Mathematics. Guide

Size: px
Start display at page:

Download "Mathematics. Guide"

Transcription

1 Mthemtics Guide - Contents Question Item Objective Type Skill 00 ALG.0.04 Multiple-choice nswer Applictions 046 ALG.0.04 Multiple-choice nswer Concepts ALG.0.03 Multiple-choice nswer Concepts ALG.0.0 Multiple-choice nswer Concepts 5 08 ALG.0.0 Multiple-choice nswer Concepts ALG.0.04 Multiple-choice nswer Applictions ALG.0.04 Multiple-choice nswer Applictions ALG.0.04 Multiple-choice nswer Concepts ALG.0.04 Multiple-choice nswer Concepts ALG.03 Short-constructed nswer Applictions 39 ALG.0.04 Short-constructed nswer Concepts 05 ALG.0 Short-constructed nswer Applictions ALG.0.04 Short-constructed nswer Applictions ALG.0.04 Short-constructed nswer Concepts 5 8 ALG.0 Short-constructed nswer Applictions 6 40 ALG.0.04 Short-constructed nswer Applictions ALG.0 Etended nswer Problem solving ALG.0 Etended nswer Problem solving 9 ALG.0 Etended nswer Problem solving ALG.0.04 Etended nswer Problem solving

2 - Correction key D C 3 B 4 A 5 B 6 B 7 A 8 D 9 A 0 The solution set of the inequlity is [7, 3[ ]43, + [. The rnge of f () is -, -5]

3 It will cost $3.5 to send the prcel f. 3 The inverse of this function is ( ) M cn hope to rech the minimum weight of 80 kg. 5 The equtions of the symptotes re -4 nd y. 6 The domin of f() is ]-, 3[ ]3, + [. The rnge of f() is ]-, -.5[ ]-.5, + [. 7 Emple of n pproprite solution Eqution of squre root function R() h + k R() 3 4 Zero of the function R() 3 4

4 Answer : The compny begn to mke profit fter.8 months. 8 Emple of n pproprite solution Eqution of the rmp f ( ) 3 + Replcing point (, 3), into the eqution f ( ) Comprison method

5 Zeroes 6 ± (-6) Reject (negtive squre root) Height of the br g 6 7 ( 7 ) 7 6 Answer: The height of the metl br is 6 units from the ground. 9 Emple of n pproprite solution Gretest integer function 0 y 3.5[ 0.05( 0) + 3] 3.5[ 3] cm Step length Lst open point is (0, 50)

6 Rtionl function y Eqution y y cm 0 Answer: To the nerest tenth of centimetre, the distnce is 5.3 cm. 0 Emple of n pproprite solution N ( p) 9 5 p N(p) - 66 N ( p) - 66 ( p 00) - 66p p? p p p p $55.90 p (00, 5) (55.90, 86) p y 86 Answer: The mimum price the school cn chrge is $55.90.

7 Nme : Group : Dte : Mthemtics Question Booklet The members of school finnce committee decided to print nd sell souvenir picture lbums s fundrising project. They estblished tht the profit P(), in dollrs, from the sle of these $5 lbums could be determined by the following eqution : P ( ) where represents the number of lbums to be printed nd sold. Wht is the minimum number of lbums the committee must sell to relize profit? A) 00 lbums C) 35 lbums B) 50 lbums D) 395 lbums

8 Given the function defined by the eqution ( ) f. Wht is the domin of this function? A) R + B) [-6, C) [-, D) [4, 3 The grph on the right represents function f y The rule of this function is of the form ( ) b( h) + k f. Which of the following sttements is FALSE? A) ]-, 0[ C) h ]0, + [ B) b ]0, + [ D) k ]0, + [

9 4 Which of grphs below represents the eqution f() [-] +? A) C) f() f() - - B) D) f() f() - -

10 5 Which of the grphs below represents the function f ( ) +? A) C) f() f() B) D) f() 4 f() The function f is defined by the following rule: f ( ) ( ) Wht re the zeros of this function? A) ].5, [ C) ]5, 8[ B) ]3, 5] D) [5, 8[

11 7 The function g is defined by the following rule: g ( ) Wht is the rule of its inverse g? A) g ( ) C) g ( ) B) g ( ) D) g ( ) Given f() nd g() 5 7. The function h is defined by h() f g ( ) ( ) where g() 0. Wht is the domin nd rnge of function h? A) Dom h R \ 3 5 C) Dom h R \ 7 5 Im h R \ -7 5 Im h R \ B) Dom h R \ 3 5 D) Dom h R \ 7 5 Im h R \ 7 5 Im h R \ - 3 5

