2.1 ANGLES AND THEIR MEASURE. y I

 Erik Jones
 4 months ago
 Views:
Transcription
1 .1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the ngle etween them. Angles re usully mesured in degrees (denoted ), rdins (denoted rd, or without unit), or sometimes grdins (denoted grd). II y I β nd re oterminl ngles euse they hve the sme initil nd sme terminl side.  β Vertex Terminl side + Initil side Rys re hlflines, tht is, portion of line tht strts vertex point nd extends indefinitely in one diretion. x III IV In retngulr oordinte system, ngles re defined s the rottion from one ry to nother. The ry t the strting position is lled the initil side nd the ry t then ending position is the terminl side. Rottion in ounterlokwise diretion is positive nd rottion in lokwise diretion is negtive. An ngle is in stndrd position if its vertex is t the origin nd its initil side oinides with the positive xxis.
2 II Qudrntl Angles One full rottion in these three mesures orresponds to 360, π rdins, or 400 grdins. Hlf full rottion is 180 nd is lled stright ngle, nd qurter of full rottion is 90 nd is lled right ngle. An ngle less thn right ngle is lled n ute ngle, nd n ngle greter thn right ngle is lled n otuse ngle. y α 90, 40, 810, 70, et. I β 180, 40,900, et. β φ α δ 0, 360,70, et. III δ IV 360 is one full revolution. A right ngle, 90, is 90/360 1/4 revolution. Φ 70, 40, 810,  90, et. Converting from Degrees, Minutes, Seonds (D, M,S ) to Deiml Form The use of degrees to mesure ngles hrks k to the Bylonins, whose sexgesiml numer system ws sed on the numer 60. likely rises from the Bylonin yer, whih ws omposed of 360 dys (1 months of 30 dys eh). The degree is further divided into 60 r minutes (denoted ), nd n r minute into 60 r seonds(denoted ). A more nturl mesure of n ngle is the rdin. It hs the property tht the r length round irle is simply given y the rdin ngle mesure times the irle rdius. The rdin is lso the most useful ngle mesure in lulus. Grdins re sometimes used in surveying (they hve the nie property tht right ngle is extly 100 grdins), ut re enountered infrequently, if t ll, in mthemtis. 1 ounterlokwise revolution minutes 1 minute seonds Exmple on p.101 Convert to deiml in degrees whih mens 1 1 / so 1 1 /60 (1 /60)/60 1 / *1 / *1 / Convert 1.6 to (D, M,S ) in degrees. Strt with the deiml prt, *60 / Tke the deiml prt of 1.36 nd onvert to seonds..36 * 60 /1 1.6 Thus,
3 Converting from Degrees, Minutes, Seonds (D, M,S ) to Deiml Form Using TI83 or TI83 plus Convert to deiml in degrees. 0 nd [ANGLE] 1 6 nd [ANGLE] 1 ALPHA ENTER Selets Degree nottion Selets Minute nottion Selets Seond nottion To onvert from deiml to DMS, just type the numer in deiml form, then press nd [ANGLE] 4 to see it in DMS form.
4 Centrl Angles nd Ar Length A entrl ngle is n ngle whose vertex is t the enter of irle. The rys of entrl ngle sutend (interset) n r on the irle. [The entrl ngle is etween 0 nd 360 degrees]. 1 rdin is defined s entrl ngle of irle with rdius r tht hs n r length of r. If the irumferene of irle is πr, how mny rdins re there in 1 full revolution of the irle? Terminl Side Vertex r 3 r r 1 1 Ar length r 1 1 Ar length r 3 1 rdin Initil side r Ar length s 1 1 r Ar length s From geometry, we know tht the rtio of the mesures of the ngles equls the rtio of the orresponding lengths of the rs sutended y these ngles. s 1 s 1 s 1 r If 1 1 rdin, then the r length s 1 r, so nd rossmultiplying givess r for ny in theirle ARC LENGTH For irle of rdius r, entrl ngle of rdins sutends n r whose length s is s r Now try #37 on p.109
5 How do you find r length if the ngle is given in degrees? Rell tht the irumferene of irle is πr. The r length of full revolution ( full ) is πr r full Therefore full π rdins 1 full revolution 360 π rdins ½ revolution 180 π rdins ¼ revolution 90 π/ rdins Exmple 4 () on p. 104 Convert 60 to rdins. 60 * π rdins/ π /180 rdins π/3 rdins Now you do #13 on p.109 Exmple () on p.10 Convert 3π/4 rdins to degrees. (3π/4)*180 / π 13 Now you do # on p.109 Ltitude is G N s AG is the r length of the entrl ngle, AG, t point G. sutended y the ry from Erth s enter to Glsgow nd the ry from Erth s enter to Aluquerque. Tht is the distne from Glsgow to Aluquerque. G s AG Step 1: A AG G A Step : Ltitude is 0 Convert from DMS to deiml degreest Equtor /(1 /60 ) Step 3: Convert from deiml degrees to rdins *(π rdins/180) 0.8 rdins. Step 4: Clulte r length s AG. s AG R Erth * AG rdins 3960* miles G We use this ft to onvert from degrees to rdins. 1 rdin 180 / π 1 degree π rdins/180 Exmple 6 on p.10 Ltitude of lotion is the ngle formed y ry drwn from the enter of Erth to the Equtor nd ry drwn from the enter of Erth to the lotion. Glsgow, Montn is due north Aluquerque, New Mexio. Glsgow s ltitude is 48 9 N nd Aluquerque s ltitude is 3 N. Find the distne etween these ities. A Rdius 3960 miles Now do #91 on p.110
6 Common Degree to Rdin Conversions Degrees Rdins 0 π/6 π/4 π/3 π/ π/3 3π/4 π/6 π 7π/6 π/4 4π/3 3π/ π/3 7π/4 11π/6 π * * * * * Memorize these
7 Are of Setor From geometry, we know tht the rtio of the mesures of the ngles equls the rtio of the orresponding res of the setors formed y these ngles. Tht is, A 1 A 1 If 1 π rdins, then A 1 re of irle. A π πr A A π πr r r Multiply oth sides y to solve for A. Are of setor formed y entrl ngle of rdins is A 1 r Now you do #4 on p.109
8 Liner nd Angulr Speed Liner speed distne trveled round irle (s) divided y the elpsed time of trvel (t). veloity v s/t Angulr speed the entrl ngle swept out in time (), divided y the elpsed time, (t). Angulr speed ω /t where ngulr speed is in rdins per unit time. Exmple: An engine is revving t 900 RPM. If you re given RPM insted of ngulr speed, you n onvert to ngulr speed y using tht ft tht 1 revolution π rdins revolutions revolutions π rdins rdins π minute minute 1 revolution minute Liner speed v s/t (r)/t r(/t) rω where ω is mesured in rdins per unit time. Exmple 8 on p.108 A hild is spinning rok t the end of foot rope t rte of 180 revolutions per minute. Find the lier speed of the rok when it releses. revolutions revolutions π rdins rdins π minute minute 1 revolution minute Liner speed v rw (ft) * (360π rdins/minute) 70π feet/min 6 feet/min feet 1mile 60 min 6 * *.7 mph min 80 feet 1hour Now you do # 87 on p.110
