CALCULUS BASIC SUMMER REVIEW


 Randolf Gibbs
 4 years ago
 Views:
Transcription
1 CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y SlopeItercept Equatio: y m b slope= m yitercept= b Geeral Form of Liear Equatio are ot zero. A By C such that A ad B both What is a Fuctio? A fuctio is a relatio that assigs a sigle elemet of R to each elemet of D. A workig defiitio of a fuctio is that it is a devise that assigs a output to every allowable iput. The iputs make up the domai of the fuctio. The outputs make up the rage. A Fuctio must pass the vertical Lie Test Vertical Lie Test by egative umber Idetifyig the Domai ad Rage: Remember, i the real umber system you ca ot divide zero or fid the eve root of a Eve ad Odd Fuctios A fuctio y = f() is a eve fuctio of if f() = f() for every i the fuctio s domai. A fuctio is a odd fuctio of if f() =  f() for every i the fuctio s domai. Page
2 Absolute Value Thik of the absolute value fuctio as a piecewise fuctio. The Greatest Iteger Fuctio f ( ) or f ( ) it( ) f ( ) if 0 if <0 The greatest iteger fuctio represets the greatest iteger less tha or equal to Compositio of Fuctios The compositio f g of the fuctios f ad g is defied by ( f g)( ) f ( g( )) The domai of domai of f. ( f g)( ) f ( g( )) cosists of those s for which g() is i the Geometric Trasformatios: Shifts, Reflectios, Stretches, ad Shriks Graph Shiftig Formulas Vertical shifts of the graph of y f () y f ( ) c Shifts graph of y f () dow c uits y f ( ) c Shifts graph of y f () up c uits Horizotal shifts of the graph of y f ( c) Shifts graph of y f () right c uits y f ( c) Shifts graph of y f () left c uits How to stretch or shrik a graph To stretch the parabola each ycoordiate by c. y vertically by a factor of c (c>0), we must multiply If you stretch the graph by a factor of two the ew equatio will be: y How to reflect a graph To reflect the graph of y=f() across the yais, we multiply each y coordiate by . Reflectio Formulas: With respect to the yais y f ( ) With respect to the ais y f () The Parabola y a b c A parabola that opes i the positive y directio if a>0 ad i the egative y directio if a<0. b b b The ais of symmetry is: The verte is at: (, f ( )) a a a POLYNOMIALS Page
3 Polyomial Epressio: a a a... a a0 Polyomial Fuctio: f ( ) a a a... a a0 Polyomial Equatio: a a a... a a0 0 Ratioal Zeros Theorem Suppose all the coefficiets i the polyomial fuctio f ( ) a a a... a a0 are itegers. c If s a ratioal zero of f, where c ad d have o commo factors, the c is a factor d of a 0, ad d is a factor of the leadig coefficiet a. How to Solve f()= 0 usig calculator or your ow brai!!!!. Fid the eact solutio algebraically (ofte by factorig). Draw a complete graph a) Use ZOOMIN b) Use SOLVE Steps for Solvig a Problem. Fid a algebraic represetatio ivolvig variables.. Draw a complete graph of the fuctio. Fid the domai ad rage 4. Determie the values that make sese i the give situatio 5. Draw a graph of the problem situatio 6. Solve the problem usig appropriate methods For istace: Solve 4 0 Factors of c:,,, 4, 6, Factors of d:, c Possible zeros:,,, 4, 6,,, d Look at the graph to see the zeroes must be betwee  ad  or ad 4. So f ( / ) 0 ( / ) ( ) So ( ) is a factor. By divisio ( )( 4) 0 Use the Quadratic Formula to fid 5 Equatios for Circles i the Plae Circle is the set of poits i a plae whose distace from a fied poit i the plae is a costat. The fied poit is the ceter of the circle. The costat distace is the radius of the circle. Equatio: ( h) ( y k) r Iverse Relatios ad Fuctios: Page
4 Iverse Relatio: Let R be a relatio. The iverse relatio R of R cosists of all those ordered pairs (b,a) for which (a,b) belogs to R. So the domai of R = the rage of R ad the rage of R = the domai of R. Horizotal Lie Test : The iverse relatio R of the relatio R is a fuctio if ad oly if every horizotal lie itersects the graph of R i at most oe poit. Notice that the iverse of f ( ) 6 is ot a fuctio sice f() fails the horizotal lie test. OetoOe: The iverse f of a fuctio f is a fuctio if ad oly if f is a oetooe fuctio. Epoetial Fuctios: Defiitio: Let a be a positive real umber other tha. The fuctio f ( ) a whose domai is (, ) ad whose rage is ( 0, ) is the epoetial fuctio with base a. Trigoometric Fuctios: Uit Circle Page 4
