# ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

Save this PDF as:

Size: px
Start display at page:

Download "ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS" ## Transcription

1 ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX Armstrong Hll Morgntown, WV 6506 Introdution Miller (011) disussed how to use the free intertive lger-geometry progrm Geoger, to investigte speil property of ui polynomils. The rtile demonstrted vrious wys to use Geoger to illustrte the speil property tht given ui polynomil with three rel zeros, tngent drwn to the urve t the point t whih the siss is the men of two of those three zeros will interset the horizontl xis t the other zero. This property ws extended y Miller nd Moseley (01) to higher degree polynomils. We will use Geoger to illustrte how this speil property of ui polynomils n e generlized to higher degree polynomils. This will e done y first reviewing the ui polynomil se through n pplet, disussing nd illustrting in Geoger the ses for fourth nd fifth degree polynomils long with derivtions of formuls. The proof for the 4 th degree polynomil se, nd the generl proof for the nth degree polynomil with zeros is shown in (Miller nd Moseley, 01). Cui Polynomils Miller (011) uses Geoger to present vriety of wys to illustrte different exmples of the speil property for ui polynomils. Insted of reiterting wht is everything desried out uis in Miller (011) nd Miller nd Moseley (01), we will onstrut n pplet in Geoger to illustrte the se where ll the zeros of the ui polynomil re rel numers nd then stte the proof. First go to nd li on the downlod t tht ppers etween Aout nd Help ner the upper right orner. Downlod free version of Geoger y liing on the Applet Strt utton. Now follow the steps elow nd refer to Miller (011) under the suheding Adding More Visul Proof for more detiled instrutions. 1. Insert three sliders y liing on the slider menu nd liing in the grphing window three seprte times (see figure 1 elow). Lel the sliders,, nd with minimum -10, mximum 10, nd inrement of Figure 1: Slider Button. In the input ox (ottom of Geoger) type in 153

2 f(x) = (x-)*(x-)*(x-). Your sreen should loo lie figure elow. Figure : Generl Cui with ll zeros t Adjust to e, to e 1, nd to e -1. Your sreen should loo similr to Figure 3 elow. Figure 3: Cui with Speified Sliders 4. Now we need to pture the x-interepts. Strt typing in roots nd when the input ox gives you drop down menu of different ommnds with root in them, selet the ommnd Roots[ <Funtion>, <Strt x-vlue>, <End x-vlue> ] nd insert f(x) for <Funtion>, -10 for <Strt x-vlue>, 10 for <End x-vlue>, nd Enter. You should get points A, B, nd C for the x-interepts of the polynomil. Your sreen should loo lie figure 4 elow. 5. Input into the input r, g(x) = Derivtive[f(x)]. Cli on the rdio utton next to g(x) in the lger ox (under the dependent ojets) to hide it. 6. Type in d = ()/ into the input ox. Repet y typing e=()/ nd h=()/. 154

3 Figure 4: Cui with Mred Zeros 7. Now type in to the input ox seprtely, y f(d)=g(d)*(x-d), y f(e)=g(e)*(x-e), nd y f(h)=g(h)*(x-h). Your sreen should loo similr to Figure 5 elow. Figure 5: Cui with Tngent Lines 8. Polish up the sreen y inserting olor, line thiness, y doing the following: Right li on f(x) nd go to ojet properties. Choose olor (I hoose lue) nd line thiness (under style) to e 4. Repet for eh tngent line nd the sliders. Under ojet properties si, you n lel f(x) nd eh tngent line y seleting the ox for show lel nd liing on the down rrow in the seletion ox nd hoosing nme & vlue option. Your sreen should loo similr figure 6 elow. 155

