10.2 The Ellipse and the Hyperbola


 Josephine Beasley
 4 years ago
 Views:
Transcription
1 CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point C is 9 meters est nd meters north of point A. Find the distnce cross the lke, from B to C. 98. A ridge constructed over ou hs supporting rch in the shpe of prol. Find n eqution of the prolic rch if the length of the rod over the rch is 00 meters nd the mimum height of the rch is 40 meters. 00 m 40 m C 0 m A (0, 0) B Use grphing clcultor to verif ech eercise. Use squre viewing window. 99. Eercise Eercise Eercise Eercise The Ellipse nd the Hperol S Define nd Grph n Ellipse. Define nd Grph Hperol. Grphing Ellipses An ellipse cn e thought of s the set of points in plne such tht the sum of the distnces of those points from two fied points is constnt. Ech of the two fied points is clled focus. (The plurl of focus is foci.) The point midw etween the foci is clled the center. An ellipse m e drwn hnd using two thumtcks, piece of string, nd pencil. Secure the two thumtcks in piece of crdord, for emple, nd tie ech end of the string to tck. Use our pencil to pull the string tight nd drw the ellipse. The two thumtcks re the foci of the drwn ellipse. Focus Center Focus Ellipse with Center (0, 0) The grph of n eqution of the form + = is n ellipse with center (0, 0). The intercepts re (, 0) nd , 0, nd the intercepts re (0, ), nd 0, . The stndrd form of n ellipse with center (0, 0) is + =.
2 Section 0. The Ellipse nd the Hperol EXAMPLE Grph 9 + =. Solution The eqution is of the form + =, with = nd = 4, so its grph is n ellipse with center (0, 0), intercepts (, 0) nd , 0, nd intercepts (0, 4) nd 0, (0, 4) (, 0) (, 0) 4 (0, ) Grph + 4 =. EXAMPLE Grph 4 + = 4. Solution Although this eqution contins sum of squred terms in nd on the sme side of n eqution, this is not the eqution of circle since the coefficients of nd re not the sme. The grph of this eqution is n ellipse. Since the stndrd form of the eqution of n ellipse hs on one side, divide oth sides of this eqution = = 4 4 Divide oth sides = Simplif. We now recognize the eqution of n ellipse with = 4 nd =. This ellipse hs center (0, 0), intercepts (4, 0) nd  4, 0, nd intercepts (0, ) nd 0, (0, ) (, 0) (4, 0) 4 (0, ) Grph =. The center of n ellipse is not lws (0, 0), s shown in the net emple. Ellipse with Center (h, k) The stndrd form of the eqution of n ellipse with center h, k is  h  k + =
3 4 CHAPTER 0 Conic Sections +  EXAMPLE Grph + =. Solution The center of this ellipse is found in w tht is similr to finding the center of circle. This ellipse hs center ,. Notice tht = nd =. To find four points on the grph of the ellipse, first grph the center, ,. Since =, count units right nd then units left of the point with coordintes ,. Net, since =, strt t , nd count units up nd then units down to find two more points on the ellipse. (, ) ( ) ( )  4 Grph =. CONCEPT CHECK In the grph of the eqution 4 + =, which distnce is longer: the distnce etween the intercepts or the distnce etween the intercepts? How much longer? Eplin. Focus Center Focus Grphing Hperols The finl conic section is the hperol. A hperol is the set of points in plne such tht the solute vlue of the difference of the distnces from two fied points is constnt. Ech of the two fied points is clled focus. The point midw etween the foci is clled the center. Using the distnce formul, we cn show tht the grph of  = is hperol with center (0, 0) nd intercepts (, 0) nd , 0. Also, the grph of  = is hperol with center (0, 0) nd intercepts (0, ) nd 0, . Hperol with Center (0, 0) The grph of n eqution of the form  = is hperol with center (0, 0) nd intercepts (, 0) nd , 0. Answer to Concept Check: intercepts, 4 units
4 Section 0. The Ellipse nd the Hperol The grph of n eqution of the form  = is hperol with center (0, 0) nd intercepts (0, ) nd 0, . The equtions  = nd  = re the stndrd forms for the eqution of hperol. Helpful Hint Notice the difference etween the eqution of n ellipse nd hperol. The eqution of the ellipse contins nd terms on the sme side of the eqution with smesign coefficients. For hperol, the coefficients on the sme side of the eqution hve different signs. (, ) (, ) (, ) (, ) Grphing hperol such s  = is mde esier recognizing one of its importnt chrcteristics. Emining the figure to the left, notice how the sides of the rnches of the hperol etend indefinitel nd seem to pproch the dshed lines in the figure. These dshed lines re clled the smptotes of the hperol. To sketch these lines, or smptotes, drw rectngle with vertices (, ), ,,,  nd , . The smptotes of the hperol re the etended digonls of this rectngle. EXAMPLE 4 Grph  =. Solution This eqution hs the form  =, with = 4 nd =. Thus, its grph is hperol tht opens to the left nd right. It hs center (0, 0) nd intercepts (4, 0) nd 4, 0. To id in grphing the hperol, we first sketch its smptotes. The etended digonls of the rectngle with corners (4, ), 4, , 4,, nd 4,  re the smptotes of the hperol. Then we use the smptotes to id in sketching the hperol. (, ) (4, ) 4 (, ) 4 (4, ) 4 Grph 9  =.