5 Graph of the Sie Curve: y = si() Graph of the Cosie Curve: y = cos() Graph of the Taget Curve: Coic Sectios y = ta() Circle Ellipse (h) Parabola (h) Hyperbola (h) Defiitio: A coic sectio is the itersectio of a plae ad a coe. Ellipse (v) Parabola (v) Hyperbola (v) By chagig the agle ad locatio of itersectio, we ca produce a circle, ellipse, parabola or Page 5
6 hyperbola; or i the special case whe the plae touches the verte: a poit, lie or itersectig lies. Poit Lie Double Lie The Geeral Equatio for a Coic Sectio: A + By + Cy + D + Ey + F = 0 psimplify the followig algebraic ad umeric epressios (9 7 5) = 5 7 = (4 5) = 8. (4i ) (i ) = 9. (4i)(i ) = 0. 5[4( y ) ( y )] =. = (5 ) a = 5. (7 y) ( 5 y) = 6. (7 y)( 5 y) = 7. (4i ) (i ) = Simplify without a calculator, givig aswer i eact form (ot decimal). I your aswer, epress all epoets as positive values ad covert ay fractioal powers to radical form y z = z y t t 5 5t 8 5t Page 6
7 y z ( ) 6 y z ( ) (6 ) ( ) 7 98 Fid the eact value without your calculator (o decimal aswers) Solve each equatio algebraically; verify your solutio by substitutig i the origial equatio. 5. ( 7) y solve for i terms of ( ) ( 7)( ) 6. c, where a b a 0, b 0, b a, solve for.. Solve by completig the square: Fid the solutio to the system of equatios. y y 9 a 7. r s, solve for r. 9. y 98y 4 Page 7
8 4. Determie the slope betwee the poits (4, ) ad (6, 4). 40. Determie the slope of the lie y = Write i slopeitercept form the equatio of the lie cotaiig the poit (, ) ad parallel to the give lie y = Write i slopeitercept form the equatio of the lie cotaiig the poit (4, 5) ad perpedicular to the give lie y = 6. You should kow how to quickly sketch the graphs of these five basic paret fuctios: a) y b) y c) y d) y e) y 4. From the paret graph of y ( ) 5 ad graph the fuctio. y describe the shift to obtai the ew graph of 44. From the paret graph of y describe the shift to obtai the ew graph of y ad graph the fuctio. 45. State whether the give set of poits is a relatio or a fuctio{(,),(,0),(,0),(,)}. 46. For the poits give F= {(,),(,0),(,0),(,)}., state the domai ad rage. 47. State the domai ad rage of the fuctio h ( ) ad vertical ad horizotal asymptotes if ay eist. 48. Fid the domai, rage, zero(s), ad yitercept of f ( ) ad verify by graphig. 49. Fid the domai, rage, zero(s), ad yitercept of g( ) 4 ad verify by graphig. 50. Give 5. Give f ( ) fid its iverse f ( ) ad the sketch the graph of both. g( ) 4 fid its iverse g ( ) ad the sketch the graph of both. For #5 ad 5 use the followig: 5. Fid f ( g( )) f g 4 ( ) ; ( ) Fid g( f ( )) Factor the followig: 58. 9y y 4z y y Page 8
9 Summer Readig AP Calculus: Complete the idicated operatio to simplify the polyomials. Ratioal aswers should have a commo deomiator What are the legths of the missig sides of a triagle if the loger leg of the triagle is 8 cetimeters? 66. The hypoteuse of a triagle is iches. Fid the measures of the other two sides. 68. What is the legth of the hypoteuse of a triagle if oe leg measures 9 cetimeters? 69. The leg of a triagle is 4 cetimeters. Fid the measures of the other two sides. 70. If the radius of a circle is 6 cetimeters, what is its eact circumferece? 7. If the radius of a circle is 6 cetimeters, what is its area? 7. What is the area of a triagle with base of 7 cm ad altitude to the base of 4 cm? 7. If the base of a parallelogram is 5 iches ad altitude to the base is oe third of the base, what is the area of this parallelogram? 9
Algebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y SlopeIntercept Equation: y m b slope=
More informationLyman Memorial High School. Honors PreCalculus Prerequisite Packet. Name:
Lyma Memorial High School Hoors PreCalculus Prerequisite Packet 2018 Name: Dear Hoors PreCalculus Studet, Withi this packet you will fid mathematical cocepts ad skills covered i Algebra I, II ad Geometry.
