I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

 Brice Carter
 10 months ago
 Views:
Transcription
1 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel world sitution. I. Equtions of Circle. At the origin center= r= b. Stndrd from: center= r= c. Generl Eqution d. Note: The leding coefficients MUST be the sme to be circle. II. Key Concepts:. To grph, locte the center, then move r units right, left, up nd down from the center. b. Midpoint =(, ). You need this if you re given the endpoints of the dimeter. c. Distnce = You need this to find the rdius given the center nd point on the circle. d. Tngent Line Touches the circle in ONE point (point of tngency) nd is perpendiculr to the dimeter. Perpendiculr lines hve slopes tht re negtive reciprocls. III. Equtions of n Ellipse (the set of ll points P in plne such tht the SUM of the distnce between P nd two fixed points, clled foci, is constnt) A. Stndrd form: ( x h ) ( y k ) + = or ( x h ) ( y k ) b 1 b + = 1 1. Center point =. > b (they look like footbll nd in footbll, size mtters) 3. Vertices units from the center point long the mjor xis (in this cse lso the Mjor xis the one with the foci) 4. Covertices  units from the center point long the minor xis (smller in length Minor xis) 5. Foci (plurl for focus) units from the center point long the trnsverse xis. c = b The sign for the foci is lwys opposite the sign between the nd b. 6. If is under the x then it is footbll on the ground (horizontl) but if it is under the y, then it is footbll on the tee (verticl) c 7. Eccentricity The mount of roundness. e = Circles e = 0, Ellipses 0 < e < 1. B. Generl Form: x + cy + dx + ey + f = 0. There is not b (it is used for bxy term). In this form, you must use completing the squre to convert to stndrd.
2 II. How to grph A. Put in stndrd form B. Determine the center point (, ) C. From the center point, move units long the xis. These re your vertices ( h ±, k ) or ( h, k ± ) D. From the center point, move units long the xis. There re your covertices or E. From the center point, move units long the xis. These re your ( h c, k ) or ( h, k c) ± ±. Use. F. From the focl points move LR to the LR endpoints LR = III. Model Problems Guided Prctice 3x + 3y 6x + 1y = 1 b On Your Own x + y + 6x + y = 4 7 9x + 4y 36x 4y + 36 = 0 4x + y 48x 14y = 0 Determine the conic section: 8x + 1y 4x + y = 6 Determine the conic section: 4x + 4y + 8x 1y =
3 Write the eqution of the circle in stndrd form: rdius=4 center t (,5) Write the eqution of the circle in stndrd form: center t (,5) contining (5,1) x + y = 4 x ( y ) + 3 = 8 ( x ) ( y ) = 4 Grph ( y ) x + =
4 ( x ) ( y + 3) + = ( x + 4) 16 Grph ( y ) = 1 11.: Hyperbol I cn Grph hyperbol nd determine ll of the criticl informtion. Determine the eqution of hyperbol given some chrcteristics. I. Hyperbol. The set of points in plne whose difference of the distnce from two fixed point (foci) is positive constnt. b. The is lwys first (not necessrily bigger) nd is on the trnsverse xis. c. Formuls i. Trnsverse xis is horizontl: ii. Trnsverse xis is verticl iii. Asymptotes of hyperbol: Note: underneth the yterm could be or b but whichever it is, it is the top prt of the slope s frction. iv. c = Notice the chnge s compred to the ellipse v. Generl Form:
5 II. How to grph A. Put in stndrd form (must equl 1) B. Determine the center point (, ). Mke sure you put h first! C. Move units long the trnsverse xis. These re the D. Move units long the xis. E. Drw box round the endpoints of nd b, then connect the digonls. These re the. F. Stying between the symptotes, connect bck to only. F. Find the LR endpoints III. Model Problems Guided Prctice b LR = On Your Own 9x y 36x 6y + 18 = 0 4x 3y + 8x + 16 = 0 Find the stndrd form of the hyperbol with foci (1,) & (5,) nd vertices (0,) & (4,). Find the stndrd form of the hyperbol with foci (,6) & (,6) nd vertices (,3) & (,3). Find the stndrd form of the hyperbol with vertices (1, ) nd (3, ) nd symptotes of y=x nd y=x+4 Find the stndrd form of the hyperbol with vertices (3,0) nd (3,6) nd symptotes of y=x nd y=x+6
6 ( y ) ( x ) = Grph 9( y + 6) 4( x + ) = 36 ( x ) ( y ) + 1 = Grph ( y ) ( x 1) + = 1 4 1
7 11.: Prbols I cn Identify nd lbel ll prts of prbol. Convert ll generl form equtions to stndrd form I. Definitions. Prbol: Grph of qudrtic eqution to which set of points in plne tht re the sme distnce from given point. b. Vertex: Midpoint of the grph; turning point c. Focus: distnce from the vertex; locted inside the prbol d. Directrix: fixed line used to define its shpe; locted outside the prbol e. Axis of Symmetry: A line tht divides plne figure or grph into congruent reflected hlves. f. Ltus Rectum: A line segment through the foci of the shpe g. Eccentricity: rtio to describe the shpe of the conic, e=1 II. Horizontl Stndrd From:. Horizontl Stndrd From Eqution: b. Focus Point: c. Directrix: d. Axis of Symmetry: e. L.R. endpoints: f. Note: If p is + it opens right nd if p is it opens left III. Verticl Stndrd From:. Verticl Stndrd From Eqution: b. Focus Point: c. Directrix: d. Axis of Symmetry: e. L.R. endpoints: f. Note: If p is + it opens up nd if p is it opens down IV. All stndrd form equtions. Center: b. Length of L.R.: c. Eccentricity:
8 III. Model Problems Guided Prctice y x y = 0 On Your Own x x y = 0 y x y = 0 Find the stndrd form of the prbol where the xis of symmetry is t y=1, directrix is t x = nd the focus (4,1) Write in stndrd form the eqution of prbol with the vertex t the origin nd the focus is t (,0) Write in stndrd form the eqution of prbol with the directrix y=6 nd focus point (0,6). Write the Stndrd form eqution of prbol where the vertex is (7,3) nd focus point (,3). Find the stndrd form of the prbol with vertex (,1) nd the directrix is t x=1
