Algebra II Notes Unit Ten: Conic Sections
|
|
- Catherine Ball
- 6 years ago
- Views:
Transcription
1 Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting the points M,. E: Find the midpoint of the line segment joining Let, 7,nd,, nd,, 7, nd,. is. Sustitute the vlues into the formul nd simplif. 7 9 M,,,3 Conic Sections: curves tht re formed the intersection of plne nd doulenpped cone Prol: the conic formed connecting ll points equidistnt from point (the focus) k 4 p h or nd line (the directri) with the eqution of the form h 4 p k Verte: the point, hk,, tht lies on the is of smmetr hlfw etween the focus nd directri Ais of Smmetr: the line of smmetr of prol tht psses through the verte. For k 4 p h, the is of smmetr is horizontl, nd hs the eqution h. For h 4 p k, the is of smmetr is verticl, nd hs the eqution k. p : the distnce the focus nd directri re from the verte Focus: point on the is of smmetr of prol equidistnt from the verte s the directri Directri: line perpendiculr to the is of smmetr equidistnt from the verte s the focus Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
2 Prol 4 p, p 0 Prol 4 p, p 0 Focus 0, p Directri p Verte 0,0 Directri p Verte 0,0 Focus 0, p Prol 4 p, p 0 Prol 4 p, p 0 Directri p Directri p Verte 0,0 Focus p,0 Focus p,0 Verte 0,0 E: Sketch the grph of the prol 3. Verte: 0,0 Becuse is squred, the is of smmetr is horizontl: 0 Find p: 3 4p 3 3 p 4 Becuse p is negtive, the prol will open to the left. Directri: 3 Focus: 4 3, Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
3 Stndrd Form of the Eqution of Prol: Horizontl Ais (opens left or right): k 4 p h Verte: hk, Ais of Smmetr: k Verticl Ais (opens up or down): h 4 p k Verte: hk, Ais of Smmetr: h Focus: h p, k Focus: hk, p E: Sketch the grph of the prol directri. Directri: h p Directri: k p. Identif the verte, focus, nd Becuse it is in the form h 4 p k verticl is., the prol hs 0 Find p: 4 p 8 p Verte: 7,3 p is positive, so the prol opens up Focus: p units up from the verte 7,3 7, - Directri: horizontl line p units down from the verte 3 E: Write n eqution of the prol whose verte is t 3, nd whose focus is t 4,. -0 Begin with sketch. The prol opens towrd the focus, so it opens right. Find p: The distnce from the focus to the verte is. The prol opens right, so p =. Becuse the prol hs horizontl is, we will use the eqution k 4 p h The verte 3, hk,, nd p = E: A store uses prolic mirror to see ll of the isles in the store. A cross section of the mirror is shown. Write n eqution for the cross section of the mirror nd identif the focus. Pge 3 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
4 Becuse the prol hs verticl is, we will use the eqution h 4 p k 0,0 hk,, so we now hve 4 p. The prol psses through the point 8,, p p p p Use this to find p.. The verte is The eqution is The focus is p units up from the verte: 0,8 You Tr: Grph the eqution Identif the verte, focus, nd directri. QOD: Prols cn e found mn plces in rel life. Find t lest three rel-life emples of prols. Wht is the significnce of the focus in these rel-life emples? Smple CCSD Common Em Prctice Question(s): Which eqution represents the grph elow? A. B. C. D. Pge 4 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
5 Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (CIRCLES) Review: The Distnce Formul The distnce d etween the points nd, is d,. Use the Pthgoren Theorem to show this on the coordinte plne forming right tringle with the points,,,, nd,. E: Find the distnce etween the points, 4 Let,, nd,, 4 nd,.. Sustitute the vlues into the formul nd simplif. d Circle: the conic formed connecting ll points equidistnt from point with the eqution of h k r the form Center of Circle: the point, hk,, tht is equidistnt from ll points on the circle Rdius: the distnce r etween the center nd n point on the circle E: Show tht the eqution of circle with center hk, nd rdius r is h k r. Let one point on the circle e, nd the center e hk,. The rdius, r, is the distnce etween the center nd point on the circle. r h h d r h k. Squring oth sides, we hve E: Drw the circle given the eqution 4. Write in stndrd form. 4 Identif the center nd the rdius. Center: 0,0 Rdius: r 4 r - Sketch the grph plotting the center then plotting points on the circle units ove, elow, nd to the left nd right of the center. - Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