More informationA.1 Algebra Review: Polynomials/Rationals. Definitions:
MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole
More informationUnit 4: Polynomial and Rational Functions
48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationn m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 31 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a
CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial
More informationSolving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots)
Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3
More informationFUNCTIONS (11 UNIVERSITY)
FINAL EXAM REVIEW FOR MCR U FUNCTIONS ( UNIVERSITY) Overall Remiders: To study for your eam your should redo all your past tests ad quizzes Write out all the formulas i the course to help you remember
More informationP.3 Polynomials and Special products
Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +
More information( ) 2 + k The vertex is ( h, k) ( )( x q) The xintercepts are x = p and x = q.
A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The xcoordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright  For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk  For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk  For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationLESSON 2: SIMPLIFYING RADICALS
High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6
More informationWe will conclude the chapter with the study a few methods and techniques which are useful
Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs
More informationG r a d e 1 1 P r e  C a l c u l u s M a t h e m a t i c s ( 3 0 S )
G r a d e 1 1 P r e  C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Grade 11 PreCalculus Mathematics (30S) is desiged for studets who ited to study calculus ad related mathematics as part of postsecodary
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationCurve Sketching Handout #5 Topic Interpretation Rational Functions
Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials
More informationSubstitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
More informationThe ztransform. 7.1 Introduction. 7.2 The ztransform Derivation of the ztransform: x[n] = z n LTI system, h[n] z = re j
The Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discretetime LTI systems. 7.
More informationBITSAT MATHEMATICS PAPER III. For the followig liear programmig problem : miimize z = + y subject to the costraits + y, + y 8, y, 0, the solutio is (0, ) ad (, ) (0, ) ad ( /, ) (0, ) ad (, ) (d) (0, )
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationMA Lesson 26 Notes Graphs of Rational Functions (Asymptotes) Limits at infinity
MA 1910 Lesso 6 Notes Graphs of Ratioal Fuctios (Asymptotes) Limits at ifiity Defiitio of a Ratioal Fuctio: If P() ad Q() are both polyomial fuctios, Q() 0, the the fuctio f below is called a Ratioal Fuctio.
