, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse.
|
|
- Domenic Marsh
- 5 years ago
- Views:
Transcription
1 8.5 The Ellipse Kidne stones re crstl-like 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. To remove the stones, doctors cn use lithotripter, which mens stone crusher in Greek. A lithotripter is ellipticll-shped, nd its design mkes use of the properties of the ellipse to provide sfer method for removing kidne stones. This method will e eplined in Emple 5. An ellipse is the set or locus of points P in the plne such tht the sum of the distnces from P to two fied points nd F is constnt. P + F P = k We Connection To lern more out lithotripters, visit the ove we site. Go to Mth Resources, then to MATHEMATICS 11, to find out where to go net. Write rief report out how lithotripters work. The two fied points, nd F, re clled the foci (plurl of focus) of the ellipse. The line segments P nd F P re clled focl rdii of the ellipse. INVESTIGATE & INQUIRE You will need two cler plstic rulers, sheet of pper, nd pencil for this investigtion. Step 1 Drw 10-cm line segment ner the centre of the piece of pper. Lel the two endpoints nd F. F Step Choose length, k centimetres, which is greter thn the length F. You m wnt to mke k less thn 0 cm. Step 3 Choose pir of lengths, centimetres nd centimetres, where k = The Ellipse MHR 619
2 Step 4 Use oth rulers to mrk two points tht re centimetres from nd centimetres from F. F Step 5 Repet steps 3 nd 4 using different vlues for nd until ou hve mrked enough points to define complete curve. Step 6 Drw smooth curve through the points. The curve is n emple of n ellipse. 1. How mn es of smmetr does the ellipse hve?. In reltion to nd F, where is the point of intersection of the es of smmetr? 3. ) Wh must k e greter thn the length of the line segment F? ) Wht digrm would result if k equlled F? 4. In step 4, ou locted two points for the chosen vlues of nd. Are there n vlues of nd for which onl one point cn e mrked? If so, wht re the vlues of nd? Where is the loction of the point on the ellipse? The digrm t the right shows nother method for drwing n ellipse. Pushpins t nd F re used to fsten loop of string, which is over twice s long s F. If ou hold pencil tight ginst the string, nd move the pencil long the string, ou will trce n ellipse. Since the length of string is constnt, the sum of the distnces P nd F P is constnt for ll positions of P. P string F 60 MHR Chpter 8
3 EXAMPLE 1 Finding the Eqution of n Ellipse From its Locus Definition Use the locus definition of the ellipse to find n eqution of n ellipse with foci ( 4, 0) nd F (4, 0), nd with the constnt sum of the focl rdii equl to 10. SOLUTION Let P(, ) e n point on the ellipse. The locus definition of the ellipse cn e stted lgericll s P + F P = 10. Use the formul for the length of line segment, l = ( 1 ) + (, 1 ) to rewrite P nd F P. l = ( 1 ) + ( 1 ) P = ( 4)) ( + ( 0) = ( + 4) + F P = ( 4) + ( 0) = ( 4) + Sustitute : ( + 4) + + ( 4) + = 10 Isolte rdicl: ( + 4) + = 10 ( 4) + Squre oth sides: ( + 4) + = ( 4) + + ( 4) + Simplif: = ( 4) Isolte the rdicl: = 0 ( 4) + Divide oth sides 4: 4 5 = 5 ( 4) + Squre oth sides: = 5(( 4) + ) Simplif: = 5( 4) = 5( ) = = Divide ech side 5: 1 = + The eqution of the ellipse is + = The Ellipse MHR 61
4 An ellipse hs two es of smmetr. The longer line segment is clled the mjor is, nd the shorter line segment is clled the minor is. The endpoints on the mjor is re the vertices of the ellipse. The endpoints of the minor es re the co-vertices of the ellipse. verte co-verte minor is mjor is co-verte verte The ellipse in Emple 1 cn e modelled grphicll, s shown. Note tht the coordintes of the vertices re (5, 0) nd ( 5, 0). The length of the mjor is is 10. The coordintes of the co-vertices re (0, 3) nd (0, 3). The length of the minor is is 6. The eqution cn e written s + = In this form of the eqution, notice tht 5 is hlf the length of the mjor is, or hlf the constnt sum of the focl rdii, nd 3 is hlf the length of the minor is. In the digrm, hlf the length of the mjor is is denoted. Hlf the length of the minor is is denoted. Hlf the distnce etween the two foci, which re on the mjor is, is denoted c. ( 5, 0) =1 5 9 ( 4, 0) c 4 (0, 3) (0, 3) F (4, 0) (5, 0) If P is point on the ellipse when = 0, the focl rdii P = c nd F P = + c. P + F P = c + + c = So, the constnt sum of the distnces from point on the ellipse to the two foci is, which is the length of the mjor is. P + c F c 6 MHR Chpter 8