12 9 3 Given the rtionl function ƒ() + 5. Wht re the equtions of the symptotes of this function? A) y 3 C) 3 y B) y -5 D) 3 - y 0 Wht is the solution set of the following inequlity? < A rocket is shot into the ir by submrine locted t the verte of the following squre root function: f ( ) 3-5 The rocket follows the pth of the squre root function. Wht is the rnge of f ()? The cost C, in dollrs, to send prcel is given by the function C() [.75] +.5 where is the mss in kg. How much will it cost Dnielle to send prcel tht weighs 4.4 kg?

13 3-0 Given the function f ( ) Wht is the rule of correspondence of the inverse of this function? 4 M's New Yer's resolution ws to lose weight by pying strict ttention to the food he te. Since then, his weight hs vried ccording to the function whose rule is: M ( t) t where t represents the number of dys gone by since Jnury st, nd m(t) represents M's weight in kilogrms. According to the rule of this function, wht minimum weight, rounded to the nerest unit, cn M hope to rech? 5 Given: ( ) + f nd g() Wht re the equtions of the verticl nd horizontl symptotes of ( g)( ) f? 6 Given the rtionl function ( ) ( ) - 3 f. 6 Wht re the domin nd rnge of f()?

14 7 After one month in opertion, compny's revenue hs grown ccording to the formul ( ) h k R + where R() is the profit in millions of dollrs fter period of months. R() Revenue in millions ($) (0, 5) (, -4) months Losses mounted to $400 the first month. After ten months, $5000 in profits ws recorded. After how mny months in opertion did the compny begin to mke profit? 8 The curve of sktebord rmp is defined by squre root function whose verte is (3, ) nd which psses through the point (, 3). In order to mke the rmp more secure, metl br ws propped up under it. The rule of the line representing the metl br is 6 g( ). 7 f() (3, ) rmp At wht height from the ground is this metl br ttched to the rmp? metl br (, 3)

15 9 A designer is prepring model of children's slide. She begn by drwing the steps nd the slide on Crtesin plne scled in cm, s shown in the digrm below. y 80 cm The steps of the slide re represented by the reltion y 3.5[ ] The top step begins on the y-is. The slide is ttched to the other end of the top step. The slide is represented by rtionl function with the eqution y The end of the slide is 80 cm from the origin of the Crtesin plne. d To the nerest tenth of centimetre, wht is the distnce (d) from the ground to the end of the slide? 0 A mthemticl reltionship eists between the number of students who prticipte in the end-of-yer school ctivities nd the mount of money students will be chrged to prticipte. The greter the number of prticipnts, the less ech hs to py. This reltion is represented by the following function: N ( p ) p 300 where p is the price of the ctivities; N(p) is the number of students prticipting. At lest 70% of the 30 students in County High must gree to prticipte, before the end-of-yer ctivities will be pproved. Wht is the mimum price the school cn chrge for the end-of-yer ctivities?

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1 MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

HQPD - ALGEBRA I TEST Record your answers on the answer sheet.

HQPD - ALGEBRA I TEST Record your answers on the answer sheet. HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property

More information

0.1 THE REAL NUMBER LINE AND ORDER

0.1 THE REAL NUMBER LINE AND ORDER 6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

More information

Mathematics Extension 2

Mathematics 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 information

MAC 1105 Final Exam Review

MAC 1105 Final Exam Review 1. Find the distnce between the pir of points. Give n ect, simplest form nswer nd deciml pproimtion to three plces., nd, MAC 110 Finl Em Review, nd,0. The points (, -) nd (, ) re endpoints of the dimeter

More information

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chapter 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 information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List 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 information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 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 information

Lesson 5.3 Graph General Rational Functions

Lesson 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 information

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If

More information

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer.

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points

More information

Math Sequences and Series RETest Worksheet. Short Answer

Math Sequences and Series RETest Worksheet. Short Answer Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning

More information

Mathematics Extension 1

Mathematics 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

Before 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!!!

Before 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 information

HIGHER 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 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 information

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x). Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.