5 CHAPTER 0 Conic Sections EXAMPLE Grph 49 =. Solution Since this is difference of squred terms in nd on the sme side of the eqution, its grph is hperol s opposed to n ellipse or circle. The stndrd form of the eqution of hperol hs on one side, so divide oth sides of the eqution. 49 = 49 = Divide oth sides. 94 = Simplif. The eqution is of the form  =, with = nd =, so the hperol is centered t (0, 0) with intercepts (0, ) nd 0, . The sketch of the hperol is shown Grph 9  =. Although this is eond the scope of this tet, the stndrd forms of the equtions of hperols with center (h, k) re given elow. The Concept Etensions section in Eercise Set 0. contins some hperols of this form. Hperol with Center (h, k) Stndrd forms of the equtions of hperols with center (h, k) re:  h   k =  k   h = Grphing Clcultor Eplortions To grph n ellipse using grphing clcultor, use the sme procedure s for grphing circle. For emple, to grph + =, first solve for. =  0 Y Y =  = { B Net, press the Y = ke nd enter Y = nd Y = . B B (Insert two sets of prentheses in the rdicnd s  / so tht the desired grph is otined.) The grph ppers s shown to the left.
6 Section 0. The Ellipse nd the Hperol 7 Use grphing clcultor to grph ech ellipse =. + =. 0 + = = = = 0.8 Voculr, Rediness & Video Check Use the choices elow to fill in ech lnk. Some choices will e used more thn once nd some not t ll. ellipse 0, 0, 0 nd , 0 0, nd 0,  focus hperol center, 0 nd , 0 0, nd 0, . A(n) is the set of points in plne such tht the solute vlue of the differences of their distnces from two fied points is constnt.. A(n) is the set of points in plne such tht the sum of their distnces from two fied points is constnt. For eercises nd ove,. The two fied points re ech clled. 4. The point midw etween the foci is clled the.. The grph of  = is (n) with center nd intercepts of.. The grph of + = is (n) with center nd intercepts of. MrtinG Interctive Videos Wtch the section lecture video nd nswer the following questions. 7. From Emple, wht informtion do the vlues of nd give us out the grph of n ellipse? Answer this sme question for Emple. 8. From Emple, we know the points (, ),, , ,, nd ,  re not prt of the grph. Eplin the role of these points. See Video Eercise Set Identif the grph of ech eqution s n ellipse or hperol. Do not grph. See Emples through.. + =. 44 =.  = =.  + =. + = Sketch the grph of ech eqution. See Emples nd = = = =. 9 + =. + 4 =. 4 + = = Sketch the grph of ech eqution. See Emple = = = =
7 8 CHAPTER 0 Conic Sections Sketch the grph of ech eqution. See Emples 4 nd = 0.  =.  = =.  4 = =.  =. 4  = 00 MIXED Grph ech eqution. See Emples through. 7. =  8. = = = =. + = = = 9 MIXED SECTIONS 0., 0. Identif whether ech eqution, when grphed, will e prol, circle, ellipse, or hperol. Sketch the grph of ech eqution. If prol, lel the verte. If circle, lel the center nd note the rdius. If n ellipse, lel the center. If hperol, lel the  or intercepts = 4. = = = = = = 4. + = 4. = = = = = 48. = = 0. = REVIEW AND PREVIEW Perform the indicted opertions. See Sections. nd CONCEPT EXTENSIONS The grph of ech eqution is n ellipse. Determine which distnce is longer, the distnce etween or the distnce etween How much longer? See the Concept Check in this section.. + = = = = 9. If ou re given list of equtions of circles, prols, ellipses, nd hperols, eplin how ou could distinguish the different conic sections from their equtions. 0. We know tht + = is the eqution of circle. Rewrite the eqution so tht the right side is equl to. Which tpe of conic section does this eqution form resemle? In fct, the circle is specil cse of this tpe of conic section. Descrie the conditions under which this tpe of conic section is circle. The orits of strs, plnets, comets, steroids, nd stellites ll hve the shpe of one of the conic sections. Astronomers use mesure clled eccentricit to descrie the shpe nd elongtion of n oritl pth. For the circle nd ellipse, eccentricit e is clculted with the formul e = c d, where c = 00 nd d is the lrger vlue of or. For hperol, eccentricit e is clculted with the formul e = c d, where c = + nd the vlue of d is equl to if the hperol hs intercepts or equl to if the hperol hs intercepts. Use equtions A H to nswer Eercises 70. A.  D.  9 = B. 4 + = C. 4 + = = E = F. + = G.  = H =. Identif the tpe of conic section represented ech of the equtions A H.. For ech of the equtions A H, identif the vlues of nd.. For ech of the equtions A H, clculte the vlue of c nd c. 4. For ech of the equtions A H, find the vlue of d.. For ech of the equtions A H, clculte the eccentricit e.. Wht do ou notice out the vlues of e for the equtions ou identified s ellipses? 7. Wht do ou notice out the vlues of e for the equtions ou identified s circles? 8. Wht do ou notice out the vlues of e for the equtions ou identified s hperols? 9. The eccentricit of prol is ectl. Use this informtion nd the oservtions ou mde in Eercises, 7, nd 8 to descrie w tht could e used to identif the tpe of conic section sed on its eccentricit vlue. 70. Grph ech of the conic sections given in equtions A H. Wht do ou notice out the shpe of the ellipses for incresing vlues of eccentricit? Which is the most ellipticl? Which is the lest ellipticl, tht is, the most circulr? 7. A plnet s orit out the sun cn e descried s n ellipse. Consider the sun s the origin of rectngulr coordinte sstem. Suppose tht the intercepts of the ellipticl pth of the plnet re {0,000,000 nd tht the intercepts re {,000,000. Write the eqution of the ellipticl pth of the plnet.
8 Integrted Review 9 7. Comets orit the sun in elongted ellipses. Consider the sun s the origin of rectngulr coordinte sstem. Suppose tht the eqution of the pth of the comet is ,78,000,000 ,400,000.4 * * 0 = Find the center of the pth of the comet. 7. Use grphing clcultor to verif Eercise Use grphing clcultor to verif Eercise. For Eercises 7 through 80, see the emple elow. Emple   Sketch the grph of  =. 9 Solution This hperol hs center (, ). Notice tht = nd = Sketch the grph of ech eqution. (, ) ( ) ( ) = = = = = = Integrted Review GRAPHING CONIC SECTIONS Following is summr of conic sections. Conic Sections Stndrd Form Grph Prol =  h + k (h, k) 0 0 (h, k) Prol =  k + h (h, k) 0 0 (h, k) Circle  h +  k = r (h, k) r Ellipse center (0, 0) + = Hperol center (0, 0)  = Hperol center (0, 0)  =
9 0 CHAPTER 0 Conic Sections Identif whether ech eqution, when grphed, will e prol, circle, ellipse, or hperol. Then grph ech eqution = 4. = + 4. = =. 99 =.  4 = = 8. + = 9. = = = = = 4. = = 0. Solving Nonliner Sstems of Equtions S Solve Nonliner Sstem Sustitution. Solve Nonliner Sstem Elimintion. In Section 4., we used grphing, sustitution, nd elimintion methods to find solutions of sstems of liner equtions in two vriles. We now ppl these sme methods to nonliner sstems of equtions in two vriles. A nonliner sstem of equtions is sstem of equtions t lest one of which is not liner. Since we will e grphing the equtions in ech sstem, we re interested in rel numer solutions onl. Solving Nonliner Sstems Sustitution First, nonliner sstems re solved the sustitution method. EXAMPLE Solve the sstem e  =  = Solution We cn solve this sstem sustitution if we solve one eqution for one of the vriles. Solving the first eqution for is not the est choice since doing so introduces rdicl. Also, solving for in the first eqution introduces frction. We solve the second eqution for.  = Second eqution  = Solve for. Replce with  in the first eqution, nd then solve for.  = First eqution $%&   = Replce with =  + = = 0 = or = Let = nd then let = in the eqution =  to find corresponding vlues. Let =. Let =. =  =  = =  =  = 0 The solutions re (, ) nd (, 0), or the solution set is,,, 0. Check oth solutions in oth equtions. Both solutions stisf oth equtions, so oth re solutions
8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
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 informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 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
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More information, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse.
8.5 The Ellipse Kidne stones re crstllike ojects tht cn form in the kidnes. Trditionll, people hve undergone surger to remove them. In process clled lithotrips, kidne stones cn now e removed without surger.
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationIntroduction. Definition of Hyperbola
Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rellife problems. Clssif
More information10.5. ; 43. The points of intersection of the cardioid r 1 sin and. ; Graph the curve and find its length. CONIC SECTIONS
654 CHAPTER 1 PARAETRIC EQUATIONS AND POLAR COORDINATES ; 43. The points of intersection of the crdioid r 1 sin nd the spirl loop r,, cn t be found ectl. Use grphing device to find the pproimte vlues of
More informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationSECTION 94 Translation of Axes
94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationSketch graphs of conic sections and write equations related to conic sections
Achievement Stndrd 909 Sketch grphs of conic sections nd write equtions relted to conic sections Clculus.5 Eternll ssessed credits Sketching Conics the Circle nd the Ellipse Grphs of the conic sections
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 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 informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationNat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS
Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X Section 4.4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether  is root of 0. Show
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationDA 3: The Mean Value Theorem
Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More informationJEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationM344  ADVANCED ENGINEERING MATHEMATICS
M3  ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
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 informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationAPPLICATIONS OF DEFINITE INTEGRALS
Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
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 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 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 informationC Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
More informationChapter 3 Exponential and Logarithmic Functions Section 3.1
Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re nonlgebric functions. The re clled trnscendentl functions. The eponentil
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
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
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationLesson 8.1 Graphing Parametric Equations
Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More information1. Twelve less than five times a number is thirty three. What is the number
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
More informationBME 207 Introduction to Biomechanics Spring 2018
April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts e,
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationThe Trapezoidal Rule
SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationChapter 7: Applications of Integrals
Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As
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 informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More informationONLINE PAGE PROOFS. Antidifferentiation and introduction to integral calculus
Antidifferentition nd introduction to integrl clculus. Kick off with CAS. Antiderivtives. Antiderivtive functions nd grphs. Applictions of ntidifferentition.5 The definite integrl.6 Review . Kick off
More information1. Extend QR downwards to meet the xaxis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
More informationCh AP Problems
Ch. 7.7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationChapter 3: Polynomial and Rational Functions
Chpter 3: Polynomil nd Rtionl Functions Section 3. Power Functions & Polynomil Functions... 94 Section 3. Qudrtic Functions... 0 Section 3.3 Grphs of Polynomil Functions... 09 Section 3.4 Rtionl Functions...
More information5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More information50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
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 information7.8 IMPROPER INTEGRALS
7.8 Improper Integrls 547 the grph of g psses through the points (, ), (, ), nd (, ); the grph of g psses through the points (, ), ( 3, 3 ), nd ( 4, 4 );... the grph of g n/ psses through the points (
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationLesson5 ELLIPSE 2 1 = 0
Lesson5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
More informationChapter 9: Conics. Photo by Gary Palmer, Flickr, CCBY, https://www.flickr.com/photos/gregpalmer/
Chpter 9: Conics Section 9. Ellipses... 579 Section 9. Hperbols... 597 Section 9.3 Prbols nd NonLiner Sstems... 67 Section 9.4 Conics in Polr Coordintes... 630 In this chpter, we will eplore set of shpes
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More information8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive stepystep
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 00 CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationMEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
More informationTImath.com Algebra 2. Constructing an Ellipse
TImth.com Algebr Constructing n Ellipse ID: 9980 Time required 60 minutes Activity Overview This ctivity introduces ellipses from geometric perspective. Two different methods for constructing n ellipse
More information8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationSections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
More informationCalculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More informationBelievethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
More informationGrade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations
Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions
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 informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More informationImproper Integrals with Infinite Limits of Integration
6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl
More informationFaith Scholarship Service Friendship
Immcult Mthemtics Summer Assignment The purpose of summer ssignment is to help you keep previously lerned fcts fresh in your mind for use in your net course. Ecessive time spent reviewing t the beginning
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationIntroduction to Algebra  Part 2
Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger  Prt Sttement of Prerequisite
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
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 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 informationEssential Question What conjectures can you make about perpendicular lines?
3. roofs with erpendiculr Lines Essentil Question Wht conjectures cn ou ke out perpendiculr lines? Writing onjectures Work with prtner. Fold piece of pper in hlf twice. Lel points on the two creses, s
More informationThe semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer.
ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationMA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1
MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]
More informationMATH 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 informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proilit Distriutions 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
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 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 informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More information