5 Let P e point on the ellipse when = 0. Since P + F P =, nd P = F P, then P = nd F P =. P Using the Pthgoren theorem, = + c, with >. Notice tht, in Emple 1, = 5, = 3, nd c = 4, nd tht 5 = c 0 c F Therefore, the mjor is hs length of units, the minor is hs length of units, nd the distnce etween the two foci is c units. The following digrms show how the ke points of ellipses centred t the origin re lelled. Horizontl mjor is Co-verte (0, ) Verticl mjor is Verte (0, ) Verte (, 0) Minor is Mjor is 0 Focus ( c, 0) Focus F (c, 0) Co-verte (0, ) Verte (, 0) Co-verte (, 0) Minor Mjor is 0 Focus (0, c) Focus F (0, c) is Co-verte (, 0) Verte (0, ) 8.5 The Ellipse MHR 63
6 These results cn e summrized s follows. Ellipse with centre t the origin nd mjor is long the -is. The stndrd form of the eqution of n ellipse centred t the origin, with the mjor is long the -is is + = 1, > > 0 V 1 (, 0) Mjor is V (, 0) F 1 ( c, 0) 0 F (c, 0) The length of the mjor is is. The length of the minor is is. The vertices re V 1 (, 0) nd V (, 0). The co-vertices re (0, ) nd (0, ). The foci re ( c, 0) nd F (c, 0). = + c Minor is (0, ) (0, ) Ellipse with centre t the origin nd mjor is long the -is. The stndrd form of the eqution of n ellipse centred t the origin, with the mjor is long the -is is + = 1, > > 0 (, 0) Minor Mjor is V (0, ) 0 F (0, c) (0, c) is V 1 (0, ) (, 0) The length of the mjor is is. The length of the minor is is. The vertices re V 1 (0, ) nd V (0, ). The co-vertices re (, 0) nd (, 0). The foci re (0, c) nd F (0, c). = + c 64 MHR Chpter 8
7 EXAMPLE Sketching the Grph of n Ellipse With Centre (0, 0) Sketch the grph of the ellipse 4 + = 36. Lel the foci. SOLUTION Rewrite 4 + = 36 in stndrd form. 4 + = 36 Divide oth sides 36: + = Since the denomintor of is less thn the denomintor of, the eqution is in the form + = 1. The ellipse is centred t the origin nd the mjor is is on the -is. Since = 36, = 6, nd the vertices re V 1 (0, 6) nd V (0, 6). Since = 9, = 3, nd the co-vertices re ( 3, 0) nd (3, 0). = + c 6 = 3 + c 36 9 = c 7 = c 3 3 = c The foci re (0, 3 3 ) nd F (0, 3 3 ), or V (0, 6) 6 pproimtel (0, 5.) nd (0, 5.). 4 + = 36 F (0, 3 3) 4 Plot the vertices nd co-vertices. Drw smooth curve through the points. Lel the foci nd the grph. ( 3, 0) (3, 0) (0, 3 3) V 1 (0, 6) An ellipse m not e centred t the origin. As in the cse of the circle, n ellipse cn hve centre (h, k). The trnsltion rules tht ppl to the circle lso ppl to the ellipse. 8.5 The Ellipse MHR 65
8 The stndrd form of the eqution of n ellipse with centre (h, k) nd the mjor is prllel to the -is is ( h) ( k) + = 1, > > 0 k + (h, k + ) The stndrd form of the eqution of n ellipse with centre (h, k) nd the mjor is prllel to the -is is ( h) ( k) + = 1, > > 0 k + V (h, k + ) F (h, k + c) V 1 (h, k) k (h c, k) (h, k) F (h + c, k) V (h +, k) k (h, k) (h, k) (h +, k) k h 0 (h, k ) h + The length of the mjor is is. The length of the minor is is. The vertices re V 1 (h, k) nd V (h +, k). The co-vertices re (h, k ) nd (h, k + ). The foci re (h c, k) nd F (h + c, k). = + c h k 0 h h (h, k c) V 1 (h, k ) h + The length of the mjor is is. The length of the minor is is. The vertices re V 1 (h, k ) nd V (h, k + ). The co-vertices re (h, k) nd (h +, k). The foci re (h, k c) nd F (h, k + c). = + c EXAMPLE 3 Sketching the Grph of n Ellipse With Centre (h, k) Sketch the grph of the ellipse ( + 1) ( 3) + = 1. Lel the foci SOLUTION Since the denomintor of ( + 1) is greter thn the denomintor of ( 3), the eqution ( + 1) ( 3) + = 1 is in the form 5 16 ( h) ( k) + = 1. The ellipse is centred t (h, k), or ( 1, 3), nd the mjor is is prllel to the -is. 66 MHR Chpter 8
