# Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.

Size: px
Start display at page:

Download "Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear."

Transcription

1 Section.8 Grphs of Polr Equtions Points:,,,,.,... The points re colliner. 9. Points:.,,.,,.,... not colliner. Section.8 Grphs of Polr Equtions When grphing polr equtions:. Test for symmetry. () ) or (r, ). Polr xis: Rlce (r, ) by (r, ) or (r, ). Pole: Rlce (r, ) by (r, ) or (r, ). (d) r f (sin ) is symmetric with respect to the line (e) r f (cos ) is symmetric with respect to the polr xis.. Find the vlues for which r is mximum.. Find the vlues for which r.. Know the different types of polr grphs. () Limçons <, < b Rose curves, n Circles (d) Lemnisctes r ± b cos r cos n r cos r cos r ± b sin r sin n r sin r sin : Rlce (r, ) by (r,. Plot dditionl points.. r Vocbulry Check.. polr xis. convex limçon. circle. lemniscte. crdioid. r cos Rose curve with petls. r sin Crdioid. r cos Limçon with inner loop. r cos Lemniscte. r sin Rose curve with petls. r cos Circle 7. r cos Polr xis: Pole: : Answer: r cos() r cos Not n equivlent eqution r cos() r cos Equivlent eqution r cos Not n equivlent eqution polr xis 8. r cos r cos : r cos r cos Not n equivlent eqution Polr r cos xis: r cos r cos Equivlent eqution Pole: r cos Not n equivlent eqution Answer: polr xis

2 98 Chpter Topics in Anlytic Geometry 9. r sin r : sin( ) r sin cos cos sin r sin Equivlent eqution Polr xis: r sin() r sin Not n equivlent eqution Pole: r sin Answer:. r cos Polr xis: Pole: r : Not n equivlent eqution r cos cos Equivlent eqution r cos Not n equivlent eqution Answer: polr xis. r cos r cos : r cos Equivlent eqution Polr xis: r cos Pole: r cos Equivlent eqution r cos r cos Equivlent eqution Answer: the, polr xis, nd the pole. r sin Polr xis: Pole: r sin : Not n equivlent eqution r sin Not n equivlent eqution r sin Equivlent eqution Answer: pole. r ( sin ) sin sin sin sin Mximum: r () or sin sin Not possible. r cos cos cos 8 cos Mximum: r 8 cos ( sin ) cos sin Zero: r, Zero: r,

3 Section.8 Grphs of Polr Equtions 987. cos r cos cos cos ±,,. r sin sin sin sin ±,,, 7 Mximum: r,, Mximum: r,,, 7 cos sin cos,, Zero: r,, sin,,, Zero: r,,, 7. Circle: r 8. Circle: r 9. Circle: r. r. r sin Circle with rdius of. r cos polr xis Circle with rdius. r cos the polr xis b Crdioid r r. r sin b r 8 Crdioid r

4 988 Chpter Topics in Anlytic Geometry. r sin 7.. r cos b Crdioid r 8 r polr xis b Crdioid r r r sin Limçon with b < inner loop r 9 r 7, 8 8. r sin, b b r 7 Dimpled limçon 9. r sin Limçon with b < inner loop r r,. r cos the polr xis b r r Limçon with inner loop,. r cos the polr xis Limçon with inner loop b < r 7 r cos or.7,.

5 Section.8 Grphs of Polr Equtions 989. r cos the polr xis b > Dimpled limçon r 7. r sin nd the pole Rose curve n with petls r r,,,,,, 7 the polr xis, 8. r cos the polr xis Rose curve n with four petls r,,, r,,, 7. r sec r cos r cos x Line. r csc r sin y Line 7. r r(sin cos ) y x sin cos y x Line 8. r r sin cos y x sin cos y x Line

6 99 Chpter Topics in Anlytic Geometry 9. r 9 cos the polr xis, nd the pole Lemniscte,. r sin r sin r sin. r 8 cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r ( sin ). min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl = r cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r 8 sin cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r csc sin min = mx = st = Xmin = -9 Xmx = 9 Xscl = Ymin = - Ymx = 8 Yscl = 7. r cos < 8. r cos < 7 9. r cos < 7 7