4 Figure 6: Polished Version of the Cui Polynomil with Sliders Notie tht s you move the sliders to different vlues, the tngent line t the verge of two of the zeros lwys hs n x-interept t the other zero. This shows ll the different exmples for rel zeros (even in the trivil se when two or three of the zeros re the sme), exept when two of the zeros re omplex onjugtes. The proof elow shows why this lwys wors for ui polynomils with zeros,, nd nd n e found in (Miller nd Moseley, 01). Students elieve tht they n sy something is true (proved) if they show some exmples. Therefore it might e est to emphsize why the proof is neessry. Given ny ui polynomil, we n ftor it y the fundmentl theorem of lger into the form ) )( )( ( ) ( x x x x f = where,, nd re omplex numers (notie in the Geoger pplet, = 1. One ould insert nother slider to inorporte the sling ftor ). Assume tht. Otherwise the derivtion is trivil when = or = =. Evluting the funtion t the verge of nd, we hve = = f. Evluting the derivtive t the verge of nd, we hve = = f So the tngent line is = x y Sustituting y = 0 nd solving for x we hve 156

5 0= x 0= 0= x x This lst eqution simplifies to 0 = x nd so x=, the other zero. Sine there ws no preferene why we pied the zeros nd to verge, we would otin the sme result if we hose ny two distint zeros. We see n wesome property of ui polynomils, ut we re left with feeling tht there is more to this tht wht meets the eye. Is there some hidden explntion for why it wors? Does the tngent line to n nth degree polynomil t the verge of ll ut one of the zeros lwys hve n x-interept t the other zero, for? If not, is there nother formul in terms of ll ut one of the zeros, esides verging, tht would yield the sme result? Although we do not nswer these questions expliitly in this rtile, the reder n referene (Miller nd Moseley, 01) to get more detils of the formuls, underlying property from Clulus, nd generl proofs. We will onentrte on uilding Geoger pplets with sliders to illustrte how the ui property n e generlized to higher degree polynomils nd use the formuls in the pplets. Fourth Degree Polynomil Exmple Now we will loo t uilding exmples of fourth nd fifth degree polynomils. Let,,, nd d e the rel zeros of the fourth degree polynomils (zeros ould e omplex, ut n not e visulized in Geoger). The formul for fourth degree polynomils f(x) is derived in (Miller nd Moseley, 01). Given three of the zeros of f(x), sy,, nd, nd the solutions, nd, of the following qudrti eqution 3x ( ) x = 0, the tngent line t nd, intersets the x-xis t the other zero d. To illustrte this we will uild Geoger pplet with the following steps. 1. Insert four sliders y liing on the slider menu nd leling them,,, nd d with minimum -10, mxium 10, nd inrement of In the input ox (ottom of Geoger) type in f(x) = (x-)*(x-)*(x-)*(x-d). Your sreen should loo figure 7 elow. 3. Adjust to e -, to e -1, to e 0, nd d to e 1. Your sreen should loo lie figure 8 elow. 157

6 4. Now we need to pture the x-interepts. Strt typing in roots nd when the input ox gives you drop down menu of different ommnds with root in them, selet the ommnd Roots[ <Funtion>, <Strt x-vlue>, <End x-vlue> ] nd insert f(x) for <Funtion>, -10 for <Strt x-vlue>, 10 for <End x-vlue>, nd Enter. You should get points A, B, C, nd D for the x-interepts of the polynomil. Your sreen should loo lie figure 9 elow. 5. Input into the input r, g(x) = Derivtive[f(x)]. Cli on rdio utton next to g(x) in the lger ox (under the dependent ojets) to hide it. 6. Type in e=(sqrt(^^^-()))/3 nd h=(-sqrt(^^^-()))/3 into the input ox. 7. Now type in y-f(e)=g(e)*(x-e) to get the tngent line t e tht hs n x-interept t d. Repet for tngent line t h y inputing y-f(h))=g(h)*(x-h).your sreen should loo lie figure 10 elow. 8. Polish up the sreen y inserting olor, nd line thiness (see figure 11). FIGURE 7: Generlized 4 th degree polynomil with sliders set t 1 FIGURE 8: Generlized 4 th degree polynomil with sliders for the zeros. FIGURE 9: Generlized 4 th degree with zeros leled A,B, C, nd D 158