9 = 5, so = 5 = 16, so = 4 The mjor is, which is prllel to the -is, hs length of, or 10. The minor is, which is prllel to the -is, hs length of, or 8. The vertices re V 1 (h, k) nd V (h +, k). Sustitute the vlues of h, k, nd. The vertices re V 1 ( 1 5, 3) nd V ( 1 + 5, 3), or V 1 ( 6, 3) nd V (4, 3). The co-vertices re (h, k ) nd (h, k + ). Sustitute the vlues of h, k, nd. The co-vertices re ( 1, 3 4) nd ( 1, 3 + 4), or ( 1, 1) nd ( 1, 7). The foci re (h c, k) nd F (h + c, k). To find c, we use = + c, with = 5 nd = 4. = + c 5 = 3 + c 5 = 16 + c 5 16 = c 9 = c 3 = c The foci re ( 1 3, 3) nd F ( 1 + 3, 3), or ( 4, 3) nd F (, 3). Plot the vertices nd co-vertices. Drw smooth curve through the points. Lel the foci nd the grph. ( + 1) 5 + ( 3) 16 = 1 8 ( 1, 7) 4 ( 1, 3) V 1 ( 6, 3) V (4, 3) ( 4, 3) F (, 3) ( 1, 1) EXAMPLE 4 Writing n Eqution of n Ellipse Write n eqution in stndrd form for ech ellipse ) The coordintes of the centre re (, 1). The mjor is hs length of 16 units nd is prllel to the -is. The minor is hs length of 4 units. ) The coordintes of the centre re (, 5). The ellipse psses through the points ( 5, 5), (1, 5), (, ), nd (, 1). 8.5 The Ellipse MHR 67
10 SOLUTION ) The centre is (, 1), so h =, nd k = 1. The mjor is is prllel to the -is. The length of the mjor is is 16, so = 8. The length of the minor is is 4, so =. Sustitute known vlues into the generl formul for n ellipse whose mjor is is prllel to the -is. ( h) ( k) + = 1 ( ) 8 ( ( 1)) + = 1 ( ) + ( + 1) = ( + 1) The eqution of the ellipse is ( ) + = ) Plot the points ( 5, 5), (1, 5), (, ), nd (, 1). Drw smooth curve through the points. Lel the centre (, 5). The centre is (, 5) so h = nd k = 5. From the sketch of the ellipse, the mjor is is prllel to the -is, nd is 14 units in length, so = 7. The minor is is 6 units in length, so = 3. Sustitute known vlues into the generl formul for n ellipse whose mjor is is prllel to the -is. ( h) ( ( )) 3 ( + ) 9 ( k) + = 1 ( 5) + = 1 7 ( 5) + = 1 49 ( + ) 9 + ( 5) 49 ( 5, 5) = (, 1) 1 6 (, 5) (, ) (1, 5) The eqution of the ellipse is ( + ) ( 5) + = MHR Chpter 8
11 Note tht ellipses cn e grphed using grphing clcultor. As with circles, the equtions of ellipses must first e solved for. ( + 1) ( ) + = ( + 1) ( ) = ( + 1) + 9( ) = 36 9( ) = 36 4( + 1) ( ) = 1 9 (36 4( + 1) ) = ± 1 9 (36 4( + 1) ) = ± ( 4 + 1) Then, enter oth of the resulting equtions in the Y= editor. Y1 = ( 4 + 1) nd Y = ( 4 + 1) Adjust the window vriles nd use the Zsqure instruction. The window vriles include Xmin = 5.3, Xm = 5.3, Ymin =, nd Ym = 5. EXAMPLE 5 Lithotrips Lithotrips is eing used to provide n lterntive method for removing kidne stones. A kidne stone is crefull positioned t one focus point of the ellipticll shped lithotripter. Shock wves re sent from the other focus point. The reflective properties of the ellipse cuse the shock wves to intensif, destroing the kidne stone locted t the focus. Suppose tht the length of the mjor is of lithotripter is 60 cm, nd hlf of the length of the minor is is 5 cm. F Kidne Stone 8.5 The Ellipse MHR 69
12 ) Write n eqution of the semi-ellipse. Assume tht the centre is t the origin nd tht the mjor is is long the -is. ) How fr must the kidne stone e from the source of the shock wves, to the nerest tenth of centimetre? SOLUTION ) Drw digrm. The ellipse is centred t the origin with mjor is long the -is, so the eqution is of the form + = 1 The length of the mjor is is 60, so = 30. Hlf the length of the minor is is 5, so = 5. An eqution of the semi-ellipse is + = 1, cm F 5 cm or + = 1, ) Find the coordintes of the foci. The foci re ( c, 0) nd F (c, 0). = + c 30 = 5 + c = c 75 = c 75 = c 5 11 = c The foci re ( 5 11, 0) nd F (5 11, 0). The distnce etween the two foci is = 33. The kidne stone must e 33. cm from the source of the shock wves, to the nerest tenth of centimetre. 630 MHR Chpter 8