7 Section.8 Grphs of Polr Equtions 99. r sin. < r 9 sin <. r < <..... r sec cos r cos cos r(r cos ) r cos r ±x y x x ±x y ±x y (x ) x ±x y x x x y x (x ) y x x (x ) x x (x ) x x x x (x ) (x ) x x x x x x (x ) (x ) y ± x x x (x ) ± x x x x The grph hs n symptote t x.. ±x y y y r csc sin r sin sin rr sin r sin r ±x y y y ±x y ±x y x y y y y y x y y y y x ± y y y y The grph hs n symptote t y. ± y y y y

8 99 Chpter Topics in Anlytic Geometry. r r sin sin r sin y. r cos sec r cos sin cos cos cos y sin As, y r cos cos sin x cos sin As, x. 7. True. For grph to hve polr xis symmetry, rlce r, by r, or r,. 8. Flse. For grph to be symmetric bout the pole, one portion of the grph coincides with the other portion rotted rdins bout the pole. 9. r cos () Upper hlf of circle 7 Lower hlf of circle 7 (d) Entire circle Left hlf of circle 7 7. r cos () The ngle hs the effect of rotting the grph by the ngle. For prt, r cos sin.. Let the curve r f() be rotted by to form the curve r g(). If r, is point on r f(), then r, is on r g(). Tht is, g r f. Letting, or, we see tht g g f f. (, r θ + φ) φ (, r θ) θ

9 Section.8 Grphs of Polr Equtions 99. Use the result of Exercise. () Rottion: Originl grph: r fsin Rotted grph: r f sin Rottion: Originl grph: r fsin Rotted grph: r f sin fsin fcos Rottion: Originl grph: r fsin Rotted grph: r f sin fcos. () r sin sin cos cos sin (sin cos ) r sin( ) sin cos cos sin sin (d) r sin sin cos cos sin cos r sin sin cos cos sin cos. r sin () r sin sin cos sin cos r sin sin cos sin cos (d) r sin sin sin sin cos sin cos r sin sin sin sin cos sin cos. () r sin r sin Rotte the grph in prt () through the ngle.

10 99 Chpter Topics in Anlytic Geometry. () r sec r cos r cos x r cos r sin r sec r r cos cos cos sin sin x y (d) r cos r sin r sec r r x y r r cos r sec cos cos sin sin cos cos sin sin cos r sin y 7. r k sin 7 8. r sin k k : k : r Circle r sin Convex limçon 7 k = k = k = k = 8 () r sin. < r sin. < k : r sin k : Crdioid r sin Limçon with inner loop Yes. r sink. Find the minimum vlue of, >, tht is multiple of tht mkes multiple of. k

11 Section.9 Polr Equtions of Conics y x 9 x x 9 x x 9 x 9 x ± 7. y x No zeros 7. x x x x y x 7. y x 7 x Zero: x x x 7. Vertices:,,, Center t, nd 7. Foci:,,, ; Mjor xis of length 8 Minor xis of length : Horizontl mjor xis b b y Center: h, k, Verticl mjor xis x h x 9 y k b y x, c, b c 9 b 7 x h b x 7 y k y y 7 x Section.9 Polr Equtions of Conics The grph of polr eqution of the form r is conic, where e > is the eccentricity nd is the distnce between the focus (pole) nd the directrix. () If e <, the grph is n ellipse. If e, the grph is prbol. If e >, the grph is hyperbol. Guidelines for finding polr equtions of conics: () (d) or r ± e cos ± e sin Horizontl directrix bove the pole: r e sin Horizontl directrix below the pole: r e sin Verticl directrix to the right of the pole: r e cos p Verticl directrix to the left of the pole: r e cos

### Section 10.2 Graphing Polar Equations

Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The

### 10.4 AREAS AND LENGTHS IN POLAR COORDINATES

65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the

### 9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show

### Geometric quantities for polar curves

Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions

### Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a

Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.