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationAlgebra 1 Hour Final Exam Review Days
Semester Fial Eam Review Packet Name Algebra 1 Hour Fial Eam Review Days Assiged o Assigmet 6/6 Fial Eam Review Chapters 11 ad 1 Problems 54 7 6/7 Fial Eam Review Chapters 10 Problems 44 5 6/8 Fial Eam
More information+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More information4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M 1 M 108 M1 A1 [6] M1 A1
4755 Mark Scheme Jue 05 * Attempt to fid M or 08M  M 08 8 4 * Divide by their determiat,, at some stage Correct determiat, (A0 for det M= 08 stated, all other OR 08 8 4 5 8 7 5 x, y,oe 8 7 4xy 8xy dep*
More informationNorthwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet
Northwest High School s Algebra /Hoors Algebra Summer Review Packet This packet is optioal! It will NOT be collected for a grade et school year! This packet has bee desiged to help you review various mathematical
More informationGRAPHING LINEAR EQUATIONS. Linear Equations ( 3,1 ) _xaxis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Quadrat II Quadrat I ORDERED PAIR: The first umer i the ordered pair is the coordiate ad the secod umer i the ordered pair is the ycoordiate. (,1 ) Origi ( 0, 0 ) _ais Liear
More informationEnd of year exam. Final Exam Review. 1.What is the inverse of the function Which transformations of the graph of. x will produce the graph of
Name Date lass 1.What is the iverse of the fuctio f ( )? f 1 ( ) f 1 ( ) f ( ) ( ) 1 1. What trasformatios o the graph of f ( ) result i the graph of g( )? Traslate right by uits ad dow by uits. Stretch
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationMathematics Extension 1
016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Boardapproved
More informationCATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01
CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ
More informationJEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018)
JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART : JEE(Advaced) 08/Paper SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Istructor: Mr. Adrew Nichols Email questios to atichols@atlata.k.ga.us Help Sessio: Moday, July, :0 5:0 Due Date: Tuesday, August Prerequisites Quiz: Thursday, August Istructios:
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More informationMTH Assignment 1 : Real Numbers, Sequences
MTH 26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a oempty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (D) Cosider the lie passig through A (,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationSail into Summer with Math!
Sail ito Summer with Math! For Studets Eterig Hoors Geometry This summer math booklet was developed to provide studets i kidergarte through the eighth grade a opportuity to review grade level math objectives
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Boardapproved calculators may be used A table of stadard
More informationMathematics: Paper 1
GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists
More informationMath IIIFormula Sheet
Math IIIFormula Sheet Statistics Zscore: Margi of Error: To fid the MEAN, MAXIMUM, MINIMUM, Q 3, Q 1, ad STANDARD DEVIATION of a set of data: 1) Press STAT, ENTER (to eter our data) Put it i L 1 ) Press
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationPolynomial and Rational Functions. Polynomial functions and Their Graphs. Polynomial functions and Their Graphs. Examples
Polomial ad Ratioal Fuctios Polomial fuctios ad Their Graphs Math 44 Precalculus Polomial ad Ratioal Fuctios Polomial Fuctios ad Their Graphs Polomial fuctios ad Their Graphs A Polomial of degree is a
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationU8L1: Sec Equations of Lines in R 2
MCVU Thursda Ma, Review of Equatios of a Straight Lie (D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (,) with slope, as show i the diagram below. I poit slope form, the equatio
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More information3 Show in each case that there is a root of the given equation in the given interval. a x 3 = 12 4
C Worksheet A Show i each case that there is a root of the equatio f() = 0 i the give iterval a f() = + 7 (, ) f() = 5 cos (05, ) c f() = e + + 5 ( 6, 5) d f() = 4 5 + (, ) e f() = l (4 ) + (04, 05) f
More informationLecture 7: Polar representation of complex numbers
Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with
More informationMATHEMATICS (Three hours and a quarter)
MATHEMATICS (Three hours ad a quarter) (The first fiftee miutes of the eamiatio are for readig the paper oly. Cadidates must NOT start writig durig this time.) Aswer Questio from Sectio A ad questios from
More informationa is some real number (called the coefficient) other
Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where
More information7.1 Finding Rational Solutions of Polynomial Equations
Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationPhysicsAndMathsTutor.com
PhysicsAdMathsTutor.com physicsadmathstutor.com Jue 005 3. The fuctio f is defied by (a) Show that 5 + 1 3 f:, > 1. + + f( ) =, > 1. 1 (4) (b) Fid f 1 (). (3) The fuctio g is defied by g: + 5, R. 1 4 (c)
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4  ad g() = si(), 1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationMathematics Extension 2
004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Boardapproved calculators may be used A table of stadard
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationCourse 4: Preparation for Calculus Unit 1: Families of Functions
Course 4: Preparatio for Calculus Uit 1: Families of Fuctios Review ad exted properties of basic fuctio families ad their uses i mathematical modelig Develop strategies for fidig rules of fuctios whose
More informationVICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015
VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write
More informationOrder doesn t matter. There exists a number (zero) whose sum with any number is the number.
