Www.mathVN.com Su Dung Phan Mem Ho Tro Day Hoc Toan
of 142
-
Upload
nguyen-thai-trung -
Category
Documents
-
view
1.046 -
download
0
Transcript of Www.mathVN.com Su Dung Phan Mem Ho Tro Day Hoc Toan
www.MATHVN.com
I HC THI NGUYN TRNG I HC S PHM==========================
TRNH THANH HI
GIO TRNH S DNG PHN MM H TR DY HC TON
Thi Nguyn, 2005
www.mathvn.com
www.MATHVN.com
MC LCMC LC ...........................................................................................................................1 LI NI U .....................................................................................................................2 Chng 1:.............................................................................................................................3 NG DNG CNG NGH THNG TIN V TRUYN THNG TRONG DY HC NH TRNG PH THNG .......................................................................................3 1.1. Tc ng ca CNTT- TT ti s pht trin ca x hi ..............................................3 1.2. Nh trng hin i trong bi cnh pht trin ca CNTT- TT ................................3 1.3. ng dng CNTT-TT trong nh trng Vit Nam ................................................9 1.4. Tc ng ca CNTT- TT trong dy hc ton.....................................................10 Chng 2............................................................................................................................19 S DNG PHN MM GRAPH TRONG DY HC TON ......................................19 2.1. Gii thiu v phn mm Graph .............................................................................................19 2.2. Lm vic vi Graph .................................................................................................................19 2.3. Gii thiu h thng Menu.......................................................................................................20 2.4. Mt s chc nng c bn ........................................................................................................21 2.5. Th vin cc hm ca Graph .................................................................................................25 2.6. Khai thc phn mm Graph ...................................................................................................26 2.7 Bi tp: ..........................................................................................................................................27 Chng 3............................................................................................................................28 S DNG PHN MM HNH HC NG CABRI GEOMETRY..............................28 3.1. Tng quan v phn mm hnh hc ng Cabri Geometry .............................................28 3.2. Thao tc vi cc cng c ca Cabri Geometry..................................................................32 3.3. Vit ho giao din ca Cabri Geometry ..............................................................................48 3.4. S dng phn mm Cabri Geometry h tr dy hc .......................................................48 3.5. Phng php khai thc phn mm Cabri Geometry h tr dy hc ton ..................61 Chng 4............................................................................................................................68 S DNG PHN MM MAPLE TRONG DY HC TON ......................................68 4.1. Tng quan chung v phn mm Maple ...............................................................................68 4.2. S dng cc lnh n gin ca Maple ................................................................................74 4.3. S dng cc cu lnh ca Maple h tr dy hc kho st hm s ..............................84 4.3.1. Nhng cu lnh ca Male h tr dy hc kho st hm s .......................................84 4.4. Cc cu lnh ca Maple h tr gii cc bi ton gii tch. ............................................98 4.5. Nhm cc lnh ca Maple h tr dy hc i s tuyn tnh .......................................102 4.6. Khai thc cc th vin ca Maple trong dy hc ton ..................................................119 Ngun ti liu gio trnh trch dn, tham kho ..........................................................141
1 www.mathvn.com
www.MATHVN.com
LI NI UHin nay chng ta ang chng kin s pht trin nh v bo ca cng ngh thng tin v truyn thng (ICT). Cc nh khoa hc khng nh: cha c mt ngnh khoa hc v cng ngh no li pht trin nhanh chng, su rng v c nhiu ng dng nh tin hc. S ra i ca Internet, n m ra mt k nguyn mi: k nguyn thng tin. Trong khung cnh , o to v gio dc c coi l mnh t mu m cho cc ng dng ca tin hc pht trin. Theo cc chuyn gia, trong giai on ti s c thay i su sc trong cng ngh o to v gio dc nh c tin hc v Internet. Nhng cng ngh tin tin ca tin hc nh Internet, a phng tin, truyn thng bng rng, CD-Rom, DVD s mang n nhng bin i c tnh cch mng trn quy m ton cu trong lnh vc o to, gio dc. Vi mc tiu nng cao cht lng o to, i mi phng php dy hc th mt trong cc bin php kh thi l bit kt hp cc phng php dy hc truyn thng v khng truyn thng trong c s dng CNTT-TT ni chung, phn mm ni ring nh mt cng c c lc. Vi mc tiu khim tn l cung cp nhng thng tin ban u bn c c th khai thc cc phn mm vo cng vic ging dy, hc tp v nghin cu ton hc, chng ti mnh dn bin son gio trnh: S DNG PHN MM H TR DY HC TON Gio trnh c bin son trc mt s l ti liu hc tp cho sinh vin chuyn ngnh ton; tin sau c th lm ti liu tham kho cho gio vin THPT v hc vin cao hc v nhng ngi quan tm n vic khai thc cc phn mm ton. y l mt cng vic mi m v qu ti i chng ti nn khng th trnh c sai st. Rt mong nhn c s ng gp kin ca bn c, c bit l cc Thy, C gio v cc em hc sinh, sinh vin- y s l ngun thng tin qu gi chng ti hon thin ti liu ny. Chng ti xin trn trng cm n.
