Preparing for the HNC Electrical Maths Components. online learning. Page 1 of 15


 Gerald Eric Lewis
 1 years ago
 Views:
Transcription
1 online learning Preparing for the HNC Electrical Maths Components Page 1 of 15
2 Contents INTRODUCTION Algebraic Methods Indices and Logarithms Indices Logarithms Exponential Growth and Decay Linear Equations and Straight Line Graphs Linear Equations Straight Line Graphs Linear Simultaneous Equations Factorisation and Quadratics Multiplication by Bracketed Expressions Common Factors Grouping Quadratics and Roots of Equations Summary Page 2 of 15
3 INTRODUCTION This Workbook guides you through the learning outcomes related to: Indices and logarithms: laws of indices, laws of logarithms e.g. common logarithms (base 10), natural logarithms (base e), exponential growth and decay Linear equations and straight line graphs: linear equations; straight line graph (coordinates on a pair of labelled Cartesian axes, positive or negative gradient, intercept, plot of a straight line); experimental data e.g. Ohm s law, pair of simultaneous linear equations in two unknowns Factorisation and quadratics: multiply expressions in brackets by a number, symbol or by another expression in a bracket; by extraction of a common factor; by grouping; quadratic expressions; roots of an equation e.g. quadratic equations with real roots by factorisation, and by the use of formula Page 3 of 15
4 1 Algebraic Methods 1.1 Indices and Logarithms Indices In engineering we often use very large quantities, and very small quantities. Let s look at a couple of examples... A large power station may supply 4,000,000,000 Watts of power to the national grid. We normally term that as 4 GigaWatts (4GW). If we were manipulating that figure in mathematical calculations it would be very tiresome to have to write 4,000,000,000 each time. Wouldn t it be better if we could represent that number in a more concise way? The number contains lots of zeros, nine of them, in fact. Maybe we could express the number 4 and multiply by a series of tens. Here s a way to express the number by multiples of ,000,000, Wow, that looks even worse. However, when we notice that there are 9 multiples of 10 there then we could employ an index, as follows... 4,000,000, The signal from a remote spacecraft is picked up from Earth. The signal is much attenuated and only registers 3 picowatts (3pW) on the instrumentation. That s a really small signal, and the number may be written in decimal form as How are going to express that in index form? Well, we could express it as = = Since we try to avoid fractions we may look at that bottom part (the denominator) and bring it to the top (numerator) simply by changing the sign on the index... 3 = The best trick to remember with this number is to think of how many places do you need to move that decimal point to the right until it appears just after the 3? You need to move it 12 places, of course. Let s look at some further examples of employing this neat notation... Page 4 of 15
5 1, , ,400, (yyyyyy cccccccc mmmmmmmm tthaaaa oooooo cccccccccccccc) An extremely important index to know about is zero. When we raise a quantity to the power zero we always get 1 (there is an exception to this rule and that is when we raise zero to the power zero something which you will never do so that we leave that one to the theorists). We shall see this proven shortly. Multiplication of Indexed Quantities If we multiply 1000 by 1000 we get a million. We know that 1,000 is expressed as and we also know that 1,000,000 is expressed as Let s perform the multiplication in our new concise indexed form... 1,000 1, = = What did we do with those 3 s to get the 6? We must have added them, right? That then brings us to our first rule of indices, for any general problem... aa mm aa nn = aa mm+nn Here, a is any base (could be 10 as before, or could be a quantity merely represented as a letter, like v for voltage). The indexes m and n can be any number you like, including negative numbers. This rule can be extended further to incorporate the multiplication of three quantities... Here are some examples... Division of Indexed Quantities xx aa xx bb xx cc = xx aa+bb+cc = = 10 5+( 3) = 10 2 tt 2 tt 6 tt 3 = tt = tt 5 pp 6 pp 2 pp 4 qq 5 qq 2 = pp qq 5 2 = pp 8 qq 3 If you had 100,000 objects and divided them fairly amongst 100 people then each person would, of course, receive 1000 objects. Let s have a look at this operation in index form , = = 105 = 1000 = Page 5 of 15