### Polar Coordinates. July 30, 2014

Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes

### Translate and Classify Conic Sections

TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke

### Triangles and parallelograms of equal area in an ellipse

1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper

### (3,4) focus. y=1 directrix

Math 153 10.5: Conic Sections Parabolas, Ellipses, Hyperbolas Parabolas: Definition: A parabola is the set of all points in a plane such that its distance from a fixed point F (called the focus) is equal

### INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS

CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite

### c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem

### Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You

### Study Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms

Study Guide # 1 MA 26100 - Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,

### 10.1 Curves defined by parametric equations

Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates

### 2. Polar coordinates:

Section 9. Polar Coordinates Section 9. Polar Coordinates In polar coordinates ou do not have unique representation of points. The point r, can be represented b r, ± n or b r, ± n where n is an integer.

### b = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )

***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector

### Outline. Drawing the Graph. 1 Homework Review. 2 Introduction. 3 Histograms. 4 Histograms on the TI Assignment

Lecture 14 Section 4.4.4 on Hampden-Sydney College Fri, Sep 18, 2009 Outline 1 on 2 3 4 on 5 6 Even-numbered on Exercise 4.25, p. 249. The following is a list of homework scores for two students: Student

### (b) ( 1, s3 ) and Figure 18 shows the resulting curve. Notice that this rose has 16 loops.

SECTIN. PLAR CRDINATES 67 _ and so we require that 6n5 be an even multiple of. This will first occur when n 5. Therefore we will graph the entire curve if we specify that. Switching from to t, we have

### On the. Geometry. of Orbits

On the Geometry of Orbits The Possible Orbits The Possible Orbits circle The Possible Orbits ellipse The Possible Orbits parabola The Possible Orbits hyperbola Speed and Distance 4000 mi 17,600 mph 1.4

### FP2 POLAR COORDINATES: PAST QUESTIONS

FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)

### RECTANGULAR EQUATIONS OF CONICS. A quick overview of the 4 conic sections in rectangular coordinates is presented below.

RECTANGULAR EQUATIONS OF CONICS A quick overview of the 4 conic sections in rectangular coordinates is presented below. 1. Circles Skipped covered in MAT 124 (Precalculus I). 2. s Definition A parabola

### You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas.

You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas. conic section degenerate conic locus parabola focus directrix

### REVIEW, pages

REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:

### CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... 8 Section. Parametric Equations and

### CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... Section. Parametric Equations and Calculus............

### Objectives. Materials

. Objectives Activity 8 To plot a mathematical relationship that defines a spiral To use technology to create a spiral similar to that found in a snail To use technology to plot a set of ordered pairs

### Algebra II B Review 3

Algebra II B Review 3 Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the equation. Describe the graph and its lines of symmetry. 1. a. c. b. graph

### Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor

Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor As we studied last section points may be described in polar form or rectangular form. Likewise an equation may be written using either

### 10.3 Polar Coordinates

.3 Polar Coordinates Plot the points whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > and one with r

### The GC s standard graphing window shows the x-axis from -10 to 10 and the y-axis from -10 to 10.

Name Date TI-84+ GC 17 Changing the Window Objectives: Adjust Xmax, Xmin, Ymax, and/or Ymin in Window menu Understand and adjust Xscl and/or Yscl in Window menu The GC s standard graphing window shows

### The Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2

Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Focus 1 Focus 2 Major Axis Point PF

### Pre-Calculus Notes: Chapter 6 Graphs of Trigonometric Functions

Name: Pre-Calculus Notes: Chapter Graphs of Trigonometric Functions Section 1 Angles and Radian Measure Angles can be measured in both degrees and radians. Radian measure is based on the circumference

### Do You See What I See?

Concept Geometry and measurement Activity 5 Skill Calculator skills: coordinate graphing, creating lists, ' Do You See What I See? Students will discover how pictures formed by graphing ordered pairs can

### MAT01B1: Calculus with Polar coordinates

MAT01B1: Calculus with Polar coordinates Dr Craig 23 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h30 12h55 Friday (this week) 11h20 12h25 Office C-Ring 508

### Section 16.3 Double Integrals over General Regions

Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is

### Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is

### STUDY GUIDE, CALCULUS III, 2017 SPRING

TUY GUIE, ALULU III, 2017 PING ontents hpter 13. Functions of severl vribles 1 13.1. Plnes nd surfces 2 13.2. Grphs nd level curves 2 13.3. Limit of function of two vribles 2 13.4. Prtil derivtives 2 13.5.