6 E: Write n eqution of the circle with center, 4 nd rdius 3. Center:, 4 hk, Rdius: r 3 Eqution: h k r E: Write n eqution of the circle shown. The center of the circle is 0,0 hk, Eqution: r. To find r, we will use the point given on the circle,. r r r The eqution of the circle is 6. E: A street light cn e seen on the ground within 30 d of its center. You re driving nd re 0 d est nd d south of the light. Write n inequlit to descrie the region on the ground tht is lit the light. Is the street light visile? Write the eqution of the circle (use the center 0,0 ) The region lit the light is the region inside the circle, so we wnt to include ll distnces less thn or equl to the rdius. 900 To check if the street light is visile, sustitute the point 0, into the inequlit true YES, the street light is visile. You Tr: Grph the circle QOD: Descrie how to grph circle on the grphing clcultor. Pge 6 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
7 Smple CCSD Common Em Prctice Question(s): Wht grph represents? A. B. C. D. Pge 7 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
8 Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (ELLIPSES) Ellipse: the set of ll points P such tht the sum of the distnces etween P nd two distinct fied points is constnt. Foci: the two fied points tht crete n ellipse Vertices: the two points t which the line through the foci intersect the ellipse Mjor Ais: the line segment joining the two vertices Center of the Ellipse: the midpoint of the mjor is Co-Vertices: the two points t which the line perpendiculr to the mjor is t the center intersects the ellipse Minor Ais: the line segment joining the co-vertices Eqution of n Ellipse: h k Center: : Horizontl Mjor Ais; hk, Vertices: h, k Co-Vertices: hk, Foci: h c, k h k Center: : Verticl Mjor Ais; hk, Vertices: hk, Co-Vertices: h, k Foci: hk, c Note: The foci of the ellipse lie on the mjor is, c units from the center where c. Ellipse centered t the origin with horizontl mjor is: Pge 8 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
9 E: Drw the ellipse given Identif the foci. Write the eqution in stndrd form. (Must e set equl to.) Center: 0,0 4 Length of Mjor Ais (horizontl) = = 0 Length of Minor Ais (verticl) = = 4 - c 4 c Foci:,0,,0 - E: Write n eqution of the ellipse with center t, 0. The verte is t 0, 3 0,0, verte t 0, 3, nd co-verte t h k, so the ellipse hs verticl mjor is. We will use. The center is 0,0 hk,. The distnce etween the center nd the verte is 3, so 3. The distnce etween the center nd the co-verte is, so E: Grph the ellipse foci. Center: 6, Length of Mjor Ais (verticl) = = 0 Length of Minor Ais (horizontl) = = 0. Identif the center, vertices, co-vertices, nd 00 Pge 9 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
10 Vertices: 6, 06, 8 nd 6, Co-Vertices: 6,, nd, c 00 7 c 7 3 Foci: 6, 36, 6.66 nd 6, E: The plnet Jupiter rnges from 460. million miles w to 07.0 million miles w from the sun. The center of the sun is focus of the orit. If Jupiter s orit is ellipticl, write n eqution for its orit in millions of miles. c 460. c c c 07.0 c 3.4 c You Tr: Write n eqution of the ellipse with the center t 0,0, verte t 4,0, nd focus t QOD: How cn ou tell from the eqution of n ellipse whether the mjor is is horizontl or verticl?,0. Pge 0 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
11 Smple CCSD Common Em Prctice Question(s): Which is the eqution for the grph elow? E F G H Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
12 Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (HYPERBOLAS) Hperol: the set of ll points P such tht the difference of the distnces from P to two fied points, clled the foci is constnt Vertices: the two points t which the line through the foci intersects the hperol Trnsverse Ais: the line segment joining the vertices Center: the midpoint of the trnsverse is Eqution of Hperol h k Center: : Horizontl Trnsverse Ais hk, Vertices: h, k k h Center: : Verticl Trnsverse Ais hk, Vertices: hk, Note: The foci of the hperol lie on the trnsverse is, c units from the center where c. Hperols hve slnt smptotes. Drw the rectngle formed the vertices nd the points hk, for horizontl nd h, k for verticl. The lines tht pss through the corners of this rectngle re the slnt smptotes of the hperol. E: Drw the hperol given the eqution smptotes. Write the eqution in stndrd form (set equl to ) Find the vertices, foci, nd Center: 0,0 Vertices: 4,0 nd 4,0 6 4 Foci: c c 6 9 c,0 nd,0 Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