2 www.mathvn.com
www.MATHVN.com
Chng 1:NG DNG CNG NGH THNG TIN V TRUYN THNG TRONG DY HC NH TRNG PH THNG1.1. Tc ng ca CNTT- TT ti s pht trin ca x hi Trong nhng nm gn y, loi ngi c chng kin mt k nguyn gn lin vi s pht trin nhanh chng ca CNTT-TT. Internet, cng ngh truyn thng a phng tin (Multimedia) mang li nhiu ng dng trong i sng x hi nh: trao i th tn qua mng Internet: e-mail; chnh ph in t: e-government; gio dc in t: e-education; dy hc qua mng: e-learning; th vin in t: e-library; vn ho s hay vn ho in t: e-culture. Tt c u c mt c im chung l d liu c s ho v vic trao i thng tin c thc hin trn mng. Nh vy CNTT-TT xm nhp vo mi ngc ngch ca cuc sng v tr thnh mt cng c c lc khng th thiu trong cuc sng hin i. Con ngi tip xc vi kho kin thc khng l ca nhn loi qua mn hnh my tnh v giao tip vi nhau qua mng Internet, khi mi cn tr v khng gian, thi gian tr nn khng ng k. Nhng thnh tu ca CNTT-TT to ra mt cuc cch mng trong hu ht cc lnh vc x hi, kinh t... S thay i khng ch thy trong cc ngnh sn xut cng nghip, in t, vin thng m ngay trong cc lnh vc nh y t, ti chnh, ngn hng, thng mi, qun l nh nc... th CNTT-TT cng thc s mang li cho cc ngnh ny cc cng c mi cho php y nhanh gp bi tc x l nghip v. C th k ra rt nhiu thnh tu khoa hc mi ra i da trn c s ng dng CNTT-TT nh cc thnh tu trong y hc (chp ct lp, m ni soi, chn on bnh v iu tr t xa...), trong sinh hc (cc nghin cu mi v gen, cy ghp t bo...). Trong bi cnh ny, gio dc khng th l trng hp ngoi l, sm hay mun th gio dc cng phi chu tc ng su sc bi cc thnh tu ca CNTT-TT. 1.2. Nh trng hin i trong bi cnh pht trin ca CNTT- TT CNTT-TT mang li nhng trin vng mi cho ngnh gio dc ch CNTT-TT khng ch thay i cn bn phng thc iu hnh v qun l gio dc (Education Management Technology) m cn tc ng mnh m lm thay i ni dung v phng
3 www.mathvn.com
www.MATHVN.com
php dy hc. CNTT-TT tr thnh mt b phn gio dc v khoa hc, cng ngh cho mi HS. K nng v MTT tr thnh mt trong nhng k nng thit yu ca HS. 1.2.1. CNTT-TT gp phn i mi ni dung, phng php dy hc Ngay t khi MTT ra i, cc chuyn gia gio dc ch khai thc th mnh ca MTT trong lnh vc GD&T. Ti Hi ngh quc t v gio dc i hc th k 21 Tm nhn v hnh ng ti Paris din ra t ngy 5 n 9 thng 10 nm 1998 do UNESCO t chc a ra ba m hnh gio dc: M hnh Truyn thng Thng tin Tri thc Vai tr trung tm GV Ngi hc Nhm HS Vai tr ngi hc Th ng Ch ng Thch nghi cao Cng ngh s dng Bng, tivi, radio, n chiu MTT MTT v mng
MTT ng vai tr quyt nh trong vic chuyn t m hnh truyn thng sang m hnh thng tin v s xut hin ca mng my tnh l tc ng chnh chuyn t m hnh thng tin sang m hnh tri thc. Nh vy, t nhng hnh thc n gin ban u, vic ng dng CNTT-TT trong GD&T ngy cng khng nh c tnh u vit vt tri so vi cc phng tin, dng dy hc truyn thng v CNTT-TT khng ch l mt cng c h tr dy hc m cn l tc nhn gp phn to ra mt cuc cch mng trong GD&T. Nhng thnh tu ca CNTT-TT c th khai thc trong dy hc Trong thp nin va qua, CNTT-TT c tc pht trin rt nhanh. Bn cnh cng ngh phn cng lin tc pht trin th cng ngh phn mm cng khng ngng a ra th trng nhng ng dng mi trong nhiu lnh vc. Trong cc thnh tu , c rt nhiu kt qu c th khai thc trong dy hc: Cng ngh ho 2 chiu, 3 chiu trn my tnh thit k cc PMDH, cc th nghim o hay mt qu trnh khoa hc no c thu gn... Mt khc thng qua giao din ha cc PMDH tr nn rt thn thin vi ngi s dng, y l mt trong cc l do ph cp vic s dng PMDH cho GV v HS. Cng ngh a phng tin (multimedia) cho php tch hp nhiu dng d liu nh vn bn, biu , th, m thanh, hnh nh, video... vo bi ging nhm gip HS c iu kin tip thu bi hc qua nhiu knh thng tin khc nhau.
4 www.mathvn.com
www.MATHVN.com
-
Vic trao i thng tin gia GV vi HS, gia HS vi HS c thc hin trc tip hoc gin tip qua mng v Internet. S pht trin ca cc ngnh khoa hc trong lnh vc tin hc nh tr tu nhn to, h chuyn gia, mng noron, x l tri thc cho php ch to v iu khin MTT bt chc suy ngh v nhng hnh ng ca con ngi. Trong thi gian gn y vic s dng MTT trong cc cng vic i hi suy lun nh chng minh cc mnh ton hc tr thnh hin thc. Nh vy, qua nhng ng dng trnh by s lc trn chng ta c th hnh dung
c hiu qu v tim nng ng dng cc thnh tu ca CNTT-TT trong dy hc l rt ln. CNTT-TT to ra mt mi trng dy hc mi CNTT-TT to ra mt mi trng dy hc hon ton mi so vi mi trng dy hc truyn thng bi cc yu t sau: Ti nguyn hc tp phong ph. Ngoi sch gio khoa, ti liu tham kho, cn c Sch gio khoa in t" di dng CD-ROM, DVD... HS c tip cn bi hc qua nhiu knh thng tin a dng nh vn bn, hnh nh tnh, hnh nh ng, th, biu , m thanh, video... HS c c hi quan st, tm hiu v hnh thnh cc khi nim phc tp trong cuc sng thng qua cc m hnh o do MTT cung cp. PMDH to ra mi trng thun li t chc cc hot ng hc tp hng vo vic lnh hi tri thc, khuyn khch HS tm ti, luyn tp nhng k nng cn thit v nng lc s dng thng tin gii quyt vn , gp phn pht trin tnh sng to, kh nng t duy c lp, phng php hc tp v cch thc lm vic hp tc trong vic x l thng tin mt phn c thc hin nh MTT v nh vy CNTT-TT tr thnh mt b phn ca bi hc. Tng tc, trao i thng tin a chiu gia GV v HS, gia HS vi HS, gia gia nh v nh trng... c thc hin qua mng v Internet, nh vy Internet va l kho thng tin khng l cha ng tri thc nhn loi va l chic cu ni mi ngi li vi nhau. CNTT-TT cho php c th ho dy hc mc cao. Nh cc PMDH m ngi GV c th thng qua MTT a ra khi lng kin thc ph hp vi c im ring ca tng HS. Trong qu trnh hc tp vi s tr gip ca CNTT-TT, mi HS nhn
5 www.mathvn.com
www.MATHVN.com
c mt nhim v ring tu theo tin ca mnh. Nh vy, CNTT-TT cho php thc hin phng thc dy hc mt-mt (iu ny rt kh thc hin trong cc mi trng dy hc khc). Khai thc CNTT-TT thay th GV trong mt s khu ca qu trnh dy hc (xt ton b qu trnh th CNTT-TT ch l cng c ca GV). Vai tr ca CNTT-TT trong vic to ra mt mi trng dy hc mi cng c nhiu chuyn gia gio dc nh Nguyn B Kim, Quch Tun Ngc, o Thi Lai v Sheldon Shaefer... khng nh. thy: Cung cp cho GV nhiu phng tin dy hc mi nh MTT, my chiu a nng, bng in t... H tr GV gia tng gi tr lng thng tin n HS, hnh thnh nhiu knh trao i thng tin hai chiu gia GV v HS. a ra nhiu la chn GV chun b bi ging v tin hnh ln lp sao cho pht huy cao nht tnh tch cc ch ng ca HS. Cho php GV thc hin vic phn ho cao trong dy hc. Ngoi vic dy hc trn lp cn c th dy hc t xa qua mng LAN, WAN v Internet. Trong mi trng a phng tin cho php thc hin hnh thc dy hc hp tc. CNTT-TT tc ng mt cch tch cc ti qu trnh hc tp ca HS, to ra mt mi trng thun li cho vic hc tp m c bit l t hc ca HS: Bn cnh vic tip nhn kin thc t GV, sch gio khoa, ti liu tham kho th HS cn c th tip cn vi kin thc, vi th gii khch quan qua sch gio khoa in t, CD-ROM, Internet... Cc PMDH gia s s tr gip, khuyn khch mt cch kp thi ti cc thi im cn thit khng ch trong cc gi hc ti trng m c trong thi gian t hc nh, gip HS hon thnh nhim v chim lnh kin thc v c iu kin pht trin ti a nng lc ca bn thn. Mt khc vic thc hin nhim v hc tp ca mi HS khng lm nh hng ti cc HS khc, nhng HS hon thnh sm nhim v hc tp c th tip tc tip cn vi cc ni dung mi, nhim v mi pht huy ht kh nng ca bn thn. CNTT-TT gp phn i mi vic dy hc CNTT-TT l cng c c lc gp phn i mi vic chun b v ln lp ca ngi
6 www.mathvn.com
www.MATHVN.com
-
Cc PMDH vi th gii to ra mt mi trng thun li, mt th gii sinh ng thu nh kch thch tr t m, gi nhu cu tm hiu, khm ph... gip HS ch ng, sng to trong qu trnh tip cn v chim lnh tri thc.
