Identifying and Classifying 2D Shapes

 Matthew Doyle
 3 months ago
 Views:
Transcription
1 Ientifying n Clssifying D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte elow. Cirle Squre Retngle Otgon Tringle Knowlege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes /6
2 The tle shows how mny sies eh shpe hs. Shpe Numer of sies Cirle Tringle Retngle Squre Shpes tht hve more thn one sie re lle polygons. A stop sign is n 8sie polygon lle n otgon n yiel sign is sie polygon lle tringle. Qurilterls Retngles n squres hve four sies n re prt of lrger group of foursie shpes lle qurilterls. The ngles of qurilterls totl 60. A qurilterl is ny foursie figure. quri four lterl sie There re mny types of qurilterls, n they re lssifie oring to their shpes, ngles n lengths of sies, s shown elow. squre retngle prllelogrm rhomus trpezoi Know lege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes /6
3 Shpe Squre Retngle Prllelogrm Rhomus Trpezoi Chrteristis All sies equl in length All ngles 90 Prllel sies equl in length, ut not the sme length s the other prllel sies All ngles 90 Two prllel sies re equl in length, ut other two prllel sies re ifferent length Two ifferent sets of ngle mesurements All sies equl in length Two ifferent sets of ngle mesurements Lengths of sies my not e equl Opposite sies re not prllel Angles vry n my iffer from eh other Tringles There re three types of tringles. Tringles re lssifie oring to the length of their sies n the mesurement of their lrgest ngle. The ngles of tringles totl 80.. Length of sies Isoseles Tringle Equilterl Tringle Slene Tringle Two sies re equl in length. line of symmetry All sies re equl in length. lines of symmetry All sies re ifferent in length. 0 lines of symmetry Mesurement of the lrgest ngle Right Tringle Aute Tringle Otuse Tringle Lrgest ngle is extly 90 Lrgest ngle is <90 Lrgest ngle is >90 Knowlege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes /6
4 Think Aout Humnme strutures suh s uilings n riges re reting using shpes suh s irles, retngles n squres. Shpes re lso foun in nture, for exmple, the sun is irle n the entre of flower is irle. Wht other shpes re foun in nture? Prtie: Mesuring n Clssifying Qurilterls. Use ruler n protrtor to mesure n reor the lengths of eh sie n ngle. Then lssify eh qurilterl. Shpe Length of eh sie Angles.. ) ) ). ) ). ) Knowlege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes /6
5 Mesure n Clssify Qurilterls (ontinue) Shpe Length of eh sie Angles.. ) ) ). ) ). ).. ) ) ) With prtner, lote vriety of lrge n smll qurilterls in your lssroom or shool, suh s oors, winows, tles n tiles. Use pproprite instruments to mesure ngles n sies (in metri n/or imperil units). Knowlege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes 5/6
6 Prtie: Mesuring n Clssifying Tringles. For eh tringle, mesure the lengths of eh sie n ngle using ruler n protrtor. Clssify eh tringle oring to the length of sies n ngles. Shpe Length of eh sie Angles. ) ). ) ). ) ) Mny is esigning tringulr errings. She mkes eh sie of the tringles ifferent length. Two of the sies join together to form n ngle of. Clssify, in two wys, the types of tringles Mny esigne. Knowlege n Employility Stuio Mthemtis Shpe n Spe: Ojets n Shpes: Alert Eution, Alert, Cn (www.lernalert.) Ientifying n Clssifying D Shpes 6/6
We use metres to measure length. There are 100 centimetres in a metre. a 6 m = cm b 3 m = cm c 9 m = cm
Units of length metres We use metres to mesure length. There re 00 entimetres in metre. 00 m = m Convert these metres to entimetres: 6 m = m 3 m = m 9 m = m 600 300 900 Estimte nd then mesure the length
More informationGeometry of the Circle  Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.
Geometry of the irle  hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in nonright ngled tringles. This unit
More informationProving 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 informationBasic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.
si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)
More information4. Statements Reasons
Chpter 9 Answers PrentieHll In. Alterntive Ativity 9. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students
More informationART LESSONS & EXERCISES
RT LESSONS & EXERISES Rmón Gllego. www.diujormon.wordpress.com www.diujormon.wordpress.com English  Pge 1 www.diujormon.wordpress.com English  Pge 2 STRTING UP WITH GEOMETRY. DEFINITION OF GEOMETRY Geometry
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationUNCORRECTED PAGE PROOFS
mesurement mesurement n n geometry geometry UNORRETE PGE PROOFS topi 10 edutive geometry 10.1 Overview Why lern this? Lerning out geometry inludes eing le to reson dedutively nd to prove logilly tht ertin
More information3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h
PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h
More informationCS 360 Exam 2 Fall 2014 Name
CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singlylinke list n stk. Write three ifferent O(n)time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More information8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter
8A Length nd perimeter 8E Are of irle 8B Cirumferene of irle 8F Surfe re 8C Are of retngles nd tringles 8G Volume of prisms 8D Are of other qudrilterls 8H Are nd volume onversions SA M PL E Mesurement
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 informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationNumbers 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 informationfor 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 informationPythagoras theorem and surds
HPTER Mesurement nd Geometry Pythgors theorem nd surds In IEEM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of rightngled tringle. This result is known s Pythgors
More informationGRADE 4. Division WORKSHEETS
GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x  4 +x xmple 3: ltitude of the
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationChapter 12. Lesson Geometry WorkedOut Solution Key. Prerequisite Skills (p. 790) A 5 } perimeter Guided Practice (pp.
