAL MATHEMATICS GCE Ordinary Level (Syllabus 4018) CONTENTS Page NOTES 1 GCE ORDINARY LEVEL ADDITIONAL MATHEMATICS 4018 2 MATHEMATICAL NOTATION 7 4018 ADDITIONAL MATHEMATICS O LEVEL (2009) NOTES

More information

CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS Objectives To introduce students to a number of topics which are fundamental to the advanced study of mathematics. Assessment Examination (72 marks) 1 hour

More information

A. Evaluate log Evaluate Logarithms

A. Evaluate log Evaluate Logarithms A. Evaluate log 2 16. Evaluate Logarithms Evaluate Logarithms B. Evaluate. C. Evaluate. Evaluate Logarithms D. Evaluate log 17 17. Evaluate Logarithms Evaluate. A. 4 B. 4 C. 2 D. 2 A. Evaluate log 8 512.

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Differential Calculus 2 Contents Limits..5 Gradients, Tangents and Derivatives.6 Differentiation from First Principles.8 Rules for Differentiation..10 Chain Rule.12

More information

G H. Extended Unit Tests A L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests

G 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 information

Higher Mathematics Course Notes

Higher Mathematics Course Notes Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that

More information

Final Problem Set. 2. Use the information in #1 to show a solution to the differential equation ), where k and L are constants and e c L be

Final Problem Set. 2. Use the information in #1 to show a solution to the differential equation ), where k and L are constants and e c L be Final Problem Set Name A. Show the steps for each of the following problems. 1. Show 1 1 1 y y L y y(1 ) L.. Use the information in #1 to show a solution to the differential equation dy y ky(1 ), where

More information

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear

More information

Learning Objectives These show clearly the purpose and extent of coverage for each topic.

Learning Objectives These show clearly the purpose and extent of coverage for each topic. Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus

More information

MLC Practice Final Exam

MLC Practice Final Exam Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response

More information

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Winter 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Winter 01/01! This problems in this packet are designed to help you review the topics from

More information

Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point

Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the

More information

Integrated Calculus II Exam 1 Solutions 2/6/4

Integrated Calculus II Exam 1 Solutions 2/6/4 Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (

More information

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA CHAPTER 7 MAXIMA AND MINIMA 7.1 INTRODUCTION The notion of optimizing functions is one of the most important application of calculus used in almost every sphere of life including geometry, business, trade,

More information

Part A: Short Answer Questions

Part A: Short Answer Questions Math 111 Practice Exam Your Grade: Fall 2015 Total Marks: 160 Instructor: Telyn Kusalik Time: 180 minutes Name: Part A: Short Answer Questions Answer each question in the blank provided. 1. If a city grows

More information

Extra FP3 past paper - A

Extra FP3 past paper - A Mark schemes for these "Extra FP3" papers at https://mathsmartinthomas.files.wordpress.com/04//extra_fp3_markscheme.pdf Extra FP3 past paper - A More FP3 practice papers, with mark schemes, compiled from

More information

2 If ax + bx + c = 0, then x = b) What are the x-intercepts of the graph or the real roots of f(x)? Round to 4 decimal places.

2 If ax + bx + c = 0, then x = b) What are the x-intercepts of the graph or the real roots of f(x)? Round to 4 decimal places. Quadratic Formula - Key Background: So far in this course we have solved quadratic equations by the square root method and the factoring method. Each of these methods has its strengths and limitations.

More information

2013 Bored of Studies Trial Examinations. Mathematics SOLUTIONS

2013 Bored of Studies Trial Examinations. Mathematics SOLUTIONS 03 Bored of Studies Trial Examinations Mathematics SOLUTIONS Section I. B 3. B 5. A 7. B 9. C. D 4. B 6. A 8. D 0. C Working/Justification Question We can eliminate (A) and (C), since they are not to 4

More information

St. Anne s Diocesan College. Grade 12 Core Mathematics: Paper II September Time: 3 hours Marks: 150

St. Anne s Diocesan College. Grade 12 Core Mathematics: Paper II September Time: 3 hours Marks: 150 St. Anne s Diocesan College Grade 12 Core Mathematics: Paper II September 2018 Time: 3 hours Marks: 150 Please read the following instructions carefully: 1. This question paper consists of 21 pages and

More information

2. Find the midpoint of the segment that joins the points (5, 1) and (3, 5). 6. Find an equation of the line with slope 7 that passes through (4, 1).

2. Find the midpoint of the segment that joins the points (5, 1) and (3, 5). 6. Find an equation of the line with slope 7 that passes through (4, 1). Math 129: Pre-Calculus Spring 2018 Practice Problems for Final Exam Name (Print): 1. Find the distance between the points (6, 2) and ( 4, 5). 2. Find the midpoint of the segment that joins the points (5,

More information

Technical Calculus I Homework. Instructions

Technical Calculus I Homework. Instructions Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

National Quali cations

National Quali cations H 2017 X747/76/11 FRIDAY, 5 MAY 9:00 AM 10:10 AM National Quali cations Mathematics Paper 1 (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given

More information