-
HS ch ng ln k hoch, trin khai vic t hc ca mnh ti bt k mt thi im no m bn thn c nhu cu nh cc chng trnh hng dn trn MTT hoc cc chng trnh dy hc t xa qua mng.
-
Song song vi vic khai thc CNTT-TT nhm c nhn ho vic hc tp ca mi HS, th vic giao cho mt nhm HS cng s dng mt my tnh gp phn hnh thnh v pht trin nng lc lp k hoch, hot ng hp tc gia cc HS trong nhm (y l mt phm cht khng th thiu ca con ngi lao ng trong k nguyn ca cng ngh cao). Nh vy, CNTT-TT lm cho qu trnh dy hc khng cn b rng buc bi khng
gian v thi gian. HS c th hc mi ni, hc mi lc, hc sut i. Vic hc tp tr nn uyn chuyn, linh hot, cn c vo nhu cu ca HS. HS c php la chn nhng phng thc hc tp c hiu qu, la chn ni dung bi ging v cc ti liu c lin quan ph hp vi nng lc bn thn. HS ch ng trao i v khai thc cc thng tin trn Internet nhm p ng nhu cu v kin thc lin quan n ni dung hc tp ca mnh. CNTT-TT cng to ra mt mi trng tng tc ngi hc hot ng v thch nghi trong mi trng v nh vy CNTT-TT to iu kin cho ngi hc c lp vi mc cao v h tr cho ngi hc vn ln trong qu trnh hc tp. CNTT-TT to ra cc m hnh dy hc mi Dy hc c s tr gip ca my tnh (Computer Based Training - CBT). Dy hc trn nn website (Web Based Training -WBT). Dy hc qua mng (Online LearningTraining- OLT). Dy hc t xa: GV v hc vin khng cng mt v tr, khng cng thi gian (Distance Learning). S dng CNTT-TT to ra mt mi trng o dy hc (E-learning). L Cng Trim, Nguyn Quang Lc, Nguyn B Kim a ra cc hnh thc s dng MTT nh mt cng c dy hc nh sau: GV trnh by bi ging vi s h tr ca CNTT-TT. HS s dng cc phn mm ci trn MTT hoc trn CD-ROM di s hng dn v kim sot cht ch ca GV.
7 www.mathvn.com
www.MATHVN.com
-
HS s dng cc phn mm ci trn MTT hoc trn CD-ROM mt cch c lp hoc theo nhm ti nh trng hoc ti nh ring theo nhng nh hng c. HS tra cu, tm kim thng tin v ti nguyn phc v hc tp trn mng hoc trn Internet. Trong qu trnh ny, HS c th tin hnh c lp hoc giao lu, trao i vi nhau thng qua dch v chat hoc E-mail. L Thun Vng cng a ra mt s m hnh:
-
Gio dc na tp trung vi s tr gip ca MTT v PMDH. Gio dc t xa vi s tr gip ca MTT, CD-ROM, DVD, PMDH. Gio dc t xa qua mng my tnh vi s h tr ca cc PMDH thng minh, c s d liu, ti nguyn hc tp trn mng my tnh. Vi tc pht trin rt nhanh, trong thi gian ti, chc chn cc thnh tu ca
CNTT-TT s tip tc h tr chng ta pht trin cc hnh thc dy hc c v trin khai thm nhiu hnh thc dy hc mi. 1.2.2. CNTT-TT gp phn i mi kim tra nh gi C th ni vic ng dng CNTT-TT em n nhiu nt mi trong kim tra nh gi, n c: GV thit lp mt h thng ngn hng cu hi. HS c nhn mt cch ngu nhin v la chn phng n tr li thng qua vic bm chn cc biu tng trn mn hnh hoc in thng tin vo cc trng. Vic x l kt qu im s c thc hin t ng hon ton bi chng trnh ci trong MTT. HS s dng phn mm dng gia s c tch hp modul kim tra t nh gi nhn thc ca mnh mt cch thng xuyn m khng cn s c mt trc tip ca GV. HS c th gi bi kim tra qua mng cho GV bng email hoc truy cp vo website v thc hin kim tra vi hnh thc trc nghim trc tuyn. V vai tr ca CNTT-TT trong vic h tr kim tra, nh gi c nhiu chuyn gia gio dc khng nh. o Thi Lai cho rng vic s dng CNTT-TT cho php t chc v kim sot c hot ng ca HS khng ch ti lp hc m c khi HS lm vic ti nh v vic nh gi s c t chc mt cch lin tc ti mi thi im hc tp ca HS mt cch khch quan lu di. Nh MTT nn vic cng c, kim tra kin thc c c thc hin thng xuyn hn, gim thi gian cho mi kho hc do tit kim c c thi gian v chi ph.