7 FIGURE 10: Generlized 4 th degree polynomil w/ tngent lines tht interset t D FIGURE 11: Polished Up Version of 4 th degree polynomil It is to note tht we re only showing the tngent lines tht interset t d. There re six more tngents lines, two tht hve x-interepts t, two t, nd two t. The x-vlues of the point of tngeny n e lulted y formul similr to the formul in 6 ove (the formul would e in terms of,,d, nd e for the tngent lines tht interset the horizontl xis t ). If you would onstrut ll the tngent lines then you would get the follow view iin Geoger (see figure 1). Fifth Degree Polynomil Exmple Let,,, d, nd e e the rel zeros of the fifth degree polynomil f(x). The formul for f(x) is derived in (Miller nd Moseley, 01). Given four of the zeros of f(x), sy,,, nd d, the solutions of the ui eqution sy,, nd the tngent line t nd, nd intersets the x-xis t the other zero e. To illustrte this we will uild Geoger pplet with the following steps. 1. Input five sliders nd lel them,,, d, nd e. Selet Min to e -10, Mx to e 10, nd Inrement to e Input into the Input ox f(x) = (x-)*(x-)*(x-)*(x-d)*(x-e) nd slide the sliders so tht is -, is -1, is 0, d is 1, nd e is. 3. Type in Roots[ f(x), -10, 10 ]. Your sreen should loo similr to figure 13 elow. 159

8 4. Now we wnt to find the solutions to the (ui) eqution we get from the proof for higher degree polynomils. In (Miller nd Moseley, 01) we see tht for higher degree polynomils we hve n 1 n1 i= 1 j= 1 i j ( x0 x j ) = 0 where nd re four of the five zeros of the fifth degree polynomil. Expnding this out we hve Let nd (prmeters orresponding to the prmeters in the pplet). Figure 1: Qurti showing ll tngent lines. FIGURE 13: 5 th degree polynomil with sliders To solve this we will type in the input ox the two ommnds nd Roots[ m(x), -10, 10 ]. 160

9 The roots re the three points where the tngent line hs n x-interept t the other zero e. Your sreen should loo similr to figure 14 elow (where the line type (under ojet properties tht you n get to y right liing on m(x)) ws hnged to dotted). 5. To find the zeros of m(x), type in Root[m(x)].You should see the points F, G, nd H pper on the grph nd in the lger ox. 6. These re the three points where the tngent line (t these points) hve x-interepts t e. To see this we first must otin the x-oordintes of F, G, nd H y typing x(f), x(g), nd x(h). These will pper in the lger window s g, h, nd I (sine f ws used for the funtion). 7. Type j(x)=derivtive[f(x)] nd hide it. Figure 14: 5 th degree polynomil with dotted derivtive 8. Now type y-f(g)=j(g)*(x-g), y-f(h)=j(h)*(x-h), nd y-f(i)=j(i)*(x-i). 9. Now polish things up y dding olor, line thiness nd leling. Your sreen should e similr to figure 15. FIGURE 15: Polished up Version 5 th degree polynomil 161

10 Conlusion This rtile hs shown how to use Geoger to extend mysterious property of ui polynomils to more generl properties for fourth nd fifth degree polynomils. In generl one n illustrte this for n nth degree polynomil to find the n- points on the x-xis in terms of n-1 zeros, tht is given y the solutions to the eqution elow, where re the n-1 zeros, n 1 n1 i= 1 j= 1 i j ( x x j ) = 0 suh tht the tngent line to the nth degree polynomil t eh intersets the x-xis t the other zero. Here for re the zeros of the resulting n- degree polynomil derived from the formul ove. See (Miller nd Moseley, 01) for more detils nd the underlying lulus onept ehind the generl property for polynomils. The reder should illustrte this for the 6 th degree polynomil nd thin out the proof efore referening the rtiles. This rtile shows some si funtions of Geoger so tht the reder n fmilirize themselves with the progrm nd is good tool for students to disover some mthemtis out polynomils in whih they n see some speifi exmples nd wor on more generl proof vi pper nd penil. Referenes Hmilton, Pete. (005). Mthemtis IB Higher Portfolio: Zeros of Cui Funtions. (essed ). Miller, D. (011). Investigting Zeros of Cuis through GeoGer. Mthemtis Teher, Vol. 105, No.. Miller, D., nd Moseley, J. (01). Extending Property of Cui Polynomils to Higher-Degree Polynomils. MthAMATYC Edutor, Vol. 3, Issue 3. Stewrt, J. (007). Essentil Clulus: Erly Trnsendentls. Thomson Broos/Cole. Sullivn nd Sullivn. (006). Alger nd Trigonometry: Enhned with Grphing Utilities. Person Prentie Hll, 4 th Edition. 16

### THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

### ( ) { } [ ] { } [ ) { } ( ] { } Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or

More information

### Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

### CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

### AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

### 1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

### 12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

### Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

### More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

### Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

### 10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis 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 information

### Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

More information

### PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

### Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

### Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

### 8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

More information

### f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

More information

### Reflection Property of a Hyperbola Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the

More information

### Learning Objectives of Module 2 (Algebra and Calculus) Notes: 67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under

More information

### 2.4 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 information

### INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

### GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

### QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

### Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

### Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

More information

### SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

### Polynomials 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 information

### Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

### for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

### Trigonometry and Constructive Geometry Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties

More information

### 4 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 information

### Math Lesson 4-5 The Law of Cosines Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude

More information

### Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

### AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

More information

### (a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

### AP Calculus AB Unit 4 Assessment Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope

More information

### Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

### 1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

### Read 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 information

### For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:

More information

### Math Lecture 23 Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

More information

### Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

### THE EXTREMA OF THE RATIONAL QUADRATIC FUNCTION f(x)=(x 2 +ax+b)/(x 2 +cx+d) Larry Larson & Sally Keely THE EXTREMA OF THE RATIONAL QUADRATIC FUNCTION f(x)(x +x+b)/(x +x+d) Lrry Lrson & Slly Keely Rtionl funtions nd their grphs re explored in most prelulus ourses. Mny websites provide intertive investigtions

More information

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

### 1 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 information

### 1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

### 5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

### Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

### 1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

More information

### Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

### 6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

### MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information

### Part I: Study the theorem statement. Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for

More information

### Unit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NON-CALCULATOR SECTION Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NON-CALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)

More information

### Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

### Logarithms 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 information

### We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

### Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

### Section 2.1 Special Right Triangles Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem

More information

### H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

### SEMI-EXCIRCLE OF QUADRILATERAL JP Journl of Mthemtil Sienes Volume 5, Issue &, 05, Pges - 05 Ishn Pulishing House This pper is ville online t http://wwwiphsiom SEMI-EXCIRCLE OF QUADRILATERAL MASHADI, SRI GEMAWATI, HASRIATI AND HESY

More information

### Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

### University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

### Now we must transform the original model so we can use the new parameters. = S max. Recruits MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with

More information

### CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

### List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

### Lab 11 Approximate Integration Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties

More information

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

More information

### 4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

### Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

### Exercise 3 Logic Control Exerise 3 Logi Control OBJECTIVE The ojetive of this exerise is giving n introdution to pplition of Logi Control System (LCS). Tody, LCS is implemented through Progrmmle Logi Controller (PLC) whih is lled

More information

### Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

### P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

### A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

### Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner

More information

### Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information

### LESSON 11: TRIANGLE FORMULAE . THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.

More information

### Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

More information

### HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots

More information

### Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

### Arrow s Impossibility Theorem Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties

More information

### PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

### Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

### Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

More information

### Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

### Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the

More information

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

More information

### Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

### Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

### 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 information

### 20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

### Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

### Objectives. Materials Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the

More information