6 We knew the answer was 1000, so what did we do with the 5 index and the 2 index to get the 3 index? We must have subtracted them. What we did was = = 10 3 It looks like we have another rule then, this time for the division of indexed quantities... Here are some examples... aa mm = aamm nn aann = = = = = ( 2) = pp 4 3pp 3 = 4pp4 3 = 4pp = = = 10 0 = 1 That last one proves that raising a quantity to the power zero gives 1, as suggested earlier. We may also combine our multiplication and division rules, as in the following examples... aa 2 aa 5 aa 3 aa 4 = aa2+( 5) aa 3+4 = aa 3 aa 7 = aa 3 7 = aa 10 cc 9 cc 4 dd 3 dd 6 = cc9+( 4) dd 3+6 = cc5 dd 3 = cc5 dd 3 In that last example we brought the dd 3 in the denominator up to the numerator, simply by changing the sign on its index. Raising Indexed Quantities to Powers What if you saw something like ? You know that the answer is 1 million but how did we get that via indexes? Let s look at the problem another way... Page 6 of 15
7 = (10 3 ) 2 = = 10 6 What we have done is to multiply the 3 by the 2. This brings us to our third rule for indices... (aa mm ) nn = aa mm nn Some examples using this rule... (10 4 ) 2 = = 10 8 (xx 2 ) 5 = xx 2 5 = xx Logarithms Many people shy away from logarithms at school. What are they for? In engineering we use logarithms to find answers to circuit problems involving capacitors and inductors (more on this in the next section). Now you know why they are needed let s look at what they are and how we use them. When we mention the term logarithm what we are really saying is what s the logarithm of a particular number? The logarithm of a particular number is the index that needs to be applied to our base to get the number we start with. For example, if we should ask what s the logarithm to base 10 of 1000? we are wondering what power we need to raise 10 by to get 1000 the answer is 3, as we already know from previous examples. We can therefore say that the log to base 10 of 1000 is 3. In engineering we tend to work either in base 10 (used for decibels) and base e (the universal constant, associated with natural growth and decay as related to capacitors and inductors). These days we tend to use calculators to work out logarithms. That is no bad thing but to be able to use and transpose many engineering expressions we need a knowledge of the rules of logarithms. Law 1: log(a) + log(b) = log(ab) Example... llllll(100) + llllll(1,000) = = 5 = log(100 1,000) = llllll(100,000) = 5 Law 2: log(a) log(b) = log(a/b) Example... llllll(1,000,000) llllll(10,000) = 6 4 = 2 = llllll(1,000,000 10,000) = llllll(100) = 2 Law 3: log(a n ) = n log (A) Example... llllll( ) = log(1,000,000) = 6 = 2 log(1000) = 2 3 = 6 Page 7 of 15
8 1.1.3 Exponential Growth and Decay We have mentioned that exponential growth and decay are related, amongst other things, to the characteristics of capacitors and inductors. Here are some characteristic features of exponential growth... The graph shows how the current in the LR circuit grows exponentially. The formula which relates the current to time, the inductor value, and the resistor value, is... ii = VV1 1 ee RRRR LL RR Notice that base 10 is not used. The base is e, a natural constant equal to (to 5 decimal places). Components behave as nature/physics dictate they should. Base 10 is a human invention, convenient because we have 10 fingers/thumbs, which we use for counting. We can also see exponential decay in circuits... The current in this RC circuit experiences exponential decay. This is given by... ii = VV bbbbbbbbbbbbbb RR ee tt RRRR Our laws of logs apply equally well to this new base of e. An example... llllll ee (4.556) 3 = 3 llllll ee (4.556) = = Page 8 of 15
9 1.2 Linear Equations and Straight Line Graphs Linear Equations A linear equation has an index of 1 on the independent variable. For example... Worked Example 1 yy = mmmm + cc Here, the independent variable is xx and mm indicates the slope of the graph s line. The constant c indicates where the line crosses the y axis. (If you let xx = 0 then y will equal c and this will indicate that crossing point) Straight Line Graphs Considering the information just discussed about linear equations, let s look at a worked example involving such a straight line graph... RMS measurements were taken for the current (i) through a resistor and the voltage (v) across it. The resistor experiences both AC and DC influences. The measurement data is as follows; v [V] i [A] a) Use the experimental data to plot a graph of voltage (v) on the vertical axis against current (i) on the horizontal axis. b) Hence, deduce the equation of the line by determining its gradient and intercept with the voltage axis. The first thing to do is to plot the graph from the table. This is most easily done by downloading and using the free Graph application. Insert the data into a point series, as follows... Page 9 of 15