13 Asmptotes: 3 3 nd Note: The hperol itself is onl the curve. (Dsh the smptotes.) E: Write n eqution of the hperol with foci t 0, nd 0,. nd 0, nd vertices t 0, The foci nd vertices lie on the -is, so the trnsverse is is verticl. We will use the eqution k h. Center hk, 0,0. This is the midpoint of the vertices. The foci re units from the center, so c. The vertices re unit from the center, so. c E: Grph the hperol. 6 hk, 4 Center,, Vertices:, 4, nd6, Note: The hperol itself is onl the curve. (Dsh the smptotes.) Pge 3 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections -0
14 E: The digrm shows the hperolic cross section of lrge hourglss. Write n eqution tht models the curved sides. Center hk, 0,0 Vertices:,0 nd,0 - The trnsverse is is horizontl, so we will use the eqution h k. Sustitute in point on the hperol 4,6, nd solve for You Tr: Grph the hperol 6 6. Identif the vertices nd foci. QOD: Wht re the smptotes of the hperol?? Wht re the smptotes of the hperol Pge 4 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
15 Smple CCSD Common Em Prctice Question(s):. Which grph est represents A. 4 4? B. C. D. Pge of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
16 Sllus Ojective: 0. The student will clssif conic section given its eqution with its center either t or not t the origin. Generl Form of Second-Degree Eqution: A B C D E F 0 Discriminnt: B 4AC Clssifing Conic from Its Eqution A B C D E F 0 If If If If B B B B 4AC 0 nd B 0 nd A C: CIRCLE 4AC 0 nd B 0 or A C: ELLIPSE 4AC 0: PARABOLA 4AC 0: HYPERBOLA Grphing from the Generl Form To grph from generl form, complete the squre for oth vriles to write in stndrd form. E: Clssif the conic given Then write in stndrd form nd grph. B 4AC Becuse A C, this is n ellipse. Complete the squre: Rewrite in stndrd form (set equl to ): Center: 0, Vertices: 0, , 0.838, 0, Co-Vertices: 0.36, 4.36, 4,.36, 4 Pge 6 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
17 E: Clssif the conic given grph Then write in stndrd form nd B 4AC This is hperol. Complete the squre: Rewrite in stndrd form (set equl to ): Center:, 4, Trnsverse Ais is horizontl. Vertices:, 0,, 4, You Tr: Clssif the conic given Then write in stndrd form nd grph. QOD: Descrie how to determine which conic section n eqution represents in generl form.. Wht tpe of conic section hs the eqution 4 7 A. circle B. ellipse C. hperol D. prol? 3 8 Pge 7 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
18 . Which is the eqution of n ellipse? A. B. C D. 0 0 Pge 8 of 8 McDougl Littell: Alg II Notes Unit 0: Conic Sections
Date: 9.1. Conics: Parabolas
Dte: 9. Conics: Prols Preclculus H. Notes: Unit 9 Conics Conic Sections: curves tht re formed y the intersection of plne nd doulenpped cone Syllus Ojectives:. The student will grph reltions or functions,
More informationGraphing Conic Sections
Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where
More informationName Date Class. cot. tan. cos. 1 cot 2 csc 2
Fundmentl Trigonometric Identities To prove trigonometric identit, use the fundmentl identities to mke one side of the eqution resemle the other side. Reciprocl nd Rtio Identities csc sec sin cos Negtive-Angle
More informationConic Sections Parabola Objective: Define conic section, parabola, draw a parabola, standard equations and their graphs
Conic Sections Prol Ojective: Define conic section, prol, drw prol, stndrd equtions nd their grphs The curves creted y intersecting doule npped right circulr cone with plne re clled conic sections. If
More informationHyperbolas. Definition of Hyperbola
CHAT Pre-Clculus Hyperols The third type of conic is clled hyperol. For n ellipse, the sum of the distnces from the foci nd point on the ellipse is fixed numer. For hyperol, the difference of the distnces
More informationSection 9.2 Hyperbolas
Section 9. Hperols 597 Section 9. Hperols In the lst section, we lerned tht plnets hve pproimtel ellipticl orits round the sun. When n oject like comet is moving quickl, it is le to escpe the grvittionl
More informationMTH 146 Conics Supplement
105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points
More informationcalled the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.