13 Ke Concepts An ellipse is the set or locus of points P in the plne such tht the sum of the distnces from P to two fied points nd F is constnt. The stndrd form of the eqution of n ellipse, centred t the origin, with > > 0, is either + = 1 (mjor is on the -is) or + = 1 (mjor is on the -is). The stndrd form of the eqution of the ellipse, with centre (h, k) nd with > > 0, is either ( h) ( k) + = 1 (mjor is prllel to the ( k) -is) or ( h) + = 1 (mjor is prllel to the -is). Communicte Your Understnding 1. If the ellipses ( + 3) ( ) + = 1 nd ( + 3) ( ) + = were grphed, wht fetures would e the sme? Wht fetures would e different? Eplin.. Stte whether ech of the following sttements is lws true, sometimes true, or never true for n ellipse. Eplin our resoning. ) The length of the mjor is is greter thn the length of the minor is. ) The ellipse is function. c) The ellipse hs infinitel mn es of smmetr. d) For the ellipse, + = 1, < 0 nd < Descrie how ou would use the locus definition of the ellipse to find n eqution of n ellipse with centre (0, 0), foci (0, 4) nd F (0, 4), nd with the sum of the focl rdii equl to Descrie how ou would sketch the grph of ( + ) ( + 4) + = The Ellipse MHR 631
14 Prctise A 1. Use the locus definition of the ellipse to write n eqution in stndrd form for ech ellipse. ) foci ( 3, 0) nd (3, 0), with sum of focl rdii 10 ) foci (0, 3) nd (0, 3), with sum of focl rdii 10 c) foci ( 8, 0) nd (8, 0), with sum of focl rdii 0 d) foci (0, 8) nd (0, 8), with sum of focl rdii 0. Sketch the grph of ech ellipse. Lel the coordintes of the centre, the vertices, the covertices, nd the foci. ) + = 1 ) + = c) + 16 = 64 d) 4 + = 36 e) = 5 3. For ech of the following ellipses, i) find the coordintes of the centre ii) find the lengths of the mjor nd minor es iii) find the coordintes of the vertices nd co-vertices iv) find the coordintes of the foci v) find the domin nd rnge vi) write n eqution in stndrd form ) ) Write n eqution in stndrd form for ech ellipse with centre (0, 0). ) The mjor is is on -is, the length of the mjor is is 14, nd the length of the minor is is 6. ) The length of the minor is is 6, nd the coordintes of one verte re ( 5, 0). c) The length of the mjor is is 1, nd the coordintes one co-verte re (5, 0). d) The coordintes of one verte re ( 8, 0), nd the coordintes of one focus re ( 55, 0). e) The coordintes of one focus re (0, 10 ), nd the length of the minor is is Sketch the grph of ech ellipse finding the coordintes of the centre, the lengths of the mjor nd minor es, nd the coordintes of the foci, the vertices, nd the co-vertices. ( + ) ( 3) ) + = ( 3) ( + 1) ) + = c) ( + 1) + 9( 3) = 36 d) 16( 3) + ( + ) = For ech of the following ellipses, i) find the coordintes of the centre ii) find the lengths of the mjor nd minor es iii) find the coordintes of the vertices nd co-vertices iv) find the coordintes of the foci v) find the domin nd rnge vi) write n eqution in stndrd form 63 MHR Chpter 8
15 ) ) 0 c) d) Write n eqution in stndrd form for ech ellipse. ) centre (, 3), mjor is of length 1, minor is of length 4 ) centre (3, ) nd pssing through ( 4, ), (10, ), (3, 1), nd (3, 5) c) centre ( 1, ) nd pssing through ( 5, ), (3, ), ( 1, 4), nd ( 1, 8) d) foci t (0, 0) nd (0, 8), nd sum of focl rdii 10 e) foci t ( 1, 1) nd (9, 1), nd sum of focl rdii 6 Appl, Solve, Communicte B 8. Whisper Chmer Sttur Hll, locted in the United Sttes Cpitol, hs ellipticl wlls. Becuse of the reflective propert of the ellipse, the hll is known s the Whisper Chmer. ) President John Quinc Adms desk ws locted t one of the focus points, nd he ws le to listen in on mn privte converstions. Where would the converstions hve to tke plce for Adms to her them? Eplin. ) Write n eqution of the ellipse tht models the shpe of Sttur Hll. Assume tht the length of the mjor is is 10 m nd the length of the semi-minor is (hlf the minor is) is 0 m. 9. Kepler s First Lw Johnnes Kepler ws phsicist who devised the three lws of plnetr motion. Kepler s First Lw sttes tht ll plnets orit the sun in ellipticl pths, with the centre of the sun t one focus. The distnce from the sun to plnet continull chnges. The Erth is closest 0 m F 10 m 8.5 The Ellipse MHR 633