10 Marker style and colour can be chosen as preferred. Then insert a curve of best fit using the icon at the top of the application. What results is shown below... We see that the graph intercepts the vertical axis at v= 6 so our equation will have a value for c of 6. We then need to complete a triangle anywhere on the graph, as shown below with the aid of the blue dotted line... To find the slope of the red line we measure the height of the triangle and divide by its width. This gives us 9/3 = 3, so m (slope) has a value of +3. We now have the equation of the straight line... vv = 3ii 6 Page 10 of 15
11 1.2.3 Linear Simultaneous Equations Such equations can be formed by analysing electrical circuits. Solution of the equations can yield unknown quantities, such as voltage or current. There are a number of ways to solve simultaneous equations. One method is by multiplication and substitution, which we shall now analyse. Worked Example 2 Solve the following pair of simultaneous linear equations = = 1111 The first thing we need to do is number the equations, so that we may refer to them easily in our developments... 2aa + 3bb = 23 [1] 6aa 2bb = 14 [2] The aim at this point is to look at the two equations and think of ways in which we may eliminate one of the unknowns ( a and b ). What is apparent from the equations is that if we multiply equation [1] by 3 we will have 6a in the new equation, which we may then add to equation [2] to eliminate the a term. Maths is much more understandable when not expressed in sentences, so let s try to do what we meant... [1] X (3): 6aa 9bb = 69 [3] The notation above is very useful. It says that when we take equation [1] and multiply it by 3 we will get what is now called equation [3]. The reason for that will quickly become apparent when we write equations [2] and [3] together... 6aa 2bb = 14 [2] 6aa 9bb = 69 [3] Notice that if we should ADD these two equations the a will disappear. That will leave one unknown ( b ), which we may easily find... [2] + [3]: 11bb = 55 [4] bb = 5 Since we now know the value of b we can substitute this value into any previous equation we like to find the value of a. Let s pick, say, equation [1]... 2aa + 3bb = 23 [1] 2aa + 3(5) = 23 2aa = = 8 aa = 4 Page 11 of 15
12 We have now solved the simultaneous equations. We have aa = 44 and bb = Factorisation and Quadratics Multiplication by Bracketed Expressions Let s look at some obvious mathematical expressions and see if we can see how to multiply by bracketed terms = = 30 2(15) = 2(7 + 8) = = 30 The term 2(7 + 8) could easily have been a bunch of letters, like aa(bb + cc) so we know we must multiply the letter on the left ( a ) by EACH of the letters in the bracket. We then have... aa(bb + cc) = aaaa + aaaa Let s look at another example, which extends this concept a little further... 8 = = (8)(18) = 144 = (3 + 5)(4 + 14) = (3)(4) + (3)(14) + (5)(4) + (5)(14) = 144 This problem could easily have been expressed in letters also... (aa + bb)(cc + dd) = aaaa + aaaa + bbbb + bbbb Common Factors Where we have common factors in an expression we tend to use efficient notation so that we are not writing out the same term multiple times. Consider... This is more efficiently written as... Again, these could have been letters... 2(8) + 2(9) 2(8 + 9) 2aa + 2bb = 2(aa + bb) We have brought out a common factor of 2 and the expression is now more elegant Grouping Consider groups of letters, such as those below... 6aaaa + 12aa 2 bb 2 What s common there, which we can group? Well, the 6 is a factor of the first and second term (two lots of six in twelve). Also, ab is a factor of both terms. We may now express this more elegantly as... Page 12 of 15