Review of conic sections Conic sections re grphs of the form REVIEW OF CONIC SECTIONS prols ellipses hperols P(, ) F(, p) O p =_p REVIEW OF CONIC SECTIONS In this section we give geometric definitions
More informationTopics in Analytic Geometry
Nme Chpter 10 Topics in Anltic Geometr Section 10.1 Lines Objective: In this lesson ou lerned how to find the inclintion of line, the ngle between two lines, nd the distnce between point nd line. Importnt
More information9.1 apply the distance and midpoint formulas
9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the
More informationObjective: Students will understand what it means to describe, graph and write the equation of a parabola. Parabolas
Pge 1 of 8 Ojective: Students will understnd wht it mens to descrie, grph nd write the eqution of prol. Prols Prol: collection of ll points P in plne tht re the sme distnce from fixed point, the focus
More informationStudy Sheet ( )
Key Terms prol circle Ellipse hyperol directrix focus focl length xis of symmetry vertex Study Sheet (11.1-11.4) Conic Section A conic section is section of cone. The ellipse, prol, nd hyperol, long with
More information8.2 Areas in the Plane
39 Chpter 8 Applictions of Definite Integrls 8. Ares in the Plne Wht ou will lern out... Are Between Curves Are Enclosed Intersecting Curves Boundries with Chnging Functions Integrting with Respect to
More informationSection 10.4 Hyperbolas
66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol
More informationClass-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts
Clss-XI Mthemtics Conic Sections Chpter-11 Chpter Notes Key Concepts 1. Let be fixed verticl line nd m be nother line intersecting it t fixed point V nd inclined to it t nd ngle On rotting the line m round
More informationEXPONENTIAL & POWER GRAPHS
Eponentil & Power Grphs EXPONENTIAL & POWER GRAPHS www.mthletics.com.u Eponentil EXPONENTIAL & Power & Grphs POWER GRAPHS These re grphs which result from equtions tht re not liner or qudrtic. The eponentil
More informationParabolas Section 11.1
Conic Sections Parabolas Section 11.1 Verte=(, ) Verte=(, ) Verte=(, ) 1 3 If the equation is =, then the graph opens in the direction. If the equation is =, then the graph opens in the direction. Parabola---
More informationEssential Question What are some of the characteristics of the graph of a rational function?
8. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..G A..H A..K Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions
More informationArea & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area:
Bck to Are: Are & Volume Chpter 6. & 6. Septemer 5, 6 We cn clculte the re etween the x-xis nd continuous function f on the intervl [,] using the definite integrl:! f x = lim$ f x * i )%x n i= Where fx
More information6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.
6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted
More informationThe Reciprocal Function Family. Objectives To graph reciprocal functions To graph translations of reciprocal functions
- The Reciprocl Function Fmil Objectives To grph reciprocl functions To grph trnsltions of reciprocl functions Content Stndrds F.BF.3 Identif the effect on the grph of replcing f () b f() k, kf(), f(k),
More informationSummer Review Packet For Algebra 2 CP/Honors
Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review
More information10.2: Parabolas. Chapter 10: Conic Sections. Conic sections are plane figures formed by the intersection of a double-napped cone and a plane.
Conic sections are plane figures formed b the intersection of a double-napped cone and a plane. Chapter 10: Conic Sections Ellipse Hperbola The conic sections ma be defined as the sets of points in the
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationANALYTICAL GEOMETRY. The curves obtained by slicing the cone with a plane not passing through the vertex are called conics.