Chpter 1 Prerequisite Skills (p. 790) 1. The re of regulr polygon is given by the formul A 5 1 p P, where is the pothem nd P is the perimeter.. Two polygons re similr if their corresponding ngles re congruent
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 information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
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 informationWhat else can you do?
Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More information4.3 The Sine Law and the Cosine Law
4.3 Te Sine Lw nd te osine Lw Te ee Tower is te tllest prt of nd s rliment uildings. ronze mst, wi flies te ndin flg, stnds on top of te ee Tower. From point 25 m from te foot of te tower, te ngle of elevtion
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 informationSeries. Teacher. Fractions
Series E Teher Frtions Copyright 009 P Lerning. All rights reserved. First edition printed 009 in Austrli. A tlogue reord for this ook is ville from P Lerning Ltd. ISBN 979090 Ownership of ontent The
More informationCIT 596 Theory of Computation 1. Graphs and Digraphs
CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More information15  TRIGONOMETRY Page 1 ( Answers at the end of all questions )
 TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationAP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight
AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More information5: 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 informationx = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given
More informationG811 Congruence Rules
G811 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th
More informationPythagoras 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 rightngled tringle is the longest side The hypotenuse is lwys opposite the rightngle 2 = 2 + 2 or 2 = 22 or 2 = 22 The re of the
More informationCS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)
CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More information10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm
HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationAngles and triangles. Chapter Introduction Angular measurement Minutes and seconds
hpter 0 ngles nd tringles 0.1 Introduction stright line which crosses two prllel lines is clled trnsversl (see MN in igure 0.1). Trigonometry is suject tht involves the mesurement of sides nd ngles of
More informationInstructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015
Instructor(s): Acost/Woodrd PHYSICS DEPATMENT PHY 049, Fll 015 Midterm 1 September 9, 015 Nme (print): Signture: On m honor, I hve neither given nor received unuthorized id on this emintion. YOU TEST NUMBE
More informationPreLie algebras, rooted trees and related algebraic structures
PreLie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A prelie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationAnswers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2
Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge
More informationVectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:
hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points
More informationMaterials Analysis MATSCI 162/172 Laboratory Exercise No. 1 Crystal Structure Determination Pattern Indexing
Mterils Anlysis MATSCI 16/17 Lbortory Exercise No. 1 Crystl Structure Determintion Pttern Inexing Objectives: To inex the xry iffrction pttern, ientify the Brvis lttice, n clculte the precise lttice prmeters.
More informationPlane Surveying Levelling
Deprtment of Civil Engineering, UC Introduction: Levelling is mens y which surveyors cn determine the elevtion of points, using other known points s references. Levelling is perhps the most sic of surveying
More informationLesson 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 informationProject 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 information200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes
PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More information2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B
Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
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 informationThe Leaning Tower of Pingala
The Lening Tower of Pingl Richrd K. Guy Deprtment of Mthemtics & Sttistics, The University of Clgry. July, 06 As Leibniz hs told us, from 0 nd we cn get everything: Multiply the previous line by nd dd
More informationQUANTITATIVE REASONING
Quntittive Resoning QUNTITTIVE RESONING The quntittive resoning section tests your bility to use numbers nd mthemticl concepts to solve quntittive problems, nd your bility to nlyze dt presented in different
More informationUnit 6 Solving Oblique Triangles  Classwork
Unit 6 Solving Oblique Tringles  Clsswork A. The Lw of Sines ASA nd AAS In geometry, we lerned to prove congruence of tringles tht is when two tringles re exctly the sme. We used severl rules to prove
More information20 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 information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationSection 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 higherdimensionl nlogues) with the definite integrls
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More information1 Online Learning and Regret Minimization
2.997 DecisionMking in LrgeScle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationAPPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS
TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex
More informationUsing integration tables
Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl
More informationUNCORRECTED. Australian curriculum MEASUREMENT AND GEOMETRY
3 3 3C 3D 3 3F 3G 3H 3I 3J Chpter Wht you will lern Pythgors theorem Finding the shorter sides pplying Pythgors theorem Pythgors in three dimensions (tending) Trigonometri rtios Finding side lengths Solving
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
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 informationCHAPTER 1 CENTRES OF MASS
1.1 Introduction, nd some definitions. 1 CHAPTER 1 CENTRES OF MASS This chpter dels with the clcultion of the positions of the centres of mss of vrious odies. We strt with rief eplntion of the mening of
More informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More information42nd International Mathematical Olympiad
nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cutengled tringle with circumcentre O. Let P on
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationSAMPLE. Ratios and similarity. 9.1 Ratios This section is revision of work of previous years. Several examples are presented.
Ojectives H P T R 9 Rtios nd similrity To divide quntity in given rtio To determine the rtio in which quntity hs een divided To pply the trnsformtions which re expnsions from the origin To define similrity
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationMA123, 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. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationNot for reproduction
AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
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 informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationContinuous probability distributions
Chpter 1 Continuous probbility distributions 1.1 Introduction We cll x continuous rndom vrible in x b if x cn tke on ny vlue in this intervl. An exmple of rndom vrible is the height of dult humn mle, selected
More informationTrigonometry. Trigonometry. Curriculum Ready ACMMG: 223, 224, 245.
Trgonometry Trgonometry Currulum Rey ACMMG: 223, 22, 2 www.mthlets.om Trgonometry TRIGONOMETRY Bslly, mny stutons n the rel worl n e relte to rght ngle trngle. Trgonometry souns ffult, ut t s relly just
More informationStudy Guide and Intervention
 Stud Guide nd Intervention with the Sme Sign The quotient of two integers with the sme sign is positive. Emple. 7 The dividend nd the divisor hve the sme sign. b. () The dividend nd divisor hve the sme
More information