Chapter 1: Logarithmic functions and indices


 Nora Stephens
 5 years ago
 Views:
Transcription
1 Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) Hint: The mth root of. Use the rule m n m n to simplify the inde. r r Rewrite the epression with the numers together nd r r the r terms together. 6 r 6 6r 5 r r r c 4 4 Use the rule m n m n to simplify the inde Use the rule m n m n d e ( ) Use the rule ( m ) n mn to simplify the inde f ( ) 4 Use the rule ( m ) n mn to simplify the inde
2 Emple Simplify: 4 _ _ c ( _ ) d Use the rule m n m n. 7 Rememer ( ). _ _ This could lso e written s. _ _ Use the rule m n m n. c ( _ ) Use the rule ( m ) n mn. _ d Use the rule m n m n. _ _ _ Emple Evlute: 9 _ 6 4 _ c 4 9 _ d 5 _ 9 _ Using m m. 9 Both nd re squre roots of 9. 9 strictly mens nd 9 ut lwys check if the negtive squre root is required nswer. 6 4 _ 64 This mens the cue root of As c 49 _ ( 49 ) n Using m m n. 4 d 5 _ Using m 5 m. ( 5 ) 5 5 ( 5) 5 This mens the squre root of 49, cued.
3 Eercise A Simplify these epressions: 4 c 4p p d 4 e k k f (y ) 5 g 0 5 h (p ) p 4 i ( ) j 8p 4 4p k 4 5 l 7 4 m 9 ( ) n 4 6 o 7 4 ( 4 ) p (4y ) y q 6 5 r 4 5 Simplify: 5 7 c _ 5_ d ( ) _ e ( ) 5_ f g 9 _ 6 h 5 _ 5 _ 5 i 4 5 Evlute: _ 5 8 _ c 7 _ d 4 e 9 _ f ( 5) g ( _ 4 ) 0 h 9 6 _ 4 i ( j ( 7) _ 8 k ( 6_ ) 5 l ( 4 9 ) _ 6 5 ) _. You cn write numer ectly using surds, e.g., 5, 9. You cnnot evlute surds ectly ecuse they give neverending, nonrepeting deciml frctions, e.g The squre root of prime numer is surd. You cn mnipulte surds using these rules: _ () You cn rtionlise the denomintor of y multiplying the top nd ottom. y
4 Emple 4 Simplify: 0 c _ 94 c (4 ) 4 Use the rule Cncel y _ is common fctor. 6 ( ) Work out the squre roots 4 nd 49. 6(5 7) (8) 8 6 Emple 5 Rtionlise the denomintor of: Multiply the top nd ottom y. ( ) Multiply the top nd ottom y. Rememer 6 Simplify your nswer 4
5 Eercise B Simplify: _ _ You need to know how to write n epression s logrithm log n mens tht n, where is clled the se of the logrithm In the IGCSE the se of the logrithm will lwys e positive integer greter thn. Emple 6 Write s logrithm So log 5 Here, 5, n. Bse Logrithm In words, you would sy the logrithm of, to se, is 5. In words, you would sy to the power 5 equls. Emple 7 Rewrite s logrithm: c 0 04 log log c log
6 log 0 Becuse 0. log Becuse. Emple 8 Find the vlue of log 8 log c log ( 5 ) log 8 4 Becuse 4 8. log Becuse 4 _ c log ( 5 ) 5 Becuse 5 5! You cn use the log key on clcultor to clculte logrithms to se 0. Emple 9 Find the vlue of for which So log Since 0 00 nd 0 000, must e log somewhere etween nd..70 (to s.f.) The log (or lg) utton on your clcultor gives vlues of logs to se 0. Eercise C Rewrite s logrithm: _ 9 c d Rewrite using power: log 6 4 log 5 5 c log 9 _ d log 5 0. e log Find the vlue of: log 8 log 5 5 c log d log e log 79 f log 0 0 g log 4 (0.5) h log ( 0 ) 4 Find the vlue of for which: log 5 4 log 8 c log 7 d log () 6
7 5 Find from your clcultor the vlue to s.f. of: log 0 0 log 0 4 c log d log Find from your clcultor the vlue to s.f. of: log 0 log 0 5. c log 0 0. d log You need to know the lws of logrithms Suppose tht Rewriting with powers: log nd log y c nd c y Multiplying: y c c (see section.) y c Rewriting s logrithm: log y c log y log log y (the multipliction lw) It cn lso e shown tht: log ( y ) log log y (the division lw) log () k k log (the power lw) Rememer: c c c Rememer: ( ) k k Note: You need to lern nd rememer the ove three lws of logrithms. Since ( ), the power rule shows tht log ( ) log ( ) log. log ( ) log And from the previous section log (since ) log 0 (since 0 ) Emple 0 Write s single logrithm: log 6 log 7 log 5 log c log 5 log 5 d log 0 4 log 0 ( _ ) log (6 7) Use the multipliction lw. log 4 log (5 ) Use the division lw. log 5 c log 5 log 5 ( ) log 5 9 First pply the power lw to oth prts of log 5 log 5 ( ) log 5 8 the epression. log 5 9 log 5 8 log 5 7 Then use the multipliction lw. d 4 log 0 ( _ ) log 0 ( _ ) 4 log 0 ( log 0 log 0 ( 6 ) log 0 ( log ) Use the power first. ) 6 Then use the division lw. 7