ANALYTICAL GEOMETRY Definition of Conic: The curves obtined by slicing the cone with plne not pssing through the vertex re clled conics. A Conic is the locus directrix of point which moves in plne, so
More information4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E
4-1 NAME DATE PERIOD Pges 142 147 Prllel Lines nd Plnes When plnes do not intersect, they re sid to e prllel. Also, when lines in the sme plne do not intersect, they re prllel. But when lines re not in
More informationSection 5.3 : Finding Area Between Curves
MATH 9 Section 5. : Finding Are Between Curves Importnt: In this section we will lern just how to set up the integrls to find re etween curves. The finl nswer for ech emple in this hndout is given for
More informationAre You Ready for Algebra 3/Trigonometry? Summer Packet **Required for all Algebra 3/Trig CP and Honors students**
Are You Red for Algebr /Trigonometr? Summer Pcket **Required for ll Algebr /Trig CP nd Honors students** Pge of The Algebr /Trigonometr course prepres students for Clculus nd college science courses. In
More information50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula:
5 AMC LECTURES Lecture Anlytic Geometry Distnce nd Lines BASIC KNOWLEDGE. Distnce formul The distnce (d) between two points P ( x, y) nd P ( x, y) cn be clculted by the following formul: d ( x y () x )
More informationZZ - Advanced Math Review 2017
ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is
More informationAPPLICATIONS OF INTEGRATION
Chpter 3 DACS 1 Lok 004/05 CHAPTER 5 APPLICATIONS OF INTEGRATION 5.1 Geometricl Interprettion-Definite Integrl (pge 36) 5. Are of Region (pge 369) 5..1 Are of Region Under Grph (pge 369) Figure 5.7 shows
More informationThe Fundamental Theorem of Calculus
MATH 6 The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus (FTC) gives method of finding the signed re etween the grph of f nd the x-xis on the intervl [, ]. The theorem is: FTC: If f is
More informationThe notation y = f(x) gives a way to denote specific values of a function. The value of f at a can be written as f( a ), read f of a.
Chpter Prerequisites for Clculus. Functions nd Grphs Wht ou will lern out... Functions Domins nd Rnges Viewing nd Interpreting Grphs Even Functions nd Odd Functions Smmetr Functions Defined in Pieces Asolute
More information1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers?
1.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Prepring for 2A.6.K, 2A.7.I Intervl Nottion nd Set Nottion Essentil Question When is it convenient to use set-uilder nottion to represent set of numers? A collection
More informationIntroduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1)
POLAR EQUATIONS AND GRAPHS GEOMETRY INU4/54 (MATHS ) Dr Adrin Jnnett MIMA CMth FRAS Polr equtions nd grphs / 6 Adrin Jnnett Objectives The purpose of this presenttion is to cover the following topics:
More information)
Chpter Five /SOLUTIONS Since the speed ws between nd mph during this five minute period, the fuel efficienc during this period is between 5 mpg nd 8 mpg. So the fuel used during this period is between
More informationB. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a
Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing
More informationNaming 3D objects. 1 Name the 3D objects labelled in these models. Use the word bank to help you.
Nming 3D ojects 1 Nme the 3D ojects lelled in these models. Use the word nk to help you. Word nk cue prism sphere cone cylinder pyrmid D A C F A B C D cone cylinder cue cylinder E B E prism F cue G G pyrmid
More information1.1 Lines AP Calculus
. Lines AP Clculus. LINES Notecrds from Section.: Rules for Rounding Round or Truncte ll finl nswers to 3 deciml plces. Do NOT round before ou rech our finl nswer. Much of Clculus focuses on the concept
More informationSECTION 8.2 the hyperbola Wake created from shock wave. Portion of a hyperbola
SECTION 8. the hperola 6 9 7 learning OjeCTIveS In this section, ou will: Locate a hperola s vertices and foci. Write equations of hperolas in standard form. Graph hperolas centered at the origin. Graph
More informationPythagoras theorem and trigonometry (2)
HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These
More informationsuch that the S i cover S, or equivalently S
MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i
More informationMATH 2530: WORKSHEET 7. x 2 y dz dy dx =
MATH 253: WORKSHT 7 () Wrm-up: () Review: polr coordintes, integrls involving polr coordintes, triple Riemnn sums, triple integrls, the pplictions of triple integrls (especilly to volume), nd cylindricl
More informationChapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
More information2 Computing all Intersections of a Set of Segments Line Segment Intersection
15-451/651: Design & Anlysis of Algorithms Novemer 14, 2016 Lecture #21 Sweep-Line nd Segment Intersection lst chnged: Novemer 8, 2017 1 Preliminries The sweep-line prdigm is very powerful lgorithmic design
More informationName: Date: Practice Final Exam Part II covering sections a108. As you try these problems, keep referring to your formula sheet.