### Example. Check that the Jacobian of the transformation to spherical coordinates is

lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.

### C.2 Equations and Graphs of Conic Sections

0 section C C. Equations and Graphs of Conic Sections In this section, we give an overview of the main properties of the curves called conic sections. Geometrically, these curves can be defined as intersections

### Hyperbolas Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas p. 583

C H A P T ER Hyperbolas Flashlights concentrate beams of light by bouncing the rays from a light source off a reflector. The cross-section of a reflector can be described as hyperbola with the light source

### Chapter 3, Part 4: Intro to the Trigonometric Functions

Haberman MTH Section I: The Trigonometric Functions Chapter, Part : Intro to the Trigonometric Functions Recall that the sine and cosine function represent the coordinates of points in the circumference

### SLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises

SLOVAK UNIVERSITY OF TECHNOLOGY Fulty of Mteril Siene nd Tehnology in Trnv ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory exerises Róbert Riedlmjer TRNAVA 00 ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory

### Pre-Calc Conics

Slide 1 / 160 Slide 2 / 160 Pre-Calc Conics 2015-03-24 www.njctl.org Slide 3 / 160 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections

### Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas

Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 160 Review of Midpoint and Distance Formulas Intro to Conic Sections

### Functions Modeling Change A Preparation for Calculus Third Edition

Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 CHAPTER

### MEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR

Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure

6.1.: Graphing Quadratic Equations 1. Obtain a pair of equations from your teacher.. Press the Zoom button and press 6 (for ZStandard) to set the window to make the max and min on both axes go from 10

### POLAR FUNCTIONS. In Precalculus students should have learned to:.

POLAR FUNCTIONS From the AP Calculus BC Course Description, students in Calculus BC are required to know: The analsis of planar curves, including those given in polar form Derivatives of polar functions

### Pre-Calc. Midpoint and Distance Formula. Slide 1 / 160 Slide 2 / 160. Slide 4 / 160. Slide 3 / 160. Slide 5 / 160. Slide 6 / 160.

Slide 1 / 160 Slide 2 / 160 Pre-alc onics 2015-03-24 www.njctl.org Slide 3 / 160 Slide 4 / 160 Table of ontents click on the topic to go to that section Review of Midpoint and istance Formulas Intro to

### UNIT I PLANE CURVES AND FREE HAND SKETCHING CONIC SECTIONS

UNIT I PLANE CURVES AND FREE HAND SKETCHING CONIC SECTIONS Definition: The sections obtained by the intersection of a right circular cone by a cutting plane in different positions are called conic sections

### Now we are going to introduce a new horizontal axis that we will call y, so that we have a 3-dimensional coordinate system (x, y, z).

Example 1. A circular cone At the right is the graph of the function z = g(x) = 16 x (0 x ) Put a scale on the axes. Calculate g(2) and illustrate this on the diagram: g(2) = 8 Now we are going to introduce

### Chapter 4: The Ellipse

Chapter 4: The Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza Chapter 4: The Ellipse Lecture 1: Introduction to Ellipse Lecture 13: Converting

### Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Idel Opertionl Amplifiers 9/11/06 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

### Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Idel Opertionl Amplifiers Fll 2009 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

### C H A P T E R 4 Trigonometric Functions

C H A P T E R Trigonometric Functions Section. Radian and Degree Measure................ 7 Section. Trigonometric Functions: The Unit Circle........ 8 Section. Right Triangle Trigonometr................

### MATH 255 Applied Honors Calculus III Winter Homework 1. Table 1: 11.1:8 t x y

MATH 255 Applied Honors Calculus III Winter 2 Homework Section., pg. 692: 8, 24, 43. Section.2, pg. 72:, 2 (no graph required), 32, 4. Section.3, pg. 73: 4, 2, 54, 8. Section.4, pg. 79: 6, 35, 46. Solutions.:

### Precalculus Second Semester Final Review

Precalculus Second Semester Final Review This packet will prepare you for your second semester final exam. You will find a formula sheet on the back page; these are the same formulas you will receive for

### Pre Calc. Conics.

1 Pre Calc Conics 2015 03 24 www.njctl.org 2 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections Parabolas Circles Ellipses Hyperbolas