P. Real Numbers ad Their Properties Natural Numbers 1,,3. Whole Numbers 0, 1,,... Itegers..., 1, 0, 1,... Real Numbers Ratioal umbers (p/q) Where p & q are itegers, q 0 Irratioal umbers otermiatig ad
More informationTaylor Series (BC Only)
Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial lookalike to a opolyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor
More informationJEE ADVANCED 2013 PAPER 1 MATHEMATICS
Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot
More informationEDEXCEL STUDENT CONFERENCE 2006 A2 MATHEMATICS STUDENT NOTES
EDEXCEL STUDENT CONFERENCE 006 A MATHEMATICS STUDENT NOTES South: Thursday 3rd March 006, Lodo EXAMINATION HINTS Before the eamiatio Obtai a copy of the formulae book ad use it! Write a list of ad LEARN
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationFLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.
Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationOptional Review for EOI & FINAL EXAM
13 7KEuXtLap vseofatzwyafreec CLXLSCK.J U oaillb 6r6i6g9hutfsO lr5ehseecrvteqdb.f 5 MUa1dzev lwyigtvhk IHffiLiiWtbe8 YAKl5gzeIbDr3aq C2Z.o Worksheet b Kuta Software LLC 04.06.10 Algebra 2 Name Optioal
More information( ) D) E) NOTA
016 MAΘ Natioal Covetio 1. Which Greek mathematicia do most historias credit with the discovery of coic sectios as a solutio to solvig the Delia problem, also kow as doublig the cube? Eratosthees Meaechmus
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More information(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.
The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as
More informationMATHEMATICS 9740 (HIGHER 2)
VICTORIA JUNIOR COLLEGE PROMOTIONAL EXAMINATION MATHEMATICS 970 (HIGHER ) Frida 6 Sept 0 8am am hours Additioal materials: Aswer Paper List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write our ame
More informationPreCalculus 12 Practice Exam 2 MULTIPLECHOICE (Calculator permitted )
Prealculus Practice Eam MULTIPLEHOIE (alculator permitted ). Solve cos = si, 0 0.9 0.40,.5 c. 0.79 d. 0.79,.8. Determie the equatio of a circle with cetre ( 0,0) passig through the poit P (,5) + = c.
More informationChapter 13: Complex Numbers
Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges  that is, lims N n
Review of Power Series, Power Series Solutios A power series i x  a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia Email : sahoopulak1@gmail.com 1 Module2: Stereographic Projectio 1 Euler
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationCalculus 2 Test File Spring Test #1
Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() =  ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad
More informationThe type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS PreCalculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationUNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Name: Date: Part I Questios UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS. For the quadratic fuctio show, the coordiates. of its verte are 0, (3) 6, (), 7 (4) 3, 6. A quadratic fuctio
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationUNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST. First Round For all Colorado Students Grades 712 November 3, 2007
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST First Roud For all Colorado Studets Grades 7 November, 7 The positive itegers are,,, 4, 5, 6, 7, 8, 9,,,,. The Pythagorea Theorem says that a + b =
More informationNorth Central Summer Assignment
North Cetral Summer Assigmet For Studets Eterig AP Calculus BC As you prepare to eter AP Calculus BC, it is crucial that you have a solid Algebra foudatio. This assigmet provides a opportuity for you to
More informationWBJEE Answer Keys by Aakash Institute, Kolkata Centre
WBJEE  7 Aswer Keys by, Kolkata Cetre MATHEMATICS Q.No. B A C B A C A B 3 D C B B 4 B C D D 5 D A B B 6 C D B B 7 B C C A 8 B B A A 9 A * B D C C B B D A A D B B C B 3 A D D D 4 C B A A 5 C B B B 6 C
More informationHonors Calculus Homework 13 Solutions, due 12/8/5
Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationUNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questios. For the quadratic fuctio show below, the coordiates of its verte are () 0, (), 7 (3) 6, (4) 3, 6. A quadratic
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More information