8 www.mathvn.com
www.MATHVN.com
1.2.3. Nhn nh chung ng dng CNTT-TT vo qu trnh dy hc to ra mt cuc cch mng trong gio dc v dn n nhng thay i trong phng php dy hc. Cng ngh Multimedia v Internet lm cho qu trnh dy hc tr nn tch cc, khuyn khch HS pht huy tnh ch ng sng to v hng say trong hc tp. Ngi GV khng cn l kho kin thc duy nht. GV phi thm chc nng t vn, t chc cho HS khai thc mt cch ti u cc ngun ti nguyn tri thc trn mng, Internet, CD-ROM v s dng PMDH. Tin trnh ln lp khng nht thit phi tun t m c th tin hnh mt cch linh hot. Pht trin cao cc hnh thc tng tc giao tip: HS GV, HS - HS, HSMTT, trong ch trng n qu trnh tm li gii, khuyn khch HS trao i, tranh lun. y l iu kin gip HS pht trin nng lc t duy. Ngi hc b thu ht bi nhng thng tin trn MTT, trn Internet. HS s kt ni li nhng tri thc c hc v thu nhn nhng thng tin phn hi t MTT i n nhng quyt nh ng n. MTT s gip HS gii quyt kh khn trc vn mi cn chim lnh v to ra mt mi trng khuyn khch tnh t m, ham mun tm hiu, khm ph, trong qu trnh hc tp i n chim lnh tri thc. Hc tp l mt hot ng x hi, qu trnh i thoi qua mng s h tr c lc cho ngi hc nm bt c kin thc khng ch trong m c ngoi trng hc. Nh vy ngoi gc l cng c h tr dy v hc, CNTT-TT tr thnh mt cng c hnh thnh v pht trin nhn thc. 1.3. ng dng CNTT-TT trong nh trng Vit Nam ng dng CNTT-TT trong dy hc tp trung vo cc lnh vc sau: S dng cc thit b (phn cng) vi vai tr l phng tin, cng c dy hc nh: MTT (PCs-Personal Computers); Thit b hin th thng tin (display): Large colour monitors, Data projectors, Interactive whiteboards, OHP displays, TV interfaces...; Cc thit b ngoi vi ghp ni vi MTT: my nh k thut s, my qut, graphic calculators... S dng cc ngn ng lp trnh nh Pascal, Logo...; Cc phn mm thng dng: Excel, Winword...; Cc phn mm ho (Graph Plotting Software-GPS); Cc phn mm s hc, h thng i s my tnh (Computer Algebra System-CAS); Cc phn
9 www.mathvn.com
www.MATHVN.com
mm hnh hc ng (Dynamic Geometry Software -DGS); Cc phn mm trnh din (Data Handling Software-DHS) Ngoi ra cn k n khai thc thng tin trn cc CD-ROM v Internet... Nhn thc r vai tr to ln ca CNTT-TT, ng v Nh nc ta c nhiu vn bn ch o v pht trin ng dng CNTT-TT trong gio dc v o to. T nm 1985, B GD&T tin hnh dy th nghim chng trnh nhp mn tin hc c s trn a bn 10 tnh v n nm 1990 trin khai vic dy th im tin hc ti hn 100 trng THPT trn phm vi ton quc. Bn cnh vic dy tin hc theo chng trnh ca B GD&T nhiu trng t tiu hc n THPT trn ton quc la chn a vo chng trnh ngoi kho mt s ni dung tin hc nh son tho vn bn, s dng cc phn mm ho, tnh ton vi bng tnh in t... Song song vi vic trin khai ca Nh nc v B GD&T, nhiu a phng ch ng y mnh a tin hc vo nh trng trn a bn ca mnh. Nh vy, vic ng dng CNTT-TT trong dy hc Vit Nam trong thi gian qua t c cc kt qu chnh sau: Nghin cu v khai thc cc PMDH trn th gii. Trin khai thit k v xy dng cc PMDH cho cc ni dung c th v d nh cc phn mm gia s v phn mm h tr kim tra nh gi. T chc dy hc vi s h tr ca MTT. Th nghim khai thc mng, Internet dy hc t xa. Tuy nhin, ng trc nhng tim nng to ln ca CNTT-TT i vi GD&T th cc thnh tu trn cn rt khim tn. Trc mt chng ta cn b ng nhiu vn c th ng dng CNTT-TT mt cch c hiu qu, c bit l vic s dng, khai thc PMDH. 1.4. Tc ng ca CNTT- TT trong dy hc ton V rt kh v khng th lit k tt c cc ng dng ca CNTT-TT trong dy hc ton nn ta ch cp n cc ng dng sau: T chc, iu khin qu trnh hc tp ca HS da trn thng tin ngc do MTT cung cp So vi cc phng php truyn thng, th r rng cc thng tin ngc do MTT cung cp s chnh xc hn, khch quan hn, nhanh chng hn v y chnh l mt yu t quan 1.4.1. ng dng CNTT-TT trong dy hc ton
10 www.mathvn.com
www.MATHVN.com
trng GV c th iu khin qu trnh hc tp ca HS cng nh HS t iu chnh li vic hc tp ca mnh. V d: GV, HS c th th, kim tra xc nh trc kt qu trn MTT, ri sau ln ngc dn dn tm ra li gii cho bi ton. Trong qu trnh dy hc ton, GV v HS c th a ra cc gi thuyt ca ring mnh ri nh MTT th nghim nhng gi thuyt c th tip tc pht trin hoc iu chnh, thay i gi thuyt ca mnh. S dng MTT xy dng cc m hnh trc quan sinh ng nghin cu mt i tng ton hc no trc ht ngi ta tm cch xy dng m hnh tng ng. Trn c s cc kt qu lm vic vi m hnh s i n vic chng minh hoc li gii trong trng hp tng qut. So vi cc phng tin dng dy hc truyn thng th MTT c kh nng ni tri hn trong vic th hin cc i tng ton hc trong th gii thc bi cc m hnh ha 2 chiu, 3 chiu. CNTT-TT c coi l mt cng c t nhin din t cc m hnh ton hc, th, biu , hnh v v qu trnh chuyn ng ca cc i tng ton hc theo mt quy lut no . V vy nhng i tng, quan h ton hc khng cn tru tng, xa l v kh nm bt i vi mt s ng HS. iu ny gip HS tip thu tt cc ni dung kh, c tnh tru tng cao trong ton hc. S dng MTT v PMDH pht hin cc tnh cht, cc mi quan h trong ton hc Ta s dng cc PMDH biu din cc m hnh, biu , hnh v... mt cch trc quan sinh ng. Ch cn mt vi thao tc n gin nh ko r chut ta c th c c nhng hnh nh v i tng cn nghin cu di cc gc khc nhau hoc c th cho mt vi thnh phn ca i tng ton hc bin i nghin cu cc thnh phn cn li t pht hin ra cc mi quan h, tnh cht ca chng. S dng kt hp cc phn mm ho v s hc, GV c th gii thch c hai trng thi hnh dng v s lng.
