When e = 0 we obtain the case of a circle.

Transcription

1 3.4 Conic sections Circles belong to specil clss of cures clle conic sections. Other such cures re the ellipse, prbol, n hyperbol. We will briefly escribe the stnr conics. These re chosen to he simple equtions, n ll other conics re rints on them. Our stnr conics re ll symmetric bout the origin n the -is. Thus the stnr circle is + y which cn be written prmetriclly s There re rious wys to efine conic sections, for emple s the cures rising from ifferent slices through cone. Ech shres the property tht: The istnce of ech point on the cure from fie point the focus) n fie stright line the irectri) is constnt rtio e the eccentricity). For circles, this hs to be interprete with cre.) The ifferent clsses correspon to ifferent rnges of the lue of e. cos θ y sin θ. nton Co City Uniersity) S05 Week 4 utumn 007 / 39 nton Co City Uniersity) S05 Week 4 utumn 007 / 39 There re two foci, t e, 0) e, 0) e /e The stnr hyperbol is gien by /e e n two irectrices, e e. We cll the shortest istnce between the two sections of the cure the mjor is, which equls. This cure hs symptotes y b y b y b with b e ) for some e >. n prmetric eqution sec θ y b tn θ. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 e e b There re two foci, t n two irectrices, e, 0) e, 0) e e. /e The stnr ellipse is gien by + y b with b e ) for some 0 < e <. /e The mimum wih of the cure long the -is is clle the mjor is, which equls, n long the y-is is clle the minor is, which equls b. Note tht the stnr ellipse is chosen such tht the mjor is is longer thn the minor is. This cure hs prmetric eqution cos θ y b sin θ. When e 0 we obtin the cse of circle. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 The stnr prbol is gien by y 4 n hs eccentricity e. There is one focus, t one irectri,, 0) n n is t y 0. This cure hs prmetric eqution t y t n the grient of the cure t t, t) is t. We cn nlyse generl conics by using chnge of rible to conert them into the stnr forms. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39

2 Emple 3.4.: Determine the foci n irectices of the ellipse ) 5 y + 3) +. 6 We compre with X + Y b. To trnsform in this wy we must he X Y y b 4. lso b e ) implies tht e 3 5. Therefore the centre of the ellipse is t, 3), the mjor is hs length 0 n the minor is hs length b 8. The foci lie on the mjor is t istnce e 3 from the centre. So the foci re 5, 3), 3). Directrices re perpeniculr to the mjor is n t istnce e 5 3 from the centre. So the irectrices re nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / Clculus I: Differentition Emple 3.4.: n ellipse hs foci t, 5) n 8, 5) n eccentricity e 4. Fin its Crtesin eqution. The centre is miwy between the foci, so lies t 5, 5). The istnce from the centre to ech focus is e 3,. Therefore 4. The eritie of function Suppose we re gien cure with point lying on it. If the cure is smooth t then we cn fin unique tngent to the cure t : b e ) 35. b c From this we see tht the eqution is gien by 5) 44 y 5) Here the cure in ) is smooth t, but the cures in b) n c) re not. nton Co City Uniersity) S05 Week 4 utumn 007 / 39 nton Co City Uniersity) S05 Week 4 utumn 007 / 39 If the tngent is unique then the grient of the cure t is efine to be the grient of the tngent to the cure t. The process of fining the generl grient function for cure is clle ifferentition. Consier the chor B. s B gets closer to, the grient of the chor gets closer to the grient of the tngent t. B For y f ), the grient function is efine by ) δy ) f + ) f ) lim lim ,y+ δy),y) We enote the grient function by or f ), n cll it the eritie of f. This is not the forml efinition of the eritie, s we he not epline ectly wht we men by the limit s 0. But this intuitie efinition will be sufficient for the bsic functions which we consier. δy nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4..: Tke f ) c, constnt function. t eery the grient is 0, so f ) 0 for ll. Or f + ) f ) Emple 4..: Tke f ). c c 0. t eery the grient is, so f ) for ll. Or f + ) f ) + ). Emple 4..3: Tke f ). Now we nee to consier the secon formultion, s we cnnot simply re the grient off from the grph. f + ) f ) + ) + + ) + ) +. The limit s tens to 0 is, so f ). nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39

3 Emple 4..4: Tke f ). f + ) f ) + ) + ) ) + ) ) + ) + ). The limit s tens to 0 is, so f ). Emple 4..5: Tke f ) n with n N n n >. Recll tht n b n b) n + n b + n 3 b + + b n ) n b n b n + n b + n 3 b + + b n where the sum hs n terms. s b we he n b n ) lim lim n + n b + n 3 b + + b n ) nb n. b b b If + n b then ) f + ) f ) lim lim 0 b n b n b ) nb n n n. Hence f ) n n. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4..6: f ) sin. We use the ientity for sin + sin B. f + ) f ) sin f + ) f ) sin ) ) cos + ) cos + ). We nee the following fct which we will not proe here): sin θ lim θ 0 θ f sin ) ) lim cos + ) cos). 0 Some stnr erities, which must be memorise: f ) f ) k k k e e ln sin cos cos sin tn sec cosec cosec cot sec sec tn cot cosec Some of these results cn be erie from the results in the following sections, or from first principles. Howeer it is much more efficient to know them. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / Differentition of compoun functions Once we know few bsic erities, we cn etermine mny others using the following rules: Let u) n ) be functions of, n n b be constnts. Function Deritie Sum n ifference u ± b u ± b Prouct u u + u u Quotient u u u Composite u)) z. z where z ). The finl rule boe is known s the chin rule n hs the following specil cse u + b) u + b) Emple 4..: Differentite Emple 4..: Differentite y y +. + ) ) 4 + ) + ). For emple, the eritie of sin + b) is cos + b). nton Co City Uniersity) S05 Week 4 utumn 007 / 39 nton Co City Uniersity) S05 Week 4 utumn 007 / 39 Emple 4..3: Differentite y ln + 3). ln + 3) Emple 4..4: Differentite y e 5. Set z 5, then z z ez 5 5e 5. Emple 4..5: Differentite y 4 sin + 3). Set z + 3, then z 4 cosz) 8 cos + 3). z s we he lre note, some of the stnr erities cn be euce from the others. Emple 4..6: Differentite y tn sin cos. cos cos sin sin ) cos cos + sin cos cos sec. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39

4 Emple 4..7: y cosec sin. sin.0). cos sin cos sin cosec cot. Emple 4..8: y ln + + ), i.e. y ln u where u + +. u u n so ) u + + ). ) Emple 4..9: y. We he y e ln ) e ln ), i.e. y e u where u ln. u eu e ln ln) + ) ln) + ). 4.3 Higher erities The eritie is itself function, so we cn consier its eritie. If y f ) then we enote the secon eritie, i.e. the eritie of with respect to, by y or f ). We cn lso clculte the higher erities n y or f n) ). n nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4.3.: y ln + ). Let z. + y z + ). ) + ) ) + ). Emple 4.3.: Show tht y e sin) stisfies y + + 5y 0. e sin + e cos e cos sin ) y e cos sin ) + e 4 sin cos ) e 3 sin 4 cos ). Writing s for sin n c for cos we he y + y + 5y e 3s 4c s + 4c + 5s) 0. Emple 4.3.3: Elute ) 3 + ) + 3) t 0. We coul use the quotient rule, but this will get complicte. Inste we use prtil frctions. y ) + 3) + + B + ) + C + 3. We obtin check!) 0, B, n C 3. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Now 4 + ) ) y + ) ) 3 Generlly it is hr to gie simple formul for the nth eritie of function. Howeer, in some cses it is possible. The following cn be proe by inuction. Emple 4.3.4: y e. 3 y ) ) 4 n substituting 0 we obtin tht 3 y 0) We cn show tht e n n y n n e. y e. nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4.3.5: y sin). We cn show tht y cos) sin + π ) y sin) sin + π) y 3 cos) 3 sin + 3π ) y i) 4 sin) 4 sin + π). n y n n sin + nπ ). 4.4 Differentiting implicit functions Sometimes we cnnot rerrnge function into the form y f ), or we my wish to consier the originl form nywy for emple, becuse it is simpler). Howeer, we my still wish to ifferentite with respect to. Gien function gy) we he from the chin rule gy)) gy)). nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39

5 Emple 4.4.: + 3y y 4. Therefore we he + 3y y 4 ) ) y ) y 4 ) 0 + 3y + 3 y ) 4y 3 + 3y + 6y 3 4y 0 0. Emple 4.4.: + 3 y. Therefore we he + 3 y ) ) ) 3 y y nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / Differentiting prmetric equtions Sometimes there is no esy wy to epress the reltionship between n y irectly in single eqution. In such cses it my be possible to epress the reltionship between them by writing ech in terms of thir rible. We cll such equtions prmetric equtions s both n y epen on common prmeter. Emple 4.5.: t 3 y t 4t +. lthough we cn write this in the form y the prmetric ersion is esier to work with. To ifferentite prmetric eqution in the rible t we use Emple 4.5.: Continue.) t 4 n t 4 3t.. 3t nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4.5.: Fin the secon eritie with respect to of We he Therefore Now y sin θ y cos θ. θ cos θ sin θ. θ sin θ 4 sin θ. cos θ ) 4 sin θ) θ 4 cos θ 4 sin θ) 4. θ cos θ Note: The rules so fr my suggest tht erities cn be trete just like frctions. Howeer y y t in generl. Moreoer y ). nton Co City Uniersity) S05 Week 4 utumn / 39 nton Co City Uniersity) S05 Week 4 utumn / 39 Emple 4.5.: Continue.) We he n Therefore θ y 4 sin θ 8 sin θ cos θ θ ) θ ) θ θ sec θ) sec θ tn θ. y θ θ 8 sin θ cos θ sec θ tn θ 8 tn θ 4 y. nton Co City Uniersity) S05 Week 4 utumn / 39