13 6aaaa + 12aa 2 bb 2 = 6aaaa(1 + 2aaaa) If you re not sure then multiply it out and confirm the result. Here are some more examples of grouping... 32ff 3 gg + 16ff 2 gg 4 = 16ff 2 gg(2ff + gg 3 ) ff 3 gg 2 ff 4 gg = ff 3 gg 2 (1 + ffff) + 8 ffff + ff 2 + ff 2 gg + ffgg = ff(gg + ff + ffff + gg 2 ) Quadratics and Roots of Equations A quadratic equation is nonlinear. What this means is that the highest index on the independent variable is 2. Such equations do not yield straight lines, they have bends. A quadratic equation has the following general form... yy = aaxx 2 + bbbb + cc You see that power of 2 there, that categorises this as a quadratic. We normally have two approaches to solving quadratic equations... Factorise to find the roots Use the quadratic formula Finding the Roots by Factorisation We always look to do this first. If it s not possible, or difficult, then we turn to the formula method, which you shall see shortly. Let s say we have the quadratic equation... yy = xx 2 + 7xx + 12 We first look at the number on the RHS and ask what factors of 12, when added together, will give that number in the middle (+7)? Let s examine the factors of x 12 = is not +71 x 12 = is not +7 2 x 6 = is not +72 x 6 = is not +7 3 x 4 = IS +7 We therefore select +3 and +4 and place these into brackets, equating to zero, as follows... Page 13 of 15
14 (xx + 3)(xx + 4) = 0 When the first bracket is zero then the expression above is true. For the first bracket to be zero the value of xx must be 3. We also look at the second bracket and do the same. In that case the value of xx needs to be 4. What we have done here is to find the roots of the given quadratic, which means we have found the points on the graph of the quadratic where the curve crosses the xx axis (i.e. where y is zero). Our proposed roots are xx = 33 and xx = 44. Let s use the Graph application to check this is so... Our calculations check out very well. Formula Method for finding the Roots of a Quadratic We have the general expression for a quadratic as follows... The formula to find the roots is... yy = aaxx 2 + bbbb + cc xx = bb ± bb2 4aaaa 2aa This formula will always find you the roots. Let s check out the previous problem this way... yy = aaxx 2 + bbbb + cc yy = xx 2 + 7xx + 12 We see that a is 1, b is 7 and c is 12. Let s use these figures in the formula... xx = bb ± bb2 4aaaa 2aa = 7 ± 72 4(1)(12) 2(1) = 7 ± = 7 ± 1 2 = 3 oooo 4 Page 14 of 15
15 Summary Well done for getting this far. If you have understood this material, it is very likely that you are ready to take on the challenge of the Edexcel HNC course in Electrical and Electronic Engineering, delivered by UniCourse. Please then go to our website and complete the short online application form. Our HNC course provides lots of workbooks, most of which are longer than this one, and useful tutorial videos which we have produced to ease your passage through the course. If you feel that you need more lessons in Maths, at any level, to build your confidence, then we highly recommend the wonderful KhanAcademy website. We look forward to welcoming you to UniCourse! Page 15 of 15
Worksheets for GCSE Mathematics. Quadratics. mrmathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mrmathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More informationTransition to College Math and Statistics
Transition to College Math and Statistics Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear College Algebra Students, This assignment
More informationdue date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)
Honors PreCalculus Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear Honors PreCalculus Students, This assignment is designed
More informationQuadratic Equations and Functions
50 Quadratic Equations and Functions In this chapter, we discuss various ways of solving quadratic equations, aaxx 2 + bbbb + cc 0, including equations quadratic in form, such as xx 2 + xx 1 20 0, and
More informationRational Expressions and Functions
Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category
More informationDefinition of a Logarithm
Chapter 17 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents
More informationDefinition of a Logarithm
Chapter 17 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationPREPARED BY: J. LLOYD HARRIS 07/17
PREPARED BY: J. LLOYD HARRIS 07/17 Table of Contents Introduction Page 1 Section 1.2 Pages 211 Section 1.3 Pages 1229 Section 1.4 Pages 3042 Section 1.5 Pages 4350 Section 1.6 Pages 5158 Section 1.7
More informationNational 5 Mathematics. Practice Paper E. Worked Solutions
National 5 Mathematics Practice Paper E Worked Solutions Paper One: NonCalculator Copyright www.national5maths.co.uk 2015. All rights reserved. SQA Past Papers & Specimen Papers Working through SQA Past
More informationMath, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD
Math, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD Outline These preliminaries serve to signal to students what tools they need to know to succeed in ECON 360 and refresh their
More informationRational Expressions and Functions
1 Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category
More informationPlease bring the task to your first physics lesson and hand it to the teacher.