11 www.mathvn.com
www.MATHVN.com
o Thi Lai, Trn Vui nhn mnh vai tr ca CNTT-TT trong vic h tr HS t khm ph v pht hin vn trong qu trnh hc ton v thng qua qu trnh ny HS c iu kin rn luyn phng php nghin cu trong hc tp, nng lc t duy sng to. Theo Phm Huy in th phn mm ton hc v MTT s h tr ging dy cc ch kh, h tr i su v hiu ng bn cht vn . Sue Johnston-Wilder, David Pimm cng khng nh CNTT-TT cung cp cho HS mt mi trng tt hc ton. Khai thc mng Internet trong dy hc ton Trc ht Internet l kho thng tin tch lu tri thc ton hc ca con ngi v y l ngun ti nguyn v cng qu gi cho nhng ngi dy v hc ton. Tip theo Internet cun g cp ph ng tin , m i tr
nh 1.1
n g
GV, HS trao i thng tin vi nhau trong qu trnh dy hc ton v dy hc ton t xa. Vi thc t h tng CNTT-TT nh ngy nay, cc nh trng, GV thm ch c HS hon ton c th thit k cc website v a ln Internet cung cp thng tin, to ra mt din n mi ngi cng khai thc thng tin, trao i v ni dung, kin thc lin quan n nhim v hc tp ca HS (nh 1.1). Dy hc ton vi my tnh Trong qu trnh nghin cu v s dng MTT dy hc ton th vic khai thc ho trn MTT c c bit quan tm v y l cng c rt hu ch trong vic biu din cc m hnh ton hc. David Tall s dng mi trng ho my tnh dy hc ton t nm 1980. Kenneth Ruthven bt u la chn, nghin cu, pht trin s dng
12 www.mathvn.com
www.MATHVN.com
ho ca my tnh vo dy hc ton t nm 1986. Theo xu hng ny, Morgan Jones, McLeay (1996), Crawford, Morrison (1998) ng dng ho trong dy hc ton. V vai tr ca ho trong dy hc ton cho HS t 11 n 16 tui cng c Arter (1993), Ruthven (1992), Graham, Galpin (1998) khng nh. Theo Colette Laborde, th MTT c kh nng to ra mi trng kch thch HS hot ng tm ti khm ph v t hnh thnh kin thc mi. John Mason khng nh rng cc PMDH ton vi mt h thng cng c c kh nng gii ton v gip HS nghin cu cc i tng tm ra cc tnh cht ton hc. Rosamund Sutherland khi nghin cu dy hc ton vi phn mm Logo c kt rng: iu quan trng nht l khi HS s dng ngn ng, k hiu my tnh th s pht trin kh nng khi qut ho ton hc. Wan Fatimah Bt Wan Ahmad, Halimah Badioze Zaman cho rng bng vic s dng MTT trong dy hc ton c th cung cp nhiu cch hc khc nhau, c bit l t chc hc nhm v PMDH gip cho kh nng suy lun ton hc ca HS THCS t hiu qu rt cao. Nhm tc gi cn dn li ca Niess (1994) cho rng khi s dng my tnh m phng cc vn v iu kin trong th gii thc th HS c th hc rt nhiu tri thc mi, cng c kin thc v nhn thy c tm quan trng ca kin thc . Tringa (1923) khng nh nhng kin thc hnh hc m HS t c khi s dng MTT s cao hn so vi phng php dy hc thng thng. Nguyn nhn chnh ca s tin b l nh vic HS s dng cc phn mm ton hc. o Thi Lai khng nh nu s dng CNTT-TT mt cch hp l trong dy hc ton th s tng c t l HS kh, gii v gim t l HS yu so vi dy hc truyn thng v GV c iu kin gip c hu ht HS rn luyn nng lc sng to, phng php nghin cu trong hc tp. Nh vy hiu qu s dng MTT trong dy hc ton c nhiu chuyn gia gio dc trn th gii v Vit Nam nghin cu v c kt mt s khng nh ng tin cy. 1.4.2. ng dng CNTT-TT trong dy hc ton v vn i mi trong h thng phng php dy hc mn ton T l lu tr thng tin trong tr nh ngi hc thng qua cc knh thng tin khc nhau c cc chuyn gia tng kt nh sau:
13 www.mathvn.com
www.MATHVN.com
Cch tip cn Li ni Hnh nh Li ni v hnh nh Li, hnh nh v hnh ng T pht hin
Sau 3 gi 30% 60% 80% 90% 99%
Sau 3 ngy 10% 20% 70% 80% 90%
Qua y ta thy c hn ch ca cc phng php dy hc th ng, nhi nht, my mc v thy c vai tr ca vic s dng hnh nh minh ho v nhu cu cp bch cn t chc cho HS hc tp trong hot ng v bng hot ng t gic, tch cc, ch ng v sng to. Mt vn c cc chuyn gia quan tm l vic ng dng CNTT-TT trong dy hc ton s tc ng n h thng phng php dy hc ton nh th no? Tc ng no mang tnh tch cc? Nhng hn ch no cn lu ? Ta s xem xt h thng phng php dy hc ton di tng gc thy c nhng tc ng tch cc do ng dng CNTT-TT mang li. Xt v vic h tr HS tm hiu su ni dung kin thc Trong hot ng ton hc, c nhng vic gm hng lot cc thao tc tnh ton, v hnh... Chng thng chim rt nhiu thi gian hc tp ca HS nhng i khi kt qu khng chnh xc. Ta c th s dng my tnh h tr HS trong cc cng on ny. V d, bn cnh vic yu cu HS nm c v thc hin chnh xc cc thao tc c bn dng mt hnh hnh hc th n mt mc no c th cho HS s dng MTT vi cc phn mm hnh hc v hnh, thm ch cho php HS s dng cc macro gm nhiu thao tc dng hnh. Khi cn v li hnh HS khng cn phi thao tc ln lt t u m ch cn gi lnh thc hin macro. Nh vy CNTT-TT tc ng trc tip dn n xu hng tng cng cc hot ng HS c iu kin hiu su hn hoc m rng hn v ni dung kin thc. Xt v vic rn luyn k nng, cng c, n tp kin thc c Ngy nay cc PMDH tr nn rt phong ph, a dng, trong c rt