Name: Date: Practice Final Eam Part II covering sections 9.1-9.4 a108 As ou tr these problems, keep referring to our formula sheet. 1. Find the standard form of the equation of the circle with center at
More informationThe Basic Properties of the Integral
The Bsic Properties of the Integrl When we compute the derivtive of complicted function, like + sin, we usull use differentition rules, like d [f()+g()] d f()+ d g(), to reduce the computtion d d d to
More information6.3 Definite Integrals and Antiderivatives
Section 6. Definite Integrls nd Antiderivtives 8 6. Definite Integrls nd Antiderivtives Wht ou will lern out... Properties of Definite Integrls Averge Vlue of Function Men Vlue Theorem for Definite Integrls
More informationCHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS
CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS Big IDEAS: 1) Writing equations of conic sections ) Graphing equations of conic sections 3) Solving quadratic systems Section: Essential Question 8-1 Apply
More information1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)
Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric
More informationMatrices and Systems of Equations
Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m............ n n n mn We will use the double subscript nottion for ech element of the mtri.
More informationGrade 7/8 Math Circles Geometric Arithmetic October 31, 2012
Fculty of Mthemtics Wterloo, Ontrio N2L 3G1 Grde 7/8 Mth Circles Geometric Arithmetic Octoer 31, 2012 Centre for Eduction in Mthemtics nd Computing Ancient Greece hs given irth to some of the most importnt
More informationAnswer Key Lesson 6: Workshop: Angles and Lines
nswer Key esson 6: tudent Guide ngles nd ines Questions 1 3 (G p. 406) 1. 120 ; 360 2. hey re the sme. 3. 360 Here re four different ptterns tht re used to mke quilts. Work with your group. se your Power
More informationOrder these angles from smallest to largest by wri ng 1 to 4 under each one. Put a check next to the right angle.
Lines nd ngles Connect ech set of lines to the correct nme: prllel perpendiculr Order these ngles from smllest to lrgest y wri ng to 4 under ech one. Put check next to the right ngle. Complete this tle
More informationApplications of the Definite Integral ( Areas and Volumes)
Mth1242 Project II Nme: Applictions of the Definite Integrl ( Ares nd Volumes) In this project, we explore some pplictions of the definite integrl. We use integrls to find the re etween the grphs of two
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationIntroduction to Integration
Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition.
More informationLesson 11 MA Nick Egbert
Lesson MA 62 Nick Eert Overview In this lesson we return to stndrd Clculus II mteril with res etween curves. Recll rom irst semester clculus tht the deinite interl hd eometric menin, nmel the re under
More informationLily Yen and Mogens Hansen
SKOLID / SKOLID No. 8 Lily Yen nd Mogens Hnsen Skolid hs joined Mthemticl Myhem which is eing reformtted s stnd-lone mthemtics journl for high school students. Solutions to prolems tht ppered in the lst
More informationChapter44. Polygons and solids. Contents: A Polygons B Triangles C Quadrilaterals D Solids E Constructing solids
Chpter44 Polygons nd solids Contents: A Polygons B Tringles C Qudrilterls D Solids E Constructing solids 74 POLYGONS AND SOLIDS (Chpter 4) Opening prolem Things to think out: c Wht different shpes cn you
More informationMA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork
MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html
More informationPrecalculus, IB Precalculus and Honors Precalculus
NORTHEAST CONSORTIUM Precalculus, IB Precalculus and Honors Precalculus Summer Pre-View Packet DUE THE FIRST DAY OF SCHOOL The problems in this packet are designed to help ou review topics from previous
More informationStained Glass Design. Teaching Goals:
Stined Glss Design Time required 45-90 minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to
More informationUnit 5 Vocabulary. A function is a special relationship where each input has a single output.