9 y = 4x x = (1/ 0) y Vertex: P: Focus point: Directrix: AOS: L.R. : Vertex: P: Focus point: Directrix: AOS: L.R. : ( y + 1) = 8( x + 1) ( x 7) = 8( y ) Vertex: P: Focus point: Directrix: AOS: L.R. : Vertex: P: Focus point: Directrix: AOS: L.R. :
MATH 115: Review for Chapter 7
MATH 5: Review for Chpter 7 Cn you stte the generl form equtions for the circle, prbol, ellipse, nd hyperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol
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 informationMATH 115: Review for Chapter 7
MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationConic Sections. Geometry  Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, ), find M 2. A(5, 7) and B( 2, ), find M 3. A( 2,0)
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 informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationCircles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
More information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of crosssection or slice. In this section, we restrict
More informationHYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..
HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of
More informationA quick overview of the four conic sections in rectangular coordinates is presented below.
MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is
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 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 informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
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 informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
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 informationRational Parents (pp. 1 of 4)
Rtionl Prents (pp of 4) Unit: 08 Lesson: 0 The grphs below describe two prent functions, ech of which is referred to s rtionl function Why do you think they re clled rtionl functions? From the grphs, provide
More informationParabola Exercise 1 2,6 Q.1 (A) S(0, 1) directric x + 2y = 0 PS = PM. x y x y 2y 1 x 2y Q.2 (D) y 2 = 18 x. 2 = 3t. 2 t 3 Q.
Prbol Exercise Q. (A) S(0, ) directric x + y = 0 PS = PM x y x y 5 5 x y y x y Q. (D) y = 8 x (t, t) t t = t t 8 4 8 t,t, 4 9 4,6 Q. (C) y 4 x 5 Eqution of directrix is x + = 0 x 0 5 Q.4 y = 8x M P t,t
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
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 informationragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This printout should hve 22 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
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 informationHonors Precalculus Chapter 8 Summary Conic Sections Parabola
Honors Precalculus Chapter 8 Summary Conic Sections Parabola Definition: Focal length: y axis P(x, y) Focal chord: focus Vertex xaxis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:
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 informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More 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 informationBRIEF NOTES ADDITIONAL MATHEMATICS FORM
BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht
More informationPhysics 9 Fall 2011 Homework 2  Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework  s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
More informationFUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x x1)(x x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = ( 1)( ) The hperbol: p q The eponentil function: b p q Importnt fetures: intercept : Let = 0 intercept : Let = 0 Turning points (Where pplicble)
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 informationAPPM 1360 Exam 2 Spring 2016
APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the
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 informationPrerequisite Knowledge Required from O Level Add Math. d n a = c and b = d
Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +,  Rtionlising the Denomintor
More informationPurpose of the experiment
Newton s Lws II PES 6 Advnced Physics Lb I Purpose of the experiment Exmine two cses using Newton s Lws. Sttic ( = 0) Dynmic ( 0) fyi fyi Did you know tht the longest recorded flight of chicken is thirteen
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice
More informationELLIPSE. Standard equation of an ellipse referred to its principal axes along the coordinate axes is. ( a,0) A'
JMthemtics LLIPS. STANDARD QUATION & DFINITION : Stndrd eqution of n ellipse referred to its principl es long the coordinte es is > & = ( e ) = e. Y + =. where where e = eccentricit (0 < e < ). FOCI
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 informationChapter 1  Functions and Variables
Business Clculus 1 Chpter 1  Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:
More informationMathematics of Motion II Projectiles
Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte
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 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 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 informationProblem Set 3
14.102 Problem Set 3 Due Tuesdy, October 18, in clss 1. Lecture Notes Exercise 208: Find R b log(t)dt,where0
More informationJune 2011 Further Pure Mathematics FP Mark Scheme
. June 0 Further Pure Mthemtics FP 6669 Mrk dy 6x dx = nd so surfce re = π x ( + (6 x ) dx B 4 = 4 π ( 6 x ) + 6 4 4π D 860.06 = 806 (to sf) 6 Use limits nd 0 to give [ ] B Both bits CAO but condone lck
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 informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