More information

MAT137 Calculus! Lecture 20

MAT137 Calculus! Lecture 20 officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find

More information

Sample pages. 9:04 Equations with grouping symbols

Sample pages. 9:04 Equations with grouping symbols Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

CHAPTER 9. Rational Numbers, Real Numbers, and Algebra

CHAPTER 9. Rational Numbers, Real Numbers, and Algebra CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers

More information

Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1

Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1 C Worksheet A Epress ech of the following in the form log = c. 0 = 000 4 = 8 c 56 = 8 d 7 0 = e = f 5 = g 7 9 = 9 h 6 = 6 Epress ech of the following using inde nottion. log 5 5 = log 6 = 4 c 5 = log 0

More information

Grade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations

Grade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

TO: Next Year s AP Calculus Students

TO: 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 information

5-A5 Using Systems of Equations to Solve Word Problems Alg 1H

5-A5 Using Systems of Equations to Solve Word Problems Alg 1H 5-A5 Using Systems of Equtions to Solve Word Problems Alg 1H system of equtions, solve the system using either substitution or liner combintions; then nswer the problem. Remember word problems need word

More information

Chapter 1 Cumulative Review

Chapter 1 Cumulative Review 1 Chpter 1 Cumultive Review (Chpter 1) 1. Simplify 7 1 1. Evlute (0.7). 1. (Prerequisite Skill) (Prerequisite Skill). For Questions nd 4, find the vlue of ech expression.. 4 6 1 4. 19 [(6 4) 7 ] (Lesson

More information

Exponents and Logarithms Exam Questions

Exponents and Logarithms Exam Questions Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the -intercept of the function ln y.. -1 b. 0 c. d.. Which eqution is represented

More information

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Lesson 1: Quadratic Equations

Lesson 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 information

Chapter 1 - Functions and Variables

Chapter 1 - Functions and Variables Business Clculus 1 Chpter 1 - Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:

More information

Equations, expressions and formulae

Equations, expressions and formulae Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b

More information

Alg 2 Honors 2018 DRHS Unit 1 Practice Problems

Alg 2 Honors 2018 DRHS Unit 1 Practice Problems Essentil Understnding: Cn ou represent liner situtions with equtions, grphs, nd inequlities, nd model constrints to optimize solutions? Assignment Clendr D Dte Assignment (Due the next clss meeting) Mond

More information

x 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

x 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 information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

A-01 Basic operations

A-01 Basic operations Appendix Clcult skills Bsic keys Key Use Key Use +,,, Bsic opertions ( ) + / Enters negtive numbers Equls sign, gives the = nswer b c Enters frctions. Deciml point ( ) Enters prentheses DEL Deletes previous

More information

Summary Information and Formulae MTH109 College Algebra

Summary 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 information

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse.

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse. 8.5 The Ellipse Kidne stones re crstl-like ojects tht cn form in the kidnes. Trditionll, people hve undergone surger to remove them. In process clled lithotrips, kidne stones cn now e removed without surger.

More information

Identify graphs of linear inequalities on a number line.

Identify 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 information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

Math 153: Lecture Notes For Chapter 5

Math 153: Lecture Notes For Chapter 5 Mth 5: Lecture Notes For Chpter 5 Section 5.: Eponentil Function f()= Emple : grph f ) = ( if = f() 0 - - - - - - Emple : Grph ) f ( ) = b) g ( ) = c) h ( ) = ( ) f() g() h() 0 0 0 - - - - - - - - - -

More information

SAINT IGNATIUS COLLEGE

SAINT 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 information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-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 information

LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS ASSIGNMENT DUE

LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS ASSIGNMENT DUE Trig/Mth Anl Nme No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS ASSIGNMENT DUE LG- 0-/0- Prctice Set E #,, 9,, 7,,, 9,, 7,,, 9, Prctice Set F #-9 odd Prctice

More information

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Eduction 006 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words SPECIALIST MATHEMATICS Written exmintion Mondy 30 October 006 Reding time: 3.00 pm to

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)

SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive) Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify

More information

Mathematics Extension Two

Mathematics 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 information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A 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 information

REVIEW SHEET FOR PRE-CALCULUS MIDTERM

REVIEW 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 information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th  Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

More information

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below. Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

More information

Each term is formed by adding a constant to the previous term. Geometric progression

Each 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 information

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+ .1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

More information

Mathematics. Guide

Mathematics. Guide 568536 Mathematics Guide Contents Question Item Objective Tpe Skill 083 ALG.0.0 Multiplechoice answer Concepts 06 ALG.03.04 Multiplechoice answer Applications 3 0506 ALG.0.04 Multiplechoice answer Applications