MODULE 3 Terms Definition Picture/Exmple/Nottion 1 Function Nottion Function nottion is n efficient nd effective wy to write functions of ll types. This nottion llows you to identify the input vlue with
More informationMath 4 Review for Quarter 2 Cumulative Test
Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry (3.1-3.3) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes by disks: volume prt ii 6 6 Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem 6) nd the ccumultion process is to determine so-clled volumes
More informationCalculus Differentiation
//007 Clulus Differentition Jeffrey Seguritn person in rowot miles from the nerest point on strit shoreline wishes to reh house 6 miles frther down the shore. The person n row t rte of mi/hr nd wlk t rte
More informationIterated Integrals. f (x; y) dy dx. p(x) To evaluate a type I integral, we rst evaluate the inner integral Z q(x) f (x; y) dy.
Iterted Integrls Type I Integrls In this section, we begin the study of integrls over regions in the plne. To do so, however, requires tht we exmine the importnt ide of iterted integrls, in which inde
More informationSolutions to Math 41 Final Exam December 12, 2011
Solutions to Mth Finl Em December,. ( points) Find ech of the following its, with justifiction. If there is n infinite it, then eplin whether it is or. ( ) / ln() () (5 points) First we compute the it:
More informationYoplait with Areas and Volumes
Yoplit with Ares nd Volumes Yoplit yogurt comes in two differently shped continers. One is truncted cone nd the other is n ellipticl cylinder (see photos below). In this exercise, you will determine the
More informationGraph and Write Equations of Hyperbolas
TEKS 9.5 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Hperbolas Before You graphed and wrote equations of parabolas, circles, and ellipses. Now You will graph and write equations of hperbolas. Wh?
More information2 b. 3 Use the chain rule to find the gradient:
Conic sections D x cos θ, y sinθ d y sinθ So tngent is y sin θ ( x cos θ) sinθ Eqution of tngent is x + y sinθ sinθ Norml grdient is sinθ So norml is y sin θ ( x cos θ) xsinθ ycos θ ( )sinθ, So eqution
More informationRewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 2 4 x 0
Pre-Calculus Section 1.1 Completing the Square Rewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 4 x 0. 3x 3y
More informationFig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1.
Answer on Question #5692, Physics, Optics Stte slient fetures of single slit Frunhofer diffrction pttern. The slit is verticl nd illuminted by point source. Also, obtin n expression for intensity distribution
More informationGRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS
GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS 1.1 DIFFERENT TYPES AND SHAPES OF GRAPHS: A graph can be drawn to represent are equation connecting two variables. There are different tpes of equations which
More information3 4. Answers may vary. Sample: Reteaching Vertical s are.
Chpter 7 Answers Alterntive Activities 7-2 1 2. Check students work. 3. The imge hs length tht is 2 3 tht of the originl segment nd is prllel to the originl segment. 4. The segments pss through the endpoints
More information8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola?
8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola? How do you know which way it opens? Given a & b, how do you find
More information9.3 Hyperbolas and Rotation of Conics
9.3 Hyperbolas and Rotation of Conics Copyright Cengage Learning. All rights reserved. What You Should Learn Write equations of hyperbolas in standard form. Find asymptotes of and graph hyperbolas. Use
More informationConic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY Joe McBride/Stone/Gett Imges Air resistnce prevents the velocit of skdiver from incresing indefinitel. The velocit pproches it, clled the terminl velocit. The development of clculus
More informationUnit #9 : Definite Integral Properties, Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties, Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationAML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces
AML7 CAD LECTURE 6 SURFACES. Anlticl Surfces. Snthetic Surfces Surfce Representtion From CAD/CAM point of view surfces re s importnt s curves nd solids. We need to hve n ide of curves for surfce cretion.