### Chapter 5 Analytic Trigonometry

Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos > sec, sec > cos, nd tn sin > cos. sin > csc nd tn > cot 3. csc > sin, cot > tn, nd cot cos

### Math 1432 DAY 37 Dr. Melahat Almus If you me, please mention the course (1432) in the subject line.

Math 1432 DAY 37 Dr. Melahat Almus almus@math.uh.edu If you email me, please mention the course (1432) in the subject line. Bubble in PS ID and Popper Number very carefully. If you make a bubbling mistake,

RAKESH JALLA B.Tech. (ME), M.Tech. (CAD/CAM) Assistant Professor, Department Of Mechanical Engineering, CMR Institute of Technology. CONICS Curves Definition: It is defined as the locus of point P moving

### 6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.

Math 160 www.timetodare.com Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such

### Polar Conics TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System

Math Objectives Students will understand that the equations for conics can be expressed in polar form. Students will be able to describe the relationship between eccentricity and the type of conic section.

### The Cartesian Coordinate System

The Cartesian Coordinate System The xy-plane Although a familiarity with the xy-plane, or Cartesian coordinate system, is expected, this worksheet will provide a brief review. The Cartesian coordinate

### TIME: 1 hour 30 minutes

UNIVERSITY OF AKRON DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING 4400: 34 INTRODUCTION TO COMMUNICATION SYSTEMS - Spring 07 SAMPLE FINAL EXAM TIME: hour 30 minutes INSTRUCTIONS: () Write your nme

### Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.

Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line

### 13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.

CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny

### Chapter 5 Analytic Trigonometry

Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos / sec, sec / cos, nd tn sin / cos. sin / csc nd tn / cot 3. csc / sin, cot / tn, nd cot cos

### Engineering Graphics, Class 5 Geometric Construction. Mohammad I. Kilani. Mechanical Engineering Department University of Jordan

Engineering Graphics, Class 5 Geometric Construction Mohammad I. Kilani Mechanical Engineering Department University of Jordan Conic Sections A cone is generated by a straight line moving in contact with

### DESIGN OF CONTINUOUS LAG COMPENSATORS

DESIGN OF CONTINUOUS LAG COMPENSATORS J. Pulusová, L. Körösi, M. Dúbrvská Institute of Robotics nd Cybernetics, Slovk University of Technology, Fculty of Electricl Engineering nd Informtion Technology

### Module 9. DC Machines. Version 2 EE IIT, Kharagpur

Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson-40) 4 40.1 Gols

### Notes on Spherical Triangles

Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The

### Math 1330 Section 8.2 Ellipses

Math 1330 Section 8.2 Ellipses To form a conic section, we ll take this double cone and slice it with a plane. When we do this, we ll get one of several different results. 1 Part 1 - The Circle Definition:

### 4-3 Trigonometric Functions on the Unit Circle

Find the exact values of the five remaining trigonometric functions of θ. 33. tan θ = 2, where sin θ > 0 and cos θ > 0 To find the other function values, you must find the coordinates of a point on the

### ALGEBRA 2 ~ Lessons 1 13

ALGEBRA 2 ~ Lessons 1 13 Remember to write the original problem and show all of your steps! All work should be done on a separate piece of paper. ASSIGNMENT 1 Arithmetic (No calculator.) Add, subtract

### Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems

hpter 12 Vectors nd the Geometry of Spce 12.1 Three-dimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3

### Synchronous Machine Parameter Measurement

Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions

### Fungus Farmers LEAF CUTTING ANTS A C T I V I T Y. Activity Overview. How much leaf do leaf cutter ants chew?

How much leaf do leaf cutter ants chew? Activity Overview Leaf cutting ants carry away leaf pieces that are up to 30 times their weight. They sometimes carry these pieces 100-200 meters (about 2 football

### Three-Phase Synchronous Machines The synchronous machine can be used to operate as: 1. Synchronous motors 2. Synchronous generators (Alternator)

Three-Phse Synchronous Mchines The synchronous mchine cn be used to operte s: 1. Synchronous motors 2. Synchronous genertors (Alterntor) Synchronous genertor is lso referred to s lterntor since it genertes