Preenrolment task for 2014 entry Physics Why do I need to complete a preenrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will
More informationA Level Maths summer preparation work
A Level Maths summer preparation work Welcome to A Level Maths! We hope you are looking forward to two years of challenging and rewarding learning. You must make sure that you are prepared to study A Level
More informationSystems of Linear Equations
Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
More informationDefinition of a Logarithm
Chapter 17 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationMathematics Revision Guide. Algebra. Grade C B
Mathematics Revision Guide Algebra Grade C B 1 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without
More informationWorksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 19. Algebra
Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 19 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets
More informationNEXTGENERATION MATH ACCUPLACER TEST REVIEW BOOKLET. Next Generation. Quantitative Reasoning Algebra and Statistics
NEXTGENERATION MATH ACCUPLACER TEST REVIEW BOOKLET Next Generation Quantitative Reasoning Algebra and Statistics Property of MSU Denver Tutoring Center 2 Table of Contents About...7 Test Taking Tips...9
More informationR.3 Properties and Order of Operations on Real Numbers
1 R.3 Properties and Order of Operations on Real Numbers In algebra, we are often in need of changing an expression to a different but equivalent form. This can be observed when simplifying expressions
More informationLesson 24: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise 1. Examine the two equation below and discuss what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. cc 14 = 5cc. Solve each equation with the
More informationEquations and inequalities
7 Stretch lesson: Equations and inequalities Stretch objectives Before you start this chapter, mark how confident you feel about each of the statements below: I can solve linear equations involving fractions.
More informationReview of Operations on the Set of Real Numbers
1 Review of Operations on the Set of Real Numbers Before we start our journey through algebra, let us review the structure of the real number system, properties of four operations, order of operations,
More informationSection 2: Equations and Inequalities
Topic 1: Equations: True or False?... 29 Topic 2: Identifying Properties When Solving Equations... 31 Topic 3: Solving Equations... 34 Topic 4: Solving Equations Using the Zero Product Property... 36 Topic
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1
Learning outcomes EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1 TUTORIAL 3  FACTORISATION AND QUADRATICS On completion of this unit a learner should: 1 Know how to use algebraic
More informationG.6 Function Notation and Evaluating Functions
G.6 Function Notation and Evaluating Functions ff ff() A function is a correspondence that assigns a single value of the range to each value of the domain. Thus, a function can be seen as an inputoutput
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationLesson 25: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise Over the years, many students and teachers have thought of ways to help us all remember the quadratic formula. Below is the YouTube link to a video created by two teachers
More informationL.2 Formulas and Applications
43 L. Formulas and Applications In the previous section, we studied how to solve linear equations. Those skills are often helpful in problem solving. However, the process of solving an application problem
More informationCHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
More informationEureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities
Eureka Lessons for 6th Grade Unit FIVE ~ Equations & Inequalities These 2 lessons can easily be taught in 2 class periods. If you like these lessons, please consider using other Eureka lessons as well.