More information

1 The Definite Integral As Area

1 The Definite Integral As Area 1 The Definite Integrl As Are * The Definite Integrl s n Are: When f () is Positive When f () is positive nd < b: Are under grph of f between nd b = f ()d. Emple 1 Find the re under the grph of y = 3 +

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

Exploring parametric representation with the TI-84 Plus CE graphing calculator

Exploring parametric representation with the TI-84 Plus CE graphing calculator Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology

More information

Precalculus Spring 2017

Precalculus 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 information

Adding and Subtracting Rational Expressions

Adding 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 information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1? 008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

More information

2.4 Linear Inequalities and Problem Solving

2.4 Linear Inequalities and Problem Solving Section.4 Liner Inequlities nd Problem Solving 77.4 Liner Inequlities nd Problem Solving S 1 Use Intervl Nottion. Solve Liner Inequlities Using the Addition Property of Inequlity. 3 Solve Liner Inequlities

More information

Equations and Inequalities

Equations 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 information

Pre-AP Geometry Worksheet 5.2: Similar Right Triangles

Pre-AP Geometry Worksheet 5.2: Similar Right Triangles ! re-a Geometr Worksheet 5.2: Similr Right Tringles Nme: te: eriod: Solve. Show ll work. Leve nswers s simplified rdicls on #1-5. For #6, round to the nerer tenth. 12!! 6! 1) =! 8! 6! 2) = 18! 8! w!+!9!

More information

4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.

4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not. 4.1 One-to-One Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches 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 information

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

More information

1.) King invests $11000 in an account that pays 3.5% interest compounded continuously.

1.) King invests $11000 in an account that pays 3.5% interest compounded continuously. DAY 1 Chpter 4 Exponentil nd Logrithmic Functions 4.3 Grphs of Logrithmic Functions Converting between exponentil nd logrithmic functions Common nd nturl logs The number e Chnging bses 4.4 Properties of

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions. TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

More information

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1 Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks

More information

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up 9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute

More information

Pre Regional Mathematical Olympiad, 2016 Delhi Region Set C

Pre Regional Mathematical Olympiad, 2016 Delhi Region Set C Pre Regionl Mthemticl Olympid, 06 Delhi Region Set C Mimum Mrks: 50 Importnt Note: The nswer to ech question is n integer between 0 nd 06. Ech Cndidte must write the finl nswer (in the spce provided) s,

More information

Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A

Time : 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 information

Study Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines

Study Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines Msschusetts Institute of Technology Deprtment of Physics 8.0T Fll 004 Study Guide Finl Exm The finl exm will consist of two sections. Section : multiple choice concept questions. There my be few concept

More information

Operations with Polynomials

Operations 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 information

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

Physics 24 Exam 1 February 18, 2014

Physics 24 Exam 1 February 18, 2014 Exm Totl / 200 Physics 24 Exm 1 Februry 18, 2014 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. The totl electric flux pssing

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

11.1 Exponential Functions

11.1 Exponential Functions . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

More information

Exponentials & Logarithms Unit 8

Exponentials & Logarithms Unit 8 U n i t 8 AdvF Dte: Nme: Eponentils & Logrithms Unit 8 Tenttive TEST dte Big ide/lerning Gols This unit begins with the review of eponent lws, solving eponentil equtions (by mtching bses method nd tril

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

Square Root Function

Square Root Function Square Root Function D Eample of an appropriate solution Equation of square root function R() = a h k 5 = a 0 4 5 = 3a 4 5 + 4 = 3a 3 = a R() = 3 4 Zero of the function R() = 3 4 0 = 3 4 4 = 3 4 = 3 6

More information

Year 2009 VCE Mathematical Methods CAS Solutions Trial Examination 2

Year 2009 VCE Mathematical Methods CAS Solutions Trial Examination 2 Yer 9 VCE Mthemticl Methods CAS Solutions Tril Emintion KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC AUSTRALIA TEL: () 987 57 FAX: () 987 kilbh@gmil.com http://kilbh.googlepges.com KILBAHA PTY LTD 9

More information

SECTION 9-4 Translation of Axes

SECTION 9-4 Translation of Axes 9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.

More information

AB Calculus Review Sheet

AB 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 information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

More information

Form 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 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 information