16 to the sun in Jnur. The closest point, or perihelion, is km from the sun. The Erth is frthest from the sun in Jul. The frthest point, or phelion, is km from the sun. Write n eqution of the ellipse tht models the Erth s orit out the sun. Assume tht the centre of the ellipse is t the origin nd tht the mjor is is long the -is. Perihelion 10. Motion in spce Hlle s Comet orits the sun out ever 76 ers. The comet trvels in n ellipticl pth, with the sun t one of the foci. At the closest point, or perihelion, the distnce of the comet to the sun is km. At the furthest point, or phelion, the distnce of the comet from the sun is km. Write n eqution of the ellipse tht models the pth of Hlle s Comet. Assume tht the sun is on the -is. 11. Appliction The Erth s moon orits the Erth in n ellipticl pth. The perigee, the point where the moon is closest to the Erth, is pproimtel km from the Erth. The pogee, the point where the moon is furthest from the Erth, is pproimtel km from the Erth. The Erth is locted t one focus. Write n eqution of the ellipse tht models the moon s orit out the Erth. Assume tht the Erth is on the -is. 1. Show tht the eqution of the of the circle + = 49 cn e written in the stndrd form of n eqution of n ellipse. 13. Sputnik I The first rtificil Erth-oriting stellite ws Sputnik I, lunched into n ellipticl orit the USSR in If this orit is modelled with the centre of the ellipse t the origin nd the mjor is long the -is, then the length of the mjor is is 1180 km, nd the length of the minor is is 935 km. The Erth is t one focus. ) Write n eqution of the ellipse tht models the orit of the stellite. ) Wht is the closest distnce of the stellite to the Erth? c) Wht is the furthest distnce of the stellite from the Erth? 14. Coin set The twelve Hopes nd Aspirtions Cndin millennium qurters cn e purchsed in n ellipticll shped souvenir set. ) If the ellipse hs vertices (0, 7) nd (30.94, 7), nd co-vertices (15.47, 0) nd (15.47, 14), write n eqution of the ellipse. ) Wht is the length of the mjor is? Sun Erth Aphelion 634 MHR Chpter 8
17 c) Wht is the length of the minor is? d) Sketch scle digrm of the souvenir set. 15. Spotlight When spotlight shines on stge, the spotlight illumintes n re in the shpe of n ellipse. Assume tht one focus of the ellipse is ( 1, 5), nd the sum of the focl rdii is 6. Write the eqution of the ellipse if the mjor is is prllel to the -is. 16. Jupiter Like ll plnets, Jupiter hs n ellipticl orit, with the centre of the sun locted t focus. The digrm gives the pproimte minimum nd mimum distnces from Jupiter to the sun, in millions of kilometres. Write n eqution of the ellipse tht models Jupiter s orit round the sun. Assume tht the centre of the sun is on the -is. 17. Covered entrnce A semi-ellipticl covering is to e uilt over n 8-m-wide rod nd the -m-wide sidewlks on either side of it tht led to n rts centre. If there is mimum clernce of 5 m over the rod, wht will e the minimum clernce over the rod, to the nerest hundredth of metre? 18. Technolog Use grphing clcultor to grph ech ellipse. ( ) ( + 1) ) + = 1 ( + 3) ( + 1) ) + = 1 ( 3) c) + = Communiction ) Use grphing clcultor to grph the fmil of ellipses + = 1 for = 1,, nd 3. 5 ) Grph the fmil for = 1, 1 3, 1 4. c) How re the grphs like? How re the different? d) Wht hppens to the ellipses s gets closer to 0? 0. Distorted circle An ellipse cn e thought of s distorted circle. ) Sketch the grphs of the circle + = 1 nd the ellipse + = on the sme grid. ) B wht fctor hs the circle epnded horizontll to form the ellipse in prt )? Sun 5 m Jupiter m rod 8 m m 8.5 The Ellipse MHR 635