More informationMathematics: Year 12 Transition Work
Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution
More informationSection 4.1: Exponential Growth and Decay
Section 4.1: Exponential Growth and Decay A function that grows or decays by a constant percentage change over each fixed change in input is called an exponential function. Exponents A quick review 1.
More informationJUST THE MATHS UNIT NUMBER 1.6. ALGEBRA 6 (Formulae and algebraic equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.6 ALGEBRA 6 (Formulae and algebraic equations) by A.J.Hobson 1.6.1 Transposition of formulae 1.6. of linear equations 1.6.3 of quadratic equations 1.6. Exercises 1.6.5 Answers
More informationEureka Math. Algebra II Module 1 Student File_A. Student Workbook. This file contains Alg IIM1 Classwork Alg IIM1 Problem Sets
Eureka Math Algebra II Module 1 Student File_A Student Workbook This file contains Alg II Classwork Alg II Problem Sets Published by the nonprofit GREAT MINDS. Copyright 2015 Great Minds. No part of
More informationLesson 15: Rearranging Formulas
Exploratory Challenge Rearranging Familiar Formulas 1. The area AA of a rectangle is 25 in 2. The formula for area is AA = llll. A. If the width ll is 10 inches, what is the length ll? AA = 25 in 2 ll
More informationMath 3 Unit 4: Rational Functions
Math Unit : Rational Functions Unit Title Standards. Equivalent Rational Expressions A.APR.6. Multiplying and Dividing Rational Expressions A.APR.7. Adding and Subtracting Rational Expressions A.APR.7.
More informationSpring 2018 Math Week Week 1 Task List
Spring 2018 Math 143  Week 1 25 Week 1 Task List This week we will cover Sections 1.1 1.4 in your ebook. Work through each of the following tasks, carefully filling in the following pages in your notebook.
More informationBefore we do that, I need to show you another way of writing an exponential. We all know 5² = 25. Another way of writing that is: log
Chapter 13 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationMATH 1130 Exam 1 Review Sheet
MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1
Learning outcomes EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1 TUTORIAL 2  LINEAR EQUATIONS AND GRAPHS On completion of this unit a learner should: 1 Know how to use algebraic
More informationTerms of Use. Copyright Embark on the Journey
Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means  electronic, mechanical, photocopies, recording, or otherwise
More informationF.1 Greatest Common Factor and Factoring by Grouping
section F1 214 is the reverse process of multiplication. polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example,
More informationF.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 ZeroProduct Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or firstdegree, equations. Solving higher
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationPrecalculus is the stepping stone for Calculus. It s the final hurdle after all those years of
Chapter 1 Beginning at the Very Beginning: PrePreCalculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationSOLUTIONS FOR PROBLEMS 130
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS  0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationAP Physics 1 Summer Assignment #
APPhysics1 SummerAssignment AP Physics 1 Summer Assignment Welcome to AP Physics 1. This course and the AP exam will be challenging. AP classes are taught as college courses not just collegelevel courses,
More informationMathematics Department. Summer Course Work. Geometry
Decatur City Schools Decatur, Alabama Mathematics Department Summer Course Work In preparation for Geometry Completion of this summer work is required on the first c l a s s day of the 20182019 school
More informationF.1 Greatest Common Factor and Factoring by Grouping
1 Factoring Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers.
More informationEdexcel AS and A Level Mathematics Year 1/AS  Pure Mathematics
Year Maths A Level Year  Tet Book Purchase In order to study A Level Maths students are epected to purchase from the school, at a reduced cost, the following tetbooks that will be used throughout their
More informationTake the Anxiety Out of Word Problems
Take the Anxiety Out of Word Problems I find that students fear any problem that has words in it. This does not have to be the case. In this chapter, we will practice a strategy for approaching word problems
More informationAQA Level 2 Further mathematics Further algebra. Section 4: Proof and sequences
AQA Level 2 Further mathematics Further algebra Section 4: Proof and sequences Notes and Examples These notes contain subsections on Algebraic proof Sequences The limit of a sequence Algebraic proof Proof
More informationLesson 23: Deriving the Quadratic Formula
: Deriving the Quadratic Formula Opening Exercise 1. Solve for xx. xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 Discussion 2. Which of these problems makes more sense to solve by completing the square? Which makes
More informationExpanding brackets and factorising
Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a
More informationMath Refresher Answer Sheet (NOTE: Only this answer sheet and the following graph will be evaluated)
Name: Score: / 50 Math Refresher Answer Sheet (NOTE: Only this answer sheet and the following graph will be evaluated) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. MAKE SURE CALCULATOR
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationAnalog Circuits Part 1 Circuit Theory
Introductory Medical Device Prototyping Analog Circuits Part 1 Circuit Theory, http://saliterman.umn.edu/ Department of Biomedical Engineering, University of Minnesota Concepts to be Covered Circuit Theory
More informationABE Math Review Package
P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationMath 95Review Prealgebrapage 1
Math 95Review Prealgebrapage 1 Name Date In order to do well in algebra, there are many different ideas from prealgebra that you MUST know. Some of the main ideas follow. If these ideas are not just
More informationP.2 Multiplication of Polynomials
1 P.2 Multiplication of Polynomials aa + bb aa + bb As shown in the previous section, addition and subtraction of polynomials results in another polynomial. This means that the set of polynomials is closed
More informationA Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:
A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,
More informationEdexcel GCSE (91) Maths for post16
Edexcel GCSE (91) Maths for post16 Fiona Mapp with Su Nicholson 227227_Front_EDEXCEL.indd 1 02/06/2017 11:34 Contents Introduction 5 How to use this book 13 1 Numbers 14 1.1 Positive and negative numbers
More information32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE
get the complete book: /getfulltextfullbook.htm 32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE classifying families of sentences In mathematics, it is common to group together sentences of the same type
More informationThe Derivative. Leibniz notation: Prime notation: Limit Definition of the Derivative: (Used to directly compute derivative)
Topic 2: The Derivative 1 The Derivative The derivative of a function represents its instantaneous rate of change at any point along its domain. There are several ways which we can represent a derivative,
More informationSecondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet
Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and
More informationSecondary Two Mathematics: An Integrated Approach Module 3  Part One Imaginary Number, Exponents, and Radicals
Secondary Two Mathematics: An Integrated Approach Module 3  Part One Imaginary Number, Exponents, and Radicals By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis
More informationElementary Algebra Study Guide Some Basic Facts This section will cover the following topics
Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics Notation Order of Operations Notation Math is a language of its own. It has vocabulary and punctuation (notation)
More informationALevel Notes CORE 1
ALevel Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationTSIA MATH TEST PREP. Math and Science, ASC 1
TSIA MATH TEST PREP Math and Science, ASC 1 Texas Success Initiative: Mathematics The TSI Assessment is a program designed to help Lone Star College determine if you are ready for collegelevel coursework
More information2. FUNCTIONS AND ALGEBRA
2. FUNCTIONS AND ALGEBRA You might think of this chapter as an icebreaker. Functions are the primary participants in the game of calculus, so before we play the game we ought to get to know a few functions.
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its yintercept. In Euclidean geometry (where
More information1.2. Indices. Introduction. Prerequisites. Learning Outcomes
Indices 1.2 Introduction Indices, or powers, provide a convenient notation when we need to multiply a number by itself several times. In this Section we explain how indices are written, and state the rules
More informationHonors Geometry Summer Packet NHHS and VHS
NAME_ STUDENT ID # Honors Geometry Summer Packet NHHS and VHS This summer packet is for all students enrolled in Honors Geometry for the upcoming school year. The entire packet will be due and collected
More informationSome examples of radical equations are. Unfortunately, the reverse implication does not hold for even numbers nn. We cannot
40 RD.5 Radical Equations In this section, we discuss techniques for solving radical equations. These are equations containing at least one radical expression with a variable, such as xx 2 = xx, or a variable
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010
Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or
More informationSelfDirected Course: Transitional Math Module 4: Algebra
Lesson #1: Solving for the Unknown with no Coefficients During this unit, we will be dealing with several terms: Variable a letter that is used to represent an unknown number Coefficient a number placed
More informationPrerequisite: Qualification by assessment process or completion of Mathematics 1050 or one year of high school algebra with a grade of "C" or higher.
Reviewed by: D. Jones Reviewed by: B. Jean Reviewed by: M. Martinez Text update: Spring 2017 Date reviewed: February 2014 C&GE Approved: March 10, 2014 Board Approved: April 9, 2014 Mathematics (MATH)
More informationAlgebra. Topic: Manipulate simple algebraic expressions.
30410 Algebra Days: 1 and 2 Topic: Manipulate simple algebraic expressions. You need to be able to: Use index notation and simple instances of index laws. Collect like terms Multiply a single term over
More informationSECTION 2: VECTORS AND MATRICES. ENGR 112 Introduction to Engineering Computing
SECTION 2: VECTORS AND MATRICES ENGR 112 Introduction to Engineering Computing 2 Vectors and Matrices The MAT in MATLAB 3 MATLAB The MATrix (not MAThematics) LABoratory MATLAB assumes all numeric variables
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010
Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationA factor times a logarithm can be rewritten as the argument of the logarithm raised to the power of that factor
In this section we will be working with Properties of Logarithms in an attempt to take equations with more than one logarithm and condense them down into just a single logarithm. Properties of Logarithms:
More informationIntermediate Tier  Algebra revision
Intermediate Tier  Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double brackets Substitution Solving equations Finding nth term
More informationEXPONENTIAL AND LOGARITHMIC FUNCTIONS
Mathematics Revision Guides Exponential and Logarithmic Functions Page 1 of 14 M.K. HOME TUITION Mathematics Revision Guides Level: ALevel Year 1 / AS EXPONENTIAL AND LOGARITHMIC FUNCTIONS Version : 4.2
More informationSome of the more common mathematical operations we use in statistics include: Operation Meaning Example
Introduction to Statistics for the Social Sciences c Colwell and Carter 206 APPENDIX H: BASIC MATH REVIEW If you are not using mathematics frequently it is quite normal to forget some of the basic principles.
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More informationRadiological Control Technician Training Fundamental Academic Training Study Guide Phase I
Module 1.01 Basic Mathematics and Algebra Part 4 of 9 Radiological Control Technician Training Fundamental Academic Training Phase I Coordinated and Conducted for the Office of Health, Safety and Security
More informationLesson 2: Put a Label on That Number!
Lesson 2: Put a Label on That Number! What would you do if your mother approached you, and, in an earnest tone, said, Honey. Yes, you replied. One million. Excuse me? One million, she affirmed. One million
More informationLesson 23: The Defining Equation of a Line
Classwork Exploratory Challenge/Exercises 1 3 1. Sketch the graph of the equation 9xx +3yy = 18 using intercepts. Then, answer parts (a) (f) that follow. a. Sketch the graph of the equation yy = 3xx +6
More informationIntroduction to Electrical Theory and DC Circuits
Introduction to Electrical Theory and DC Circuits For Engineers of All Disciplines by James Doane, PhD, PE Contents 1.0 Course Overview... 4 2.0 Fundamental Concepts... 4 2.1 Electric Charges... 4 2.1.1
More informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a twoday lesson
More informationAP Calculus AB Summer Assignment
AP Calculus AB 0809 Summer Assignment Dear Future AP Student, I hope you are ecited for the year of Calculus that we will be pursuing together! I don t know how much you know about Calculus, but it is
More information