nhiu phn mm c th khai thc rn luyn k nng thc hnh cho HS. Chng hn vi phn mm Graph, HS c th rn luyn cc k nng c bn v kho st hm s, tnh din tch ca mt min phng, xc nh gc ca tip tuyn ti mt im no trn th vi trc honh... Vi phn mm hnh hc Euclides, Geometers Sketchpad,... HS c th rn luyn k nng dng hnh, tm hiu cc bi ton qu tch mt cch rt hiu qu. Phn mm GeoSpacW c th gip HS rn luyn vic dng hnh, xc nh thit din, xc nh cc khi trn xoay v rt nhiu ni dung khc trong hnh hc khng gian. Vi cc phn mm trc nghim, HS c cung cp mt khi lng cu hi m tr li c HS phi thc s nm c kin thc c bn v t c k nng thc hnh n mt mc nht nh. Nh vy vic luyn tp v t kim tra nh gi ca HS khng cn b hn ch v mt thi gian v ni dung nh cc phng php kim tra thng thng. Xt v gc rn luyn, pht trin t duy ton hc
14 www.mathvn.com
www.MATHVN.com
Nhiu ngi lo ngi MTT vi cc chc nng "trong sut" i vi ngi s dng nn HS khng c s gn kt gia hnh tng tnh ton trong no vi thc hin tnh ton trn MTT. Mt s bc trung gian c MTT thc hin do lm mt cm gic thut ton! Ti Hi ngh nghin cu ton hc th gii ln th 3 (TIMSS) tin hnh tho lun xung quanh vn nghi ngi trn. Ann Kitchen chng minh rng trong iu kin c s dng my tnh, HS s hc ton tt hn. Cc tc gi Michael D. De Villiers, Trn Vui khi nghin cu vic dy hc ton vi phn mm The Geometers Sketchpad khng nh vai tr ca phn mm ny trong vic pht trin kh nng sng to ton hc cho HS. Phm Huy in khng nh MTT c kh nng lm sng t cc khi nim ton hc phc tp bng nhng minh ho trc quan hon ho. Nh vy dy hc ton vi s h tr ca MTT cho php GV to mi trng pht trin kh nng suy lun ton hc v t duy lgc, c bit l nng lc quan st, m t, phn tch so snh cho HS. HS s dng MTT v phn mm to ra cc i tng ton hc sau tm ti khm ph cc thuc tnh n cha bn trong i tng . Chnh t qu trnh m mm, d on HS i n khi qut ho, tng qut ho v s dng lp lun lgc lm sng t vn . V d khi s dng Graph nghin cu th ca mt hm s hoc s dng Maple v hnh bt buc HS phi tun th nghim ngt theo cc bc ca quy trnh, y l mi trng tt pht trin t duy lgc, t duy thut ton. Xt v phng php v hnh thc dy hc Khi a CNTT-TT vo nh trng s to nn mt mi trng dy hc hon ton mi, hp dn v c tnh tr gip cao... y s l iu kin thun li cho vic i mi phng php v hnh thc dy hc ton. Trc ht, CNTT-TT gp phn tng cng tnh tch cc ca HS trong hc tp. Trong nhng nm gn y, trn c s nhng thnh tu ca cng ngh phn mm, cc PMDH to ra mt mi trng hot ng thun li cho HS. Trong mi trng ny, HS l ch th hot ng, tc ng ln cc i tng v qua HS chim lnh c cc tri thc v k nng mi. Vi s pht trin ca cng ngh mng, Internet v cc ng dng trn mng to iu kin thun li cho HS tra cu, tm kim thng tin trn h thng ti nguyn gn nh v tn trn cc website, trong cc th vin in t. Vic tng cng giao lu, hp tc, trao i trong hc tp gia HS vi HS, HS vi GV khng cn b hn ch v mt thi gian v khong cch a l. Trong mi trng mi ny, GV v SGK khng cn l ngun cung cp thng tin duy nht, m HS c cung cp nhiu ngun tri thc khc nhau pht trin nng lc hot ng c lp cng nh tng cng kh nng hp tc ca bn thn. Xt v vai tr ca ngi thy trong dy hc ton
15 www.mathvn.com
www.MATHVN.com
Trc ht cn loi b t tng sai lc l MTT c th thay th hon ton ngi GV trong dy hc ton. Vic dy hc ton lun lun i hi cao vai tr m c bit l cng sc v kh nng s phm ca ngi GV. Tuy nhin vai tr ca ngi GV trong iu kin s dng MTT v PMDH cng c nhng thay i so vi truyn thng. Ngi GV phi l ngi hng dn, ch o HS pht huy c ht kh nng ca mnh trong hot ng hc tp. Ngi GV l ngi t chc, iu khin, tc ng ln HS v i khi c mi trng tin hc, chng hn: Thit k, to ra cc tnh hung HS hot ng vi MTT. Ch cho HS bit phi s dng MTT v PMDH nh th no v gip HS vt qua cc kh khn m cc em gp phi trong qu trnh ny. Thit k cc mun theo s phm khi HS s dng cc mdul ny s tip cn v t c mc ch mt cch nhanh chng. Ngoi ra GV cn l ngi ch ra a ch nhng ngun thng tin cho HS khai thc, v d khi dy nh l Py-ta-go, GV ch cho HS a ch cc website v lch s, thn th nh bc hc Py-ta-go, vic chng minh nh l Py-ta-go Xt v gc thc hin phn ho trong dy hc ton CNTT-TT to iu kin cho vic thc hin phn ho cao trong qu trnh dy hc ton. thc hin c s phn ho, GV phi nm bt c v x l kp thi mi din bin ca hot ng hc tp ca tng HS trong lp. Cng vic ny rt kh thc hin trong mi trng dy hc truyn thng mt GV m nhn vic ln lp cho ba, bn chc HS. Nu s dng CNTT-TT th chnh MTT s thay th GV trong mt thi im no a ra nhng h tr kp thi khi HS gp kh khn vi liu lng thch hp ng thi a ra nhng chng trnh, ni dung cng vic tu thuc vo mc nhn thc ca mi HS. Nu HS c MTT ti nh ring th cc PMDH li l nhng thy gio ti nh kim sot, nh gi kt qu