More informationAngle properties of lines and polygons
chievement Stndrd 91031 pply geometric resoning in solving problems Copy correctly Up to 3% of workbook Copying or scnning from ES workbooks is subject to the NZ Copyright ct which limits copying to 3%
More informationSAMPLE PREREQUISITE PROBLEMS: CALCULUS
SAMPLE PREREQUISITE PROBLEMS: CALCULUS Te following questions rise from ctul AP Clculus AB em questions; I went troug lots of questions, nd pulled out prts requiring lgebr nd trigonometr Tese problems
More informationStudy Guide for Exam 3
Mth 05 Elementry Algebr Fll 00 Study Guide for Em Em is scheduled for Thursdy, November 8 th nd ill cover chpters 5 nd. You my use "5" note crd (both sides) nd scientific clcultor. You re epected to no
More informationLecture 5: Spatial Analysis Algorithms
Lecture 5: Sptil Algorithms GEOG 49: Advnced GIS Sptil Anlsis Algorithms Bsis of much of GIS nlsis tod Mnipultion of mp coordintes Bsed on Eucliden coordinte geometr http://stronom.swin.edu.u/~pbourke/geometr/
More informationNUMB3RS Activity: Irregular Polygon Centroids. Episode: Burn Rate
Techer Pge 1 NUMB3RS Activit: Irregulr Polgon Centroids Topic: Geoetr, Points of Concurrenc Grde Level: 9-10 Ojective: Students will e le to find the centroid of irregulr polgons. Tie: 0 inutes Mterils:
More informationRigid Body Transformations
igid od Kinemtics igid od Trnsformtions Vij Kumr igid od Kinemtics emrk out Nottion Vectors,,, u, v, p, q, Potentil for Confusion! Mtrices,, C, g, h, igid od Kinemtics The vector nd its skew smmetric mtri
More informationMENSURATION-IV
MENSURATION-IV Theory: A solid is figure bounded by one or more surfce. Hence solid hs length, bredth nd height. The plne surfces tht bind solid re clled its fces. The fundmentl difference between plne
More information9.1 PYTHAGOREAN THEOREM (right triangles)
Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side
More informationMid-Chapter Quiz: Lessons 7-1 through 7-3
Write an equation for and graph a parabola with the given focus F and vertex V 1. F(1, 5), V(1, 3) Because the focus and vertex share the same x coordinate, the graph is vertical. The focus is (h, k +
More informationBefore We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):
Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters
More informationREVIEW, pages
REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in
More informationN-Level Math (4045) Formula List. *Formulas highlighted in yellow are found in the formula list of the exam paper. 1km 2 =1000m 1000m
*Formul highlighted in yellow re found in the formul lit of the em pper. Unit Converion Are m =cm cm km =m m = m = cm Volume m =cm cm cm 6 = cm km/h m/ itre =cm (ince mg=cm ) 6 Finncil Mth Percentge Incree
More informationMath 155, Lecture Notes- Bonds
Math 155, Lecture Notes- Bonds Name Section 10.1 Conics and Calculus In this section, we will study conic sections from a few different perspectives. We will consider the geometry-based idea that conics
More informationA dual of the rectangle-segmentation problem for binary matrices
A dul of the rectngle-segmenttion prolem for inry mtrices Thoms Klinowski Astrct We consider the prolem to decompose inry mtrix into smll numer of inry mtrices whose -entries form rectngle. We show tht
More informationUnit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
More informationP(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have
Rndom Numers nd Monte Crlo Methods Rndom Numer Methods The integrtion methods discussed so fr ll re sed upon mking polynomil pproximtions to the integrnd. Another clss of numericl methods relies upon using
More informationThe point (x, y) lies on the circle of radius r and center (h, k) iff. x h y k r
NOTES +: ANALYTIC GEOMETRY NAME LESSON. GRAPHS OF EQUATIONS IN TWO VARIABLES (CIRCLES). Standard form of a Circle The point (x, y) lies on the circle of radius r and center (h, k) iff x h y k r Center:
More information1.5 Extrema and the Mean Value Theorem
.5 Extrem nd the Men Vlue Theorem.5. Mximum nd Minimum Vlues Definition.5. (Glol Mximum). Let f : D! R e function with domin D. Then f hs n glol mximum vlue t point c, iff(c) f(x) for ll x D. The vlue
More information