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 informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationDE51/DC51 ENGINEERING MATHEMATICS I DEC 2013
DE5/DC5 ENGINEERING MATHEMATICS I DEC π π Q.. Prove tht cos α + cos α + + cos α + L.H.S. π π cos α + cos α + + cos α + + α + cos α + cos ( α ) + cos ( ) cos α + cos ( 9 + α + ) + cos(8 + α + 6 ) cos α
More informationz TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability
TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSPG 6. Trnsform Bsics The definition of the trnsform for digitl signl is: n X x[ n is complex vrile The trnsform
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 information4402 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
4402 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More informationWhat Is Calculus? 42 CHAPTER 1 Limits and Their Properties
60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
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 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 informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the coordinte of ech criticl vlue of g. Show
More informationCHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS
CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSSSECTIONS. A() ;, ; (digonl) ˆ Èˆ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8
More informationSUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 2012
MOCK TEST 0 SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 0 ANSWERS. () π π Tke cos  ( ) then sin [ cos  ( )]sin [ ]/. () Since sin  + sin  y + sin  z π, ; y , z  50 + y 50 + z 50  + +
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 informationMiller indices and Family of the Planes
SOLID4 Miller Indices ltest Fmily of Plnes nd Miller indices; Miller indices nd Fmily of the Plnes The geometricl fetures of the crystls represented by lttice points re clled Rtionl. Thus lttice point
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 informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
More informationConic Sections. Animation. Animation. 694 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates
9 CHAPTER Conics, Prmetric Equtions, nd Polr Coordintes Section HYPATIA (7 5 AD) The Greeks discovered conic sections sometime between nd BC B the beginning of the Alendrin period, enough ws known bout
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 informationMASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS
MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MCUPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationCalculus  Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus  Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
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 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 informationThe Quadratic Formula
LESSON 7.4 EXAMPLE A Solution The Qudrtic Formul Although you cn lwys use grph of qudrtic function to pproximte the xintercepts, you re often not ble to find exct solutions. This lesson will develop procedure
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationENGI 2422 Fundamentals Parametric Curves Page 1.A.1. 1.A Examples for the Sketching of Parametric Curves
ENGI 4 Fundmentls Prmetric Curves Pge 1.A.1 1.A Exmples for the Sketching of Prmetric Curves 3 A curve in is onedimensionl object. To locte ny point on tht curve requires the vlue of just one prmeter
More informationPhysics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:
Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationParallel Projection Theorem (Midpoint Connector Theorem):
rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length onehlf the third side. onversely, If line isects
More informationEquations, and Polar Coordinates
Prmetric Equtions, nd Polr Conics, Coordintes During the Winter Olmpic Gmes, the Olmpic rings were lighted high on mountinside in Slt Lke Cit. The volunteers who instlled the lights for the displ took
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 informationFinal Exam Study Guide
Finl Exm Study Guide Includes. Integrls & Antiderivtive Rules 2. Definite Integrls (Integrls with bounds) 3. Are Between Two Curves  Region Bounded by Two Curves 4. Consumer nd Producer Surplus. USubstitution.
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 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 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 informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
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 informationMPE Review Section I: Algebra
MPE Review Section I: lger t Colordo Stte Universit, the College lger sequence etensivel uses the grphing fetures of the Tes Instruments TI8 or TI8 grphing clcultor. Whenever possile, the questions on
More informationSimulation of Eclipsing Binary Star Systems. Abstract
Simultion of Eclipsing Binry Str Systems Boris Yim 1, Kenny Chn 1, Rphel Hui 1 Wh Yn College Kowloon Diocesn Boys School Abstrct This report briefly introduces the informtion on eclipsing binry str systems.
More informationPHYSICS 211 MIDTERM I 21 April 2004
PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front 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 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 informationMathematics. toughest areas of the 2017 exam papers. Edexcel GCSE (91) Higher. guided exam support on the top 10 toughest
toughest res of the 07 em ppers Edecel GSE (9) Mthemtics Higher guided em support on the top 0 toughest res of the 07 Higher tier ppers from Top 0 Edecel GSE Mths Higher tier 07 Help our students ctch
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 information