18 c) B wht fctor hs the circle epnded verticll to form the ellipse in prt )? d) Sketch the grphs of the circle + = 1 nd the ellipse + = on the sme grid. e) B wht fctor hs the circle epnded horizontll to form the ellipse in prt d)? f) B wht fctor hs the circle epnded verticll to form the ellipse in prt d)? g) How cn ou recognize the horizontl nd verticl stretch fctors from the eqution of the ellipse? C 1. Eccentricit Ellipses cn e long nd nrrow, or nerl circulr. The mount of elongtion, or fltness, of n ellipse is mesured numer clled the eccentricit. To clculte the eccentricit, e, use the formul e = c. ) Clculte the eccentricit of ech of the following ellipses. Round nswers to the nerest hundredth. i) + = 1 ii) + = 1 iii) + = 1 iv) circle ) Inquir/Prolem Solving Find the gretest nd lest possile eccentricities for n ellipse. Eplin our resoning.. Stndrd form Consider n ellipse with its mjor is long the -is, foci t ( c, 0) nd (c, 0), -intercepts t (, 0) nd (, 0), nd -intercepts t (0, ) nd (0, ). ) Use the distnce formul to show tht for point (, ) on the ellipse, ( c) + + ( + c) + =. ) Isolte one rdicl term in the eqution nd derive + = 1. c c) Derive + = 1, the stndrd form for the ellipse with its mjor is long the -is. A CHIEVEMENT Check Knowledge/Understnding Thinking/Inquir/Prolem Solving Communiction Appliction The roof of n ice ren is in the form of semi-ellipse. It is 100 m cross nd 1 m high. Wht is the length of stilizing em 5 m elow the roof? 636 MHR Chpter 8
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 information10.2 The Ellipse and the Hyperbola
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
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 rel-life problems. Clssif
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 information8.2: CIRCLES AND ELLIPSES
8.: CIRCLES AND ELLIPSES GEOMETRY OF AN ELLIPSE Geometry of n Ellipse Definition: An ellipse is the set of ll points in plne whose distnce from two fixed points in the plne hve constnt sum. Voculry The
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 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 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 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 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 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 informationSECTION 9-4 Translation of Axes
9-4 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 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 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 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 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 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 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 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 non-lgebric functions. The re clled trnscendentl functions. The eponentil
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 informationChapter 9: Conics. Photo by Gary Palmer, Flickr, CC-BY, https://www.flickr.com/photos/gregpalmer/
Chpter 9: Conics Section 9. Ellipses... 579 Section 9. Hperbols... 597 Section 9.3 Prbols nd Non-Liner Sstems... 67 Section 9.4 Conics in Polr Coordintes... 630 In this chpter, we will eplore set of shpes
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 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 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 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 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 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 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 informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
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 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 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 information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More information(i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation
P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se
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 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 informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
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 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 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 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 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 informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
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 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 informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level 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 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 TI-8 or TI-8 grphing clcultor. Whenever possile, the questions on
More informationLesson-5 ELLIPSE 2 1 = 0
Lesson-5 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 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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationBelievethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
More informationMath Sequences and Series RETest Worksheet. Short Answer
Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning
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 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 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 informationSESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)
Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
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 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 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 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 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 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 information2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
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 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 informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
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 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 cross-section or slice. In this section, we restrict
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 informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
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 informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More informationNAME: MR. WAIN FUNCTIONS
NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors
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 informationTime : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, 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 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 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 nth-order
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 informationFunctions and transformations
Functions nd trnsformtions A Trnsformtions nd the prbol B The cubic function in power form C The power function (the hperbol) D The power function (the truncus) E The squre root function in power form
More informationMCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x
MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If
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 informationExponents and Logarithms Exam Questions
Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the -intercept of the function ln y.. -1 b. 0 c. d.. Which eqution is represented
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 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 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 information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
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 first-degree eqution, solve for the vrible. The grph of this solution will be single point
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
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 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 informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
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 informationEllipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION
FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied
More information( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).
Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.
More information