v gip HS hc tp mt cch hiu qu. Nu GV dy hc trong phng a phng tin vi h thng Hiclass th vic thc hin phn ho trong dy hc ton c thc hin mt cch thun li. Theo o Thi Lai d c gng n u chng na trong iu kin cc dng, phng tin dy hc truyn thng th vic m bo cc nguyn tc phn ho trong dy hc ton vn b hn ch. Vi MTT v PMDH, mi HS nh c c mt tr ging ring lun sn sng gip HS vt qua cc tr ngi ti mi thi im cn thit. Vic khai thc PMDH v Internet cng ni di cnh tay ca ngi thy dy ton n tng gia nh, ti tng HS c th v ngoi vic hng dn HS hc tp, cng tc kim tra, nh gi cng c thc hin ngay ti ch. Xt v vai tr h tr kh nng i su vo cc phng php hc tp, phng php thc nghim ton hc MTT vi cc phn mm cho php GV, HS to ra cc m hnh, m t qu trnh din bin ca cc i lng ton hc hoc t chc cc thc nghim ton hc. Bng quan st trc quan qu trnh do MTT a ra, HS nu ra gi thuyt v s dng MTT kim tra gi thuyt ca mnh. y l c s cho HS s dng suy lun c l
16 www.mathvn.com
www.MATHVN.com
khng nh hoc bc b gi thuyt bc tip theo. Vn ny rt kh thc hin nu ch s dng cc phng tin dng dy hc truyn thng. Trong qu trnh hc tp, vi s h tr ca MTT v PMDH, HS tin hnh hng lot cc hot ng tm hiu, khm ph, phn tch v kim chng cc gi thuyt ca mnh, y chnh l qu trnh i ti li gii ng n cho bi ton. Qua cc hot ng ny, HS s hnh thnh, rn luyn phng php hc tp, phng php thc nghim ton hc. Xt v vic p dng cc hnh thc dy hc trong dy hc ton Cc hnh thc dy hc truyn thng nh dy hc ng lot, dy hc theo nhm, dy hc c th s c iu kin kt hp mt cch hiu qu, linh hot hn nu s dng, khai thc CNTT-TT. Hn na cc hnh thc dy hc cng m hn, chng hn khi nim dy hc ng lot khng ch l thy ln lp ti ging ng nh hnh thc truyn thng m thy ti mt a im no (chng hn ti H Ni) c th ln lp v truyn trc tip ln mng Internet v rt ng HS cng vo mng tham d lp hc ny. Hnh thc hc theo nhm c m rng bao gm cc HS cng quan tm, nghin cu v trao i vi nhau v mt ni dung c th m khng gii hn v phm vi bn b trong mt lp, mt trng hoc sinh sng gn nhau m tt c u thng qua mng Internet, thm ch mt HS cng mt lc c th tham gia nhiu hnh thc hc tp hoc tham gia hc tp theo nhiu nhm khc nhau. Xt v gc kim sot v nh gi qu trnh hc tp ca HS Vi s tr gip ca cc phn mm kim tra, nh gi, GV c iu kin kim sot cht ch ton b qu trnh hc tp ca HS. Vic kim tra nh gi c tin hnh lin tc, trong mi thi im ca qu trnh hc tp ca HS. Vi cc phn mm ghi trn a CD-ROM hay trn cc website s cung cp cc kim tra trc nghim khch quan, cc t lun gip GV, HS thc hin kim tra nh gi mt cch nhanh chng v n gin. S dng cc phn mm cng c, GV s nhn nh mt cch chnh xc v k nng tnh ton, kh nng tp trung ch , kh nng suy lun lgc... ca HS. Vi kh nng lu tr v x l gn nh v tn ca MTT, GV c th lu li ton b qu trnh hc tp ca HS c nhng nh hng ng n trong qu trnh hc tp ca tng HS. Xt v vic hnh thnh phm cht, o c, tc phong cho HS trong qu trnh dy hc ton Vic s dng CNTT-TT ngay khi ngi trn gh nh trng trc tip gp phn hnh thnh v pht trin k nng s dng thnh tho MTT v lm vic trong mi trng CNTT-TT cho HS. y l nhng c tnh khng th thiu ca con ngi lao ng trong thi i ca cng ngh cao trn c s s pht trin ca CNTT-TT. S dng CNTT-TT trong qu trnh thu thp v x l thng tin gip hnh thnh v pht trin cho HS cch gii quyt vn hon ton mi: a ra cc quyt nh trn c s kt qu x l thng tin. Cch hc ny trnh c kiu hc vt, my mc, nhi nht th ng trc y v gp phn hnh thnh cho HS mt phng php nghin cu ton hc mi, c bit l trong dy hc hnh hc. Trong qu trnh hc tp vi s tr gip ca CNTT-TT, HS c iu kin pht trin nng lc lm vic vi cng cao mt cch khoa hc, c tnh cn c, chu kh, kh nng c lp, sng to, t ch v k lut cao.
17 www.mathvn.com
www.MATHVN.com
Vic t nh gi, kim tra kin thc bn thn bng cc phn mm trn MTT cng gip HS rn luyn c tnh trung thc, cn thn, chnh xc v kin tr, kh nng quyt on Nhn nh chung - CNTT-TT khc phc c vic dy hc n thun truyn th mt chiu, HS th ng tip thu v ti hin mt cch my mc. CNTT-TT to ra mi trng thun li cha tng c gip HS hc ton mt cch tch cc, ch ng, t mnh gii quyt vn v pht trin t duy sng to, kh nng t hc. CNTT-TT gip hng ti vic khuyn khch HS bn cnh vic tch lu kin thc cn ch trng n pht trin nng lc m ch yu l nng lc gii quyt vn . CNTT-TT gip to ra cc hnh thc dy hc phong ph, hiu qu. Vic s dng CNTT-TT gp phn nng cao thc v hiu qu ca vic s dng phng tin dy hc. Vi nhng dch v phong ph ca CNTT-TT, ngi GV c iu kin la chn phng php dy hc theo ni dung, s trng, i tng HS sao cho ph hp nht. Vi s h tr c lc ca CNTT-TT, GV c mi trng v iu kin t chc cc hot ng tho lun, tranh lun ca HS c iu kin pht huy nhm tng cng kh nng hp tc trong hc tp.
18 www.mathvn.com
www.MATHVN.com
Chng 2
S DNG PHN MM GRAPH TRONG DY HC TON2.1. Gii thiu v phn mm Graph PHN MM GRAPH L MT PHN MM H TR MINH HO V GII QUYT MT S VN TRONG B MN TON PH THNG TNG I GN NH C CI T TRONG MI TRNG H IU HNH WINDOWS. TON B CHNG TRNH CHA GN TRN MT A MM 1.44 MB CA IVAN JOHANSEN. PHN MM NY HIN NAY C TH DOWNLOAD MIN PH TI A CH: HTTP://WWW.PADONWAN.DK 2.2. Lm vic vi Graph np chng trnh Graph, ta thc hin dy thao tc: Start/Programs/Graph hoc nhy chut vo biu tng ca Graph GIAO DIN CA PHN MM GRAPH GM CC THNH PHN: H THNG MENU, THANH CNG C V TRANG CNG TC C CHIA THNH 2 PHN: CA S TRI L DANH HM DANH SCH CC I TNG: DANH SCH (FUNCTIONS),
SCH CC IM (POINT SERIES), DANH SCH CC MIN C LA CHN (SHADES) V DANH SCH TN CC I TNG (LABELS), CA S
19 www.mathvn.com
www.MATHVN.com
BN PHI DNH HIN TH CC I TNG NH TH, NG THNG, IM, NHN TN I TNG,... 2.3. Gii thiu h thng Menu H thng menu ca Graph gm 6 chc nng c bn: File, Edit, Function, Zoom, Calc v Hepl. 2.3.1. Menu File: 2.3.2. Menu Edit: 2.3.3. Menu Function: (Edit...), Khi to mt hm mi( Insert function- Ins), To v tip tuyn (Insert tangent - F2), nh du mt min(Insert shade... - F3), V im trn h to ca trang cng tc (Insert point series... -F4), V h thng im (Insert trendline-Ctrl+T), t tn cho cc i tng (Insert label...), Cp nht cc i tng ang c la chn Hu b thao tc ngay trc ( Undo - Ctrl+Z), Lp li thao tc ngay trc (Redo - Ctrl+Y), Ct i tng lu vo b m (Cut - Ctrl+X), Copy i tng lu vo b m (Copy - Ctrl+C), Dn i tng t b m ra trang cng tc (Paste - Ctrl+V), Sao chp hnh nh (Copy image), Tu bin h trc to (Axes - Ctrl+A), Xc lp mi trng lm vic (Options). M mt tp mi (New - Ctrl+N), M mt tp c (Open - CTrl+O), Lu tr tp (Save - Ctrl+S, Save as), In n (Print), Kt thc phin lm vic (Exit - Alt+F4), Lu tr kt qu di dng nh (Save as image -
Ctrl+B), chc nng ny gip ta c c cc th p thit k gio n in t.
20 www.mathvn.com
www.MATHVN.com
-
Xo b cc i tng ....ang c la chn (Delete Ctrl+Del), Chn th o hm ca hm s (Insert f(x)). H thng cc chc nng ca menu Zoom gm cc lnh iu khin, thay i gc hin th ca trang lm vic, trong ch cc chc nng sau: iu chnh theo hng thu hp khong [a,b] ca trc honh c hin th trn trang cng tc (In), iu chnh theo hng gia tng khong [a,b] ca trc honh c hin th trn trang cng tc (Out),
2.3.4. Menu Zoom :
-
Chuyn v trng thi chun (Standard-Ctrl+D), Chuyn v trng thi cho php di chuyn cc i tng trn trang cng tc (Move system - Ctrl+M), Chuyn v ch hin th sao cho quan st c tt c cc im trn trang cng tc (All points).
2.3.5. Menu Calc: Xc nh di ca th f(x) trn on [a,b] no (Length of path), Tnh din tch phn gii hn bi cc ng thng x=a, x=b vi th ca f(x) ( Area), Xc nh gi tr ca f(x) ti mt im xo no (Evaluate - Ctrl+E), To bng tnh gi tr ca f(x) trong on [a,b] vi bc chia cch u (Table). 2.4.1. V TH HM F(X) khi to mt th mi, dy thao tc nh sau:-> Functiontng > Insert function (hoc chn biu trn thanh cng c). Xut hin bng
2.4. Mt s chc nng c bn
khai bo cc tham s: + Biu thc tng qut ca f(x), + Gii hn phm vi gi tr ca i s, + Kiu nt v, + rng nt v,
21 www.mathvn.com
www.MATHVN.com
+ Mu nt v, Khai bo xong, nhn OK hon tt cng vic. 2.4.2. Cp nht i tng chnh sa th ca hm s c, thao tc nh sau: Trc tin la chn th s chnh sa, tip theo chn: ->Function ->Edit (hoc bm p vo biu thc ca f(x) ca s bn tri) s xut hin ca s Edit function ta cp nht li. Ta c th khai bo li gi tr on [a,b], chn li dy nt v, nhp ni dung ghi ch cho i tng hoc mu v ca ng tip tuyn. Nhn OK hon tt cng vic. 2.4.3. V tip tuyn vi th f(x) ti im xo v tip tuyn vi th hm s f(x) ti im xo trc tin phi la chn hm s, tip theo chn: -> Function -> Insert tangent. Xut hin ca s Insert tangent. Ta nhp gi tr xo ti ca s: x=, sau chn rng, kiu ng v tip tuyn, mu v c th nhp ni dung ghi ch cho tip tuyn ti ca s: Description. Sau cng nhn OK hon tt. iu chnh tip tuyn v, bm p vo biu thc ca tip tuyn ti ca s tri, s xut hin ca s Edit tangent ta cp nht.
22 www.mathvn.com
www.MATHVN.com
2.4.4. Chn th ca o hm f(x) Graph c chc nng v cng mt h trc to th ca hm s f(x) v f(x). s dng chc nng ny, trc tin ta chn hm cn chn thm th ca o hm ca s bn tri, sau thao tc: ->Function -> Insert f(x). Xut hin ca s Insert (fx). Ta khai bo khong [a,b], kiu nt v, dy, mu v ghi ch cho th mi ny. Nhn OK hon tt. 2.4.5. Xc nh di ca th f(x) trn on [a,b] Chc nng Length of path cho php ta bit c ngay gi tr di ca th hm s f(x) trn on [a,b]. s dng chc nng ny, trc tin ta chn hm ca s bn tri sau thao tc: ->Calc -> length of path. Xut hin ca s cho ta nhp gi tr hai u mt a ti ca s From: v b ti ca s To:, ta s c kt qu c thng bo Length. C th nhp cc gi tr a, b khc nhau tnh nhiu ln. 2.4.6. Tnh din tch Graph c chc nng tnh nhanh din tch phn mt phng gii hn bi cc ng thng x=a, x=b vi th ca f(x). s dng chc nng tnh din tch hnh phng, trc tin ta chn hm ca s bn tri, tip theo ta thao tc nh sau: -> Calc -> Area. Xut hin ca s, ta nhp gi tr u mt a ti ca s From: b ti ca s To:, ta c kt qu din tch s c thng bo ti ca s Area. Trn mn hnh ho s thy phn din tch tng ng s c biu din bi cc ng gch sc. Ta c th nhp cc gi tr u mt a, b khc nhau tnh din tch cc min khc nhau.
23 www.mathvn.com
www.MATHVN.com
2.4.7. Tnh gi tr f(x), f(x), f(x) ti im xo s dng chc nng ny, trc tin ta chn hm ