Bai Tap Hinh Hoc Vi Phan

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Phptnhvi phntrn Rn1BITPCHNG1Bi tp1.1. Chohmf: R2 R,(x, y) sin x.DngnhnghachngminhDf(a, b) = ,vixcnhbi(x, y) = (cos a)x.Bitp1.2. Chohmf: RnRthamniukin[f(x)[ |x|2.Chngminhfkhvitix = 0vDf(0) = 0.Bitp1.3. Chohmf: R2Rxcnhbi:f(x, y) =

x[y[(x2+y2)2, nu (x, y) = (0, 0)0 nu (x, y) = (0, 0)(a) TnhD1f(0, 0)vD2f(0, 0).(b) Chngminhfkhngkhviti(0, 0).Bitp1.4. Tmohmcaccnhxsau:(a) f(x, y, z) = xy, x > 0.(b) f(x, y, z) (xy, x2+z), x > 0.(c) f(x, y) = sin(x sin y).(d) f(x, y) = (sin(xy), sin(x sin y), xy), x > 0.Bitp1.5. Sdngvdf(x) =

x2+x2sin

1x

, x = 00 x = 0Chngtrngiukinlintctrongnhlhmngckhngthbc.Bitp1.6. Cho hm glin tc trn ng trn n v S1tha mn iu kin

g(0, 1) = g(1, 0) = 0g(x) = g(x)2 Bitpchng1Xthmf: R2Rxcnhbi:f(x) =

|x|g

x|x|

, x = 00, x = 0vimix R2.(a) Chngminhvix R2cnhchotrc,hmsh : R R, h(t) = f(t, x)khvitrn R.(b) Chngminhfkhngkhviti(0, 0)trkhihmg= 0.Bitp1.7. Cho hm f: R2R kh vi lin tc. Chng minh rng fkhngthlnnh.Bi tp1.8. Chof: Rn Rm,g: Rm RkhvilpC.Chngminhrng(g f)= g f.Bi tp1.9. ChoL : Rn Rmlmtnhxtuyntnh,chngminhrngLlintc,khvitimiimx Rn.Bitp1.10. Chngminhrngphptnhtuynvphpvttrn Rnlccnhxlintc.Bi tp1.11. ChoUlmttpmtrong Rnvf : U Rm, m nlmtnhxthuclpC1.Gisrngflmtnnhvf1: A U,viA = f(U)cngthuclpC1.Chngminhrngmkhngthnhhnn.(ylmtnhlyucaBrouwer:Khngtnti1ngtmttpmU Rnvo Rmvim < n).Bi tp1.12. Chof: Rn Rnlmtnhxkhvi, chnhqui trn Rn,chngminhrngflmtnhxm.Bitp1.13. ChngminhrngiukincnvmtnhxtrnFlmtviphitWvoF(W)lFlmtnnhvDFkhngcimkdtrnW.Bi tp1.14. Chngminhrngkhngtnti 1vi phi tmttpmcaRnvomttpmca Rmnum < n.Lthuytng 3BITPCHNG2Bitp2.1. Hyxcnhvtcaccngthamssau:(a) (nghnhs8),xcnhbic(t) = (sin t, sin 2t)(b) (ngcubic),xcnhbic(t) = (t, t2, t3)Bi tp2.2. Tmmtngthams(t)mvtlngtrnx2+ y2= 1saocho(t)chyquanhngtrncngchiukimnghv(0) = (1, 0).Bi tp2.3. Chongtrnthams(t)khngiquagc.Gis(t0)limtrnvtcagnvi gctanht. Hychngminhrngvector(t0)trcgiaovivector

(t0).Bi tp2.4. Gis(t)lngthamsm

(t) = 0vimit.Chngtacthktlungv(t)?Bitp2.5. Chongthams : I R3v v lvectorcnh.Gisrng

(t0)trcgiaovi v vimit Iv(0)cngtrcgiaovi v .Chngminhrngvimit I,(t0)trcgiaovi v .Bi tp2.6. Chongthams: I R3, vi

(t) =0, t I. Hychngminhrng [(t)[=a(alhngskhckhng)khivchkhi(t)trcgiao

(t)vimit I.Bitp2.7. Vtcaccngthamssaunmtrnnhngmtquenthucno.(a) c : t

at cos t ,atsint ,a2t22

(b) c : t (sin2t , 1 cos 2 t ,2 cos t)Bi tp2.8. Hychngminhrngcctiptuyncangthams(t) =

3 t ,3 t2,2 t3

tomtgckhngivingthngcnhy= 0;z= x.Bi tp2.9. Mtatrnbnknh1trongmtphngOxylnkhngtrtdctheotrcOx.KhimtimnmtrnbincaavchramtngconggilngCycloid(Hnh2.0.1).(a) HytmmtthamshocangCycloidvhyxcnhccimkd.4 Bitpchng2Hnh2.0.1:ngcycloid(b) TnhdimtcangCycloid(ngvimtvngquaycaa).Bitp2.10. Tnh di ca cc ng tham s phng sau trn on [A, B](a) c : t

t ,t2

(b) c : t (t ,lnt)(c) c : t

t ,coshta

(d) c : t (asin t ,a(1 cos t)) a > 0(e) c : t

a (lntant2+ cos t) , asint

a > 0.Bitp2.11. Tnhdicaccngthamssau:(a) c :

a(t sint) , a (1 cos t) , 4 a cost2

, giahai giaoimcangvimtphngy= 0;(b) c : t

cos3t ,sin3t ,cos2t

mtvngkhpkn;(c) c : t (acosht ,asinht ,at),trongkhong[0, b];Bitp2.12. Tnhdicaphnngcong.

x3= 3a2y2xz= a2giahaimtphngy= a/3vy= 9a,via > 0.Bi tp2.13. ChoOA=2a,a>0lngknhcangtrn(S), haingOyvAV lhaitiptuynca(S)tiOvA.TiaOrctngtrn(S)tiCvAV tiB.TrnOBlyimPsaochoOP=CB.Nutaquaytia Orquanh im Oth cc im Pv nn ng cong gi l ng xixit caDiocles(cissoidofDiocles).ChnOAlmtrchonhvOyltrctung.HyLthuytng 5chngminhrng(a) Vtcang(t) =

2at21 +t2,2at31 +t2

, t RlngxixitcaDiocles(t = tan xemHnh2.0.2)Hnh 2.0.2: ng xixit ca Diocles(cissoidofDiocles)Hnh2.0.3:ngTractrix(b) GctaO(0, 0)limkdcangxixit.(c) Khi t th ngcongdnvngthngx=2av

(t) (0, 2a).Do,khit thngcongvtiptuyncandnvngthngx=2a. Tagi ngthngx=2alngtimcn(asymptote)cangxixit.6 Bitpchng2Bitp2.14. Cho : (0 ,) R2cxcnhbithams(t) =

sint , cos t + lntan

t2

(2.0.1) y t l gc gia trc Oy vi vector

(t). Vt ca c gi l ng tractrix.(Hnh2.0.3).Hychngminhrng:(a) lngthamskhvi,chnhquingoitrt = /2.(b) Khong cch t tip im n giao im ca tip tuyn vi trc Oylunbng1.Bitp2.15. Chongthams : (1 ,+) R3xcnhbi:(t) = (3at1 +t3,3at21 +t3) (2.0.2)Chngminhrng:(a) Tit = 0,

tipxcvitrcOx.(b) Khit ,th(t) (0, 0)v

(t) (0, 0).(c) Lyngcongvi hngngcli. Khi nut 1. ngcongvtiptuyncantintingthngx + y + a = 0.Hpca2ngvamtl1ngixngquangthngy=xvcgillDescartes(foliumofDescartes)(Hnh2.0.4)Hnh2.0.4:LDescartesLthuytng 7Bitp2.16. Chongthams(t) = (aebtcos t, aebtsin t),t Ravblhngs,a > 0, b < 0.(a) Hy chng t rng khi t , th (t) tin dn ti gc Ov xon quanhgcO,vthvtcan(Hnh2.0.5)cgilngxonlogarithm(loga-rithmicSpiral).(b) Hychngtrng

(t) (0, 0)khi t v limtt

t0[

(t)[dtlhuhn;nghalcdihuhntrnon[t0, ).Hnh2.0.5:ngxonlogarithmBi tp 2.17. Cho : I R3l mt ng cong n, lin tc (thuc lp C0).Chngtani rngctiptuynyu(weaktangent)ti t0nungthngxc nh bi (t0+h) v (t0) c cng mt v tr ti hn khi h 0. Chng ta nirng c tip tuyn mnh (strong tangent) ti t = t0nu ng thng xc nhbi (t0+h) v (t0+k) c cng mt v tr ti hn khi h, k 0. Chng t rng:(a) ngthams(t)=(t3, t2),t R,ctiptuynyunhngkhngctiptuynmnhtit = 0.(b) Nu ng tham s : I R3thuc lp C1v chnh qui ti t = t0khictiptuynmnhtit = t0.8 Bitpchng2(c) ngthamschobi(t) =

(t2, t2) nu t 0(t2, t2) nu t 0thuclpC1nhngkhngthuclpC2.Hyvphcthongcongvccvcttipxccan.Bitp2.18. (on thng l ngn nht). Cho c : I R3l ng tham s,ly[a, b] Ivt(a) = p,(b) = q.(a) Hychngtrngvimivcthng,nv v ([ v [ = 1),talunc(q p). v =

ba

(t). v dt

ba[

(t)[dt.(b) t v =p q[p q[vchngminhrng[(b) (a)[

ba[

(t)[dt.Cnghalcungcdingnnhtnipvqlonthng.Bi tp2.19. Chngminhrngngthamschnhquiphngvithamsdi cungccongk=const>0khi vch khi vtcanlmtngtrn(hoclmtphncangtrn).Bi tp2.20. XcnhtrngmctiuFrenetvtmcong, xontiimtucaccngthamssau:(a) c(t) = (t2, 1 t, t3)(b) c(t) = (a cosh t, a sin t, at)(c) c(t) = (et, et,2t)(d) c(t) = (cos3t, sin3t, cos 2t)(e) c(t) = (2t, ln t, t2)Bitp2.21. Chongthams(s) =

a cos sc, a sin sc, bsc

, s RLthuytng 9vic2= a2+b2.(a) Chngminhrngthamssldicung.(b) Xcnhhmcongvxonca(s).(c) Xcnhmtphngmttipca(s).(d) Chng minh rng ng php tuyn n(s) v i qua (s) ct trc Oztheomtgcbng/2.(e) ChngminhrngtiptuyncatovitrcOzmtgckhngi.Bi tp2.22. Tmccimtrnngthamsc(t)=

a(t sin t), a(1 cos t), 4a cost2

, t R,mtibnknhcongtcctraphng.Bitp2.23. Chng minh rng nu mt phng php din ca ng tham ssongchnhquitrong R3timiimuchamtvectorcnhthcungcholngphng.Bitp2.24.(a) Mtngthamschnhquylinthngphngc(t)ctnhchtlmitiptuynluni quamtimcnh. Chngminhrngvtcalmtngthnghocmtoncangthng.(b) Chngminhrngnuvectortrngphpcamtngthamssongchnh qui trong R3ti mi im l mt vector c nh th cung cho l ngphng.Bitp2.25. Vit phng trnh mt phng tip xc, mt phng php v mtphngmttipcangcongc(t) = (t3t31, t2, t2t)tiimc(2).Vitphngtrnhmtphngtipxc,mtphngphpvmtphngmttipcangcongc(t) = (t2t31, t2+t, t2t)tiim

258, 2, 94

.Bitp2.26. Chongthams(helix)c(t) = (a cos t, a sin t, bt),a > 0,b = 0.10 Bitpchng2(a) Hyvitphngtrnhtiptuyn, phptuynchnh, trngphptuyn,mtphngmttip,mtphngtrcctimtimtu.(b) Chngminhrngcctiptuyncannghingmtgckhngivimtphngz= 0,cnccphptuynchnhcttrcOz.Bi tp2.27. Chngtrngcthangthamsc : 'a, b` Rn,via, b R,vngthamstngng : '0, 1` Rn.Bi tp 2.28. Cho c : I R3,t (t, f(t), g(t)), vi f(t), g(t) l cc hm trn,lmtngthams.(a) Chngminhrngclngthamschnhqui.(b) Tmvectortipxccactrongtrnghpf(t)=sin t + t2vg(t)=et(1 t3).Bitp2.29. (iu kin cn v ng tham s nm trn mt mt cu).Gislngcongc = 0vk

= 0.ChngminhrngiukincnvvtcanmtrnmtmtculR2+ (R

)2T2= constyR = 1/k,T= 1/vR

lohmcaRtheos.Bitp2.30. (iu kin cn v ng tham s nm trn mt mt cu).Cho : I R3llngthamssongchnhquivithamsdicung.Gis = 0vk > 0(a) ChngminhrngnuC= c(I)nmtrnmtcua,bnknhr.thc a = 1k . n

1k

/.1.bTysuyrar2=1k2+

1k

/1

2(b) Ngc li, nu1k2+

1k

/1

2= const > 0 th C= c(I) nm trn mtmtcu.Bitp2.31. ChngtrngccngthamshasaukhngtngngLthuytng 11(a) c1(t) = (t, 1 t), t (0, 1);(b) c2(t) = (cos t, sin t), t (0, /2);(c) c3(t) = (t, 1 t2), t (0, 1).Bi tp2.32. Chngminhrngngcongtrongkhnggianctiptuynto vi mt ng thng c nh mt gc khng i khi v ch khi t s gia xonvcongtimtimtylhngs.Bi tp2.33. Cho:I R3lmtngcongthamshatnhinccongk(s) > 0, s I.GiPlmtphngthahaiiukinsau:(a) Pchattccctiptuyncactis0;(b) Vi mi lncnJ I cas0, luntnti nhngimcac(J)nmtrongP.ChngminhrngPlmtphngtipxccactis0.Bitp2.34. Trongtrnghptngqut,mtngthamscgilmthelix(xonc)nucctiptuyncatomtgckhngi vi mtphngcnh.Gisrng = 0,chngminhrng:(a) lmtngxoncnuvchnuk/lmthmhng.(b) l mt ng xon c nu v ch nu cc ng php tuyn ca songsongvimtmtphngcnh.(c) lmtngxoncnuvchnuccngtrngphptuyncatomtgckhngivimtphngcnh.Bi tp2.35. Cho:I R3lmtngcongthamshatnhinccongk(s) > 0, s I.Chngminhrng(a) Mt phng tip xc ca c ti s0chnh l gii hn ca cc mt phng qua3imc(s0),c(s0 +h1),c(s0 +h2)khih1, h2 0.(b) Gii hncaccngtrni qua3imc(s0), c(s0 + h1), c(s0 + h2)l mt ng trn nm trong mt phng tip xc ca c ti s0, c tm nm trnphp tuyn ti s0ca c v bn knh bng 1/k(s0). ng trn ny gi l ngtrnmttip(osculatingcircle)cactis0.Bi tp2.36. Chngminhrngdi cangcong, congvxonlcckhinimEuclide(tclnbtbinquaphpbiningc).12 Bitpchng2Bitp2.37. Gi s rng tt c cc php tuyn ca mt ng tham s chnhqui phng lun i qua mt im c nh. Chng minh rng ng l mt ngtrnhocmtphncangtrn.Bi tp 2.38. Tm cc ng tham s song chnh qui ca R3m cc mt phngmttipthamnmttrongcciukinsau:(a) Vunggcvimtphngcnh;(b) Songsongvi mtngthngcnhvtiptuynkhngsongsongvingthng;(c) iquamtimcnhvcctiptuyniquaim.Bi tp2.39. Chngminhrngcctnhchtsaucaccngsongchnhquinhhngtrong R3ltngng:(a) Tiptuyntomtgckhngiviphngcnh;(b) Phptuynchnhsongsongvimtmtphngcnh;(c) Trngphptuyntomtgckhngi vi mtphngcnh(viiukinxonkhckhngtimiim);(d) Tsgiacongvxonlmthmhng.Bi tp2.40. Mtngthamschnhqui phngctnhchtmi tiptuyn lun i qua mt im c nh. chng minh rng vt ca n l mt ngthnghocmtoncangthng.Bitp2.41. Xc nh ng tc b v ng thn khai ca cc ng thamsphngsa