8 Emple Write in terms of log, log y nd log z log ( yz ) log ( y ) c log ( z ) d log ( y 4 ) log ( yz ) c d log ( ) log y log (z ) log log y log z log ( y ) log log (y ) log log y log ( y z ) log ( y ) log z log log y log z log _ log y log z log ( 4 ) log log ( 4 ) log 4 log log 4 log. Use the power lw ( y y _ ). Eercise D Write s single logrithm: log 7 log log 6 log 4 c log 5 log 5 0 d log log 6 e log 0 5 log 0 6 log 0 ( _ 4 ) Write s single logrithm, then simplify your nswer: log 40 log 5 log 6 4 log 6 9 c log 4 log d log 8 5 log 8 0 log 8 5 e log 0 0 (log 0 5 log 0 8) Write in terms of log, log y nd log z: log ( y 4 z) log ( 5 d y ) c log ( ) log ( y z ) log 8
9 . 5 You cn use the chnge of se formule to solve equtions of the form Working in se, suppose tht: Writing this s power: Tking logs to different se : Using the power lw: log m m log ( m ) log m log log Writing m s log : This cn e written s: log log log log log log This is the chnge of se rule for logrithms. Using this rule, notice in prticulr tht log log log, ut log, so: log log Emple Solve the following equtions, giving your nswers to significnt figures. 0 8 c log 0 Use the definition of logrithms from section.. By chnge of se formul, chnging to se 0 log 0 log 0 0 log 0 log Some clcultors cn evlute log 0. If your clcultor does not hve this fcility, you cn use the chnge of se formul nd use se 0 The log utton on your clcultor uses log 0. Use this to find log 0 0 nd log 0. Give nswer to sf. 8 log 8 Use the definition from section.. Chnging to se 0 log 8 log 0 log 0 8 Evlute using clcultor nd give nswer to sf..5 c log This cn e found directly using the log utton on 0.55 clcultor. NB A logrithm cn give negtive nswer: log < 0 when 0 < < 9
10 Emple Solve the eqution log 5 6 log 5 5: 6 log 5 log 5 5 Use chnge of se rule (specil cse). Let log 5 y y 6 5 y y 6 5y Multiply y y. y 5y 6 0 (y )(y ) 0 So y or y log 5 or log 5 5 or 5 5 or 5 Eercise E Write s powers. Find, to deciml plces: log 7 0 log 45 c log 9 d log e log 6 4 Solve, giving your nswer to significnt figures: c 6 Solve, giving your nswer to significnt figures: 75 0 c 5 d Solve, giving your nswer to significnt figures: log 8 9 log log 4 log 4 0 c log log 4. 6 You need to e fmilir with the functions y nd y log nd to know the shpes of their grphs As n emple, look t tle of vlues for y : Hint: A function tht involves vrile power such s is clled n eponentil function. 0 y _ 8 _ 4 _ 4 8 Note tht 0 (in fct 0 lwys if 0) nd ( negtive inde implies the reciprocl of positive inde) 8 0
11 The grph of y looks like this: 0 Other grphs of the type y re of similr shpe, lwys pssing through (0, ). Now look t the tle of vlues of y log : _ 8 _ 4 _ 4 8 y 0 You should note tht the vlues for nd y hve swpped round. This mens tht the shpe of the curve is simply reflection in the line y =. y The grph of y log will hve similr shpe nd it will lwys pss through (, 0) since log 0 for every vlue of. y Hint: Notice tht log 0 O Hint: The y is is n symptote to the curve. Emple 4 On the sme es sketch the grphs of y y nd y.5 On nother set of es sketch the grphs of y _ ( ) nd y. For ll the three grphs, y when 0. 0 When > 0, > >.5 When < 0, < <.5 Work out the reltive positions of the three grphs y y y y.5 0
12 _ So y ( _ ) is the sme s y ( ). ( m ) n mn So the grph of y ( _ ) is reflection in the yis of the grph of y. y y ( ) y 0 Emple 5 On the sme es, sketch the grphs of y log nd y log 5. For oth grphs y = 0 when =. But log so y log psses through (, ) nd log 5 5 so y log 5 psses through (5, ). By considering the shpe of the grphs etween y = 0 nd y =, you cn see tht log > log 5 for >. Since the log grphs re reflections of the eponentil grphs then from Emple 4 you cn see tht the reverse will pply the other side of (, 0). So log < log 5 for <. Since log 0 for every vlue of y y log y log 5 Eercise F On the sme es sketch the grphs of y 4 y 6 c y ( _ 4 ) On the sme es sketch the grphs of y y log c y ( _ ) On the sme es sketch the grphs of y log 4 y log 6 4 On the sme es sketch the grphs of y y log c Write down the coordintes of the point of intersection of these two grphs.
13 Eercise G Simplify: y y 5 5 c (4 ) 5 d 4 4 Simplify: 9 (4 _ ) _ 4 d _ 6 _ Evlute: ( 8 _ 7 ) d ( 5 89 ) 4 Simplify: 6 5 Rtionlise: 5 5 _ Epress log (p q) in terms of log p nd log q. Given tht log (pq) 5 nd log (p q) 9, find the vlues of log p nd log q. 7 Solve the following equtions giving your nswers to significnt figures: Given tht log, determine the vlue of. Clculte the vlue of y for which log y log (y 4). c Clculte the vlues of z for which log z 4 log z. 9 Find the vlues of for which log log. 0 Solve the eqution log ( ) log 9 (6 9 ).
14 Chpter : Summry Logrithms You cn simplify epressions y using rules of indices (powers). m n m n m n m n m m m m n m m n ( m ) n mn 0 Chpter : Summry You cn mnipulte surds using the rules: The rule to rtionlise surds is: Frctions in the form, multiply the top nd ottom y. 4 log n mens tht n, where is clled the se of the logrithm. 5 log 0 log 6 log 0 is sometimes written s log. 7 The lws of logrithms re log y log log y log ( y ) log log y log () k k log 8 From the power lw, log ( ) log (the power lw) (the multipliction lw) (the division lw) 9 The chnge of se rule for logrithms cn e written s log log log 0 From the chnge of se rule, log log 4
Bridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel 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 informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationSUMMER 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 informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationARITHMETIC 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 informationLogarithmic Functions
Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter 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 informationChapters 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 informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More informationWorksheet 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 informationPrecalculus Chapter P.2 Part 1 of 3. Mr. Chapman Manchester High School
Preclculus Chpter P. Prt of Mr. Chpmn Mnchester High School Eponents Scientific Nottion Recll: ( ) () 5 ( )( )( ) ()()()() Consider epression n : Red s to the nth power. is clled the bse n is clled the
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationIntroduction to Algebra  Part 2
Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger  Prt Sttement of Prerequisite
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. 6 ì 4 = 6 ì 0 = 4 ì 0 = 6 ì 3 = 5 ì 3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = 5 ì 2 = 6 ì 2 = 6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity B Integer multipliction doc69 Activity B More integer multipliction doc698 Activity B Advnced integer multipliction doc699 Multiplying integers FLUENCY
More informationAP Calculus AB Summer Packet
AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationConsolidation Worksheet
Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is
More informationChapter 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 information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction
Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the
More informationMath 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 informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationNumber systems: the Real Number System
Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationSection 3.2: Negative Exponents
Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.
More informationThe 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 informationSections 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 informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationMA Lesson 21 Notes
MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot rewrite this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 00 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 informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
More informationAP Calculus AB Summer Packet
AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself
More informationthan 1. It means in particular that the function is decreasing and approaching the x
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationFUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x x1)(x x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = ( 1)( ) The hperbol: p q The eponentil function: b p q Importnt fetures: intercept : Let = 0 intercept : Let = 0 Turning points (Where pplicble)
More informationMath 017. Materials With Exercises
Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson
More informationCalculus 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 informationBob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9 # + 4 ' ) ' ( 9 + 7) ' ' Give this
More informationSection 3.1: Exponent Properties
Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More information7h1 Simplifying Rational Expressions. Goals:
h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & , no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd
More informationNat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS
Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationObj: SWBAT Recall the many important types and properties of functions
Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions OnetoOne nd Inverse Functions
More informationPreparation for A Level Wadebridge School
Preprtion for A Level Mths @ Wdebridge School Bridging the gp between GCSE nd A Level Nme: CONTENTS Chpter Removing brckets pge Chpter Liner equtions Chpter Simultneous equtions 6 Chpter Fctorising 7 Chpter
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationIn this skill we review equations that involve percents. review the meaning of proportion.
6 MODULE 5. PERCENTS 5b Solving Equtions Mening of Proportion In this skill we review equtions tht involve percents. review the mening of proportion. Our first tsk is to Proportions. A proportion is sttement
More informationExponential and logarithmic. functions. Areas of study Unit 2 Functions and graphs Algebra
Eponentil nd logrithmic functions VCE co covverge Ares of study Unit Functions nd grphs Algebr In this ch chpter pter A Inde lws B Negtive nd rtionl powers C Indicil equtions D Grphs of eponentil functions
More informationQuotient 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 informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For , () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationAlgebra 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 PrePost Assessment CRS  Algebr Comprehensive Midterm Assessment Algebr Bsics CRS  Algebr QuikPiks
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationStage 11 Prompt Sheet
Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions
More informationBy the end of this set of exercises, you should be able to. reduce an algebraic fraction to its simplest form
ALGEBRAIC OPERATIONS By the end of this set of eercises, you should be ble to () (b) (c) reduce n lgebric frction to its simplest form pply the four rules to lgebric frctions chnge the subject of formul
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, 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 informationPrerequisites CHAPTER P
CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More information