### TUTORIAL 9 OPEN AND CLOSED LOOP LINKS. On completion of this tutorial, you should be able to do the following.

TUTORIAL 9 OPEN AND CLOSED LOOP LINKS This tutorial is of interest to any student studying control systems and in particular the EC module D7 Control System Engineering. On completion of this tutorial,

### Job Sheet 2. Variable Speed Drive Operation OBJECTIVE PROCEDURE. To install and operate a Variable Speed Drive.

Job Sheet 2 Vrible Speed Drive Opertion OBJECTIVE To instll nd operte Vrible Speed Drive. PROCEDURE Before proceeding with this job, complete the sfety check list in Appendix B. 1. On the Vrible Speed

### Regular languages can be expressed as regular expressions.

Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected

### Unit 6 Task 2: The Focus is the Foci: ELLIPSES

Unit 6 Task 2: The Focus is the Foci: ELLIPSES Name: Date: Period: Ellipses and their Foci The first type of quadratic relation we want to discuss is an ellipse. In terms of its conic definition, you can

### r = (a cos θ, b sin θ). (1.1)

Peeter Joot peeter.joot@gmail.com Circumference of an ellipse 1.1 Motivation Lance told me they ve been covering the circumference of a circle in school this week. This made me think of the generalization

### Pre-Calculus Notes: Chapter 6 Graphs of Trigonometric Functions

Name: Pre-Calculus Ntes: Chapter Graphs f Trignmetric Functins Sectin 1 Angles and Radian Measure Angles can be measured in bth degrees and radians. Radian measure is based n the circumference f a unit

### Chapter 9. Conic Sections and Analytic Geometry. 9.1 The Ellipse. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Graph ellipses centered at the origin. Write equations of ellipses in standard

### Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Figure 50.1

50 Polar Coordinates Arkansas Tech University MATH 94: Calculus II Dr. Marcel B. Finan Up to this point we have dealt exclusively with the Cartesian coordinate system. However, as we will see, this is

### Algebra Practice. Dr. Barbara Sandall, Ed.D., and Travis Olson, M.S.

By Dr. Brr Sndll, Ed.D., Dr. Melfried Olson, Ed.D., nd Trvis Olson, M.S. COPYRIGHT 2006 Mrk Twin Medi, Inc. ISBN 978-1-58037-754-6 Printing No. 404042-EB Mrk Twin Medi, Inc., Pulishers Distriuted y Crson-Dellos

### CONIC SECTIONS 1. Inscribe a parabola in the given rectangle, with its axis parallel to the side AB

Inscribe a parabola in the given rectangle, with its parallel to the side AB A D 1 1 2 2 3 3 B 3 2 1 1 2 3 C Inscribe a parabola in the rectangle below, with its vertex located midway along the side PQ.

### 7KH4XLQFXQ; Earth/matriX SCIENCE IN ANCIENT ARTWORK. Charles William Johnson

Erth/mtriX SCIENCE IN ANCIENT ARTWORK 7KH4XLQFXQ; Chrles Willim Johnson Erth/mtriX P.O. Box 231126, New Orlens, Louisin, 70183-1126 2001 Copyrighted y Chrles Willim Johnson www.erthmtrix.om www.the-periodi-tle.om

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd

### Vector Calculus. 1 Line Integrals

Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q

### CHAPTER 2 LITERATURE STUDY

CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:

### Synchronous Machine Parameter Measurement

Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions

### Experiment 3: The research of Thevenin theorem

Experiment 3: The reserch of Thevenin theorem 1. Purpose ) Vlidte Thevenin theorem; ) Mster the methods to mesure the equivlent prmeters of liner twoterminl ctive. c) Study the conditions of the mximum

### M.V.S.R. ENGINEERING COLLEGE, NADERGUL HYDERABAD B.E. I/IV I - Internal Examinations (November 2014)

Sub: Engineering Graphics Branches: Civil (1&2), IT-2 Time: 1 Hr 15 Mins Max. Marks: 40 Note: Answer All questions from Part-A and any Two from Part B. Assume any missing data suitably. 1. Mention any

### Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ.

1 Radian Measures Exercise 1 Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1. Suppose I know the radian measure of the