9 Right Tringle Trigonometry is n ute ngle euse it is less thn 90 degrees. Hypotenuse Adjent to 90  Opposite to From geometry, we know tht the rtios of the sides of similr tringles re equl so: ', ' ', ' ', ' ', ' These rtios re the sme for ny right tringle with ute ngle. They re lled the trigonometri funtions of ute ngles. ' ' ' ' FUNCTION NAME ABBREV. VALUE Notie these funtions re the reiprols of sine, osine, & tngent, respetively. Sine of Cosine of Tngent of Cosent of Sent of Cotngent of sin() /opposite/hypotenuse os() /djent/hypotenuse tn() /opposite/djent s() /hypotenuse/opposite se() /hypotenuse/djent ot() /djent/opposite Rememer SOHCAHTOA! In other words: s()1/sin(), se()1/os(), ot()1/tn()
10 Finding Trig Funtions Exmple on p.114 Finding the Vlue of the Remining Trig Funtions, given sin nd os. Given sin( ), nd os( ), find the remining trig funtions of tn( ) sin( ) os( ) se( ) 1/ os( ) s( ) 1/ sin( ) 1 NOW YOU DO #11 on p11
11 Fundmentl Identities of Trigonometry We n use the Pythgoren Theorem to derive the fundmentl identities of trigonometry. We know +, right? Let s rerrnge some terms nd divide eh side y ( sin( )) + ( os( )) whih n lso e written s follows: sin Now we n get nother identity y dividing oth sides of this eqution y os sin ( ) os + os ( ) os tn sin sin + 1+ ot ( ) + os ( ) + 1 se ( ) os + ( ) sin 1 1 ( ) 1 ( ) ( ) ( ) Similrly, you hd divided eh side of + + ( ) s ( ) 1 1 os ( ) ( ) 1 ( ) sin ( ) Exmple 3 on p.11 tht eqution y sin Fundmentl Identities ( ) : ( ) tn() sin()/ os() ot() os()/ sin() s() 1/sin() se() 1/ os() ot() os()/ sin() sin () + os () 1 tn () + 1 se () 1 + ot () s ()
12 Finding the ext vlue of the trig funtions, given one. Exmple 4 Given tht sin() 1/3 nd is n ute ngle, find the ext vlue of eh of the remining five trigonometri funtions of. Rememer sin() 1/3 opposite/hypotenuse so let s mke right tringle with the opposite of to e 1 nd the hypotenuse to e 3. We n use the Pythgoren Theorem to find the djent side,. 3? Now tht you know ll three sides of the tringle, just use the definitions to find the ext vlues of the trig funtions. Trig Funtion sin() os() tn() s() se() ot() Definition /opposite/hypotenuse /djent/hypotenuse /opposite/djent /hypotenuse/opposite /hypotenuse/djent /djent/opposite Ext Vlue /1/3 / 3 / 1 /3/1 3 / 3 /
13 Complementry Angles; Cofuntions α 90 β Hypotenuse β α Opposite to β Adjent to α α nd β re omplementry ngle euse their sum is right ngle, 90. Adjent to β Opposite to α Complementry Angle Theorem Cofuntions of omplementry ngles re equl. Cofuntions re trigonometri funtions tht shre the sme ngles of right tringle. sin(β)opposite to β / hypotenuse djent to α/hypotenuse os(α)os(90 β) os(β)djent to β / hypotenuse opposite to α/hypotenuse sin(α)sin(90 β) tn(β)opposite to β / djent to α djent to α/ opposite to β ot(α) ot(90 β) Likewise, the reiprol of these properties is true. s(β) hypotenuse / opposite to β / hypotenuse /djent to α se(α)se(90 β) se(β) hypotenuse /djent to β hypotenuse/opposite to α s(α) s(90 β) ot(β) djent to α /opposite to β opposite to β / djent to α tn(90 β)
14 HOMEWORK Homework p #1, 3, 7, 39, 41, 47, 1, 3, 7, 69,7,81,8,91 p. 11 #3, 13,17, 3,37, 47, 3, 7
Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative 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 AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
More informationTopics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving rightangle triangles Sine and cosine rule
Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving rightngle tringles Sine nd osine rule Lelling rightngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite
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 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 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 informationNaming the sides of a rightangled 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 informationStandard Trigonometric Functions
CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent
More informationm A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NONCALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in northesterl diretion. Simplif:
More informationObjective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15
Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
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 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 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 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 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 informationChapter 4 Trigonometric Functions
Chter Trigonometric Functions Chter Trigonometric Functions Section. Angles nd Their Mesures Exlortion. r. rdins ( lengths of ted). No, not quite, since the distnce r would require iece of ted times s
More informationMath 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 singlevrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
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 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 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 informationReflection 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 informationMAT 1275: Introduction to Mathematical Analysis
MAT 75: Intrdutin t Mthemtil Anlysis Dr. A. Rzenlyum Trignmetri Funtins fr Aute Angles Definitin f six trignmetri funtins Cnsider the fllwing girffe prlem: A girffe s shdw is 8 meters. Hw tll is the girffe
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 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 informationH (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 informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
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 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 information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationUse of Trigonometric Functions
Unit 03 Use of Trigonometric Functions 1. Introduction Lerning Ojectives of tis UNIT 1. Lern ow te trigonometric functions re relted to te rtios of sides of rigt ngle tringle. 2. Be le to determine te
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 2942 9/11/04 Quntum Ciruit Model, SolovyKitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits  Universl Gte Sets A lssil iruit implements multioutput oolen funtion f : {0,1}
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 informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
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 informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
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 informationf (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 informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationSAMPLE. Trigonometry. Naming the sides of a rightangled triangle
H P T E R 7 Trigonometry How re sin, os nd tn defined using rightngled tringle? How n the trigonometri rtios e used to find the side lengths or ngles in rightngled tringles? Wht is ment y n ngle of elevtion
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
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 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 informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
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 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 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 informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( 5, 9 ) ) (,
More informationPhys 7221, Fall 2006: Homework # 6
Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 37 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which
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 informationApplications of trigonometry
3 3 3 3 3D 3E 3F 3G 3H Review of rightngled tringles erings Using the sine rule to find side lengths Using the sine rule to find ngles re of tringle Using the osine rule to find side lengths Using the
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 information8.3 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 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 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 informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More 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 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 informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 0506 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
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 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 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 information[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
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 information6.5 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 informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
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 informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationDrill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise  1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltusrectum is 8 nd conjugte
More informationSOLUTION OF TRIANGLES
SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, 940, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :
More informationSolving Radical Equations
Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is 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 informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More informationLecture 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 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 informationMore 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 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 informationMATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].
More informationSimilarity and Congruence
Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles
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 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 informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. OnetoOne Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More information20. Direct and Retrograde Motion
0. Direct nd Retrogrde Motion When the ecliptic longitude λ of n object increses with time, its pprent motion is sid to be direct. When λ decreses with time, its pprent motion is sid to be retrogrde. ince
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 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 informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
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 informationQUADRATIC EQUATION EXERCISE  01 CHECK YOUR GRASP
QUADRATIC EQUATION EXERCISE  0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions
More informationAnswers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017
Answers: (0 HKMO Het Events) reted y: Mr. Frncis Hung Lst updted: 5 Decemer 07  Individul  Group Individul Events 6 80 0 4 5 5 0 6 4 7 8 5 9 9 0 9 609 4 808 5 0 6 6 7 6 8 0 9 67 0 0 I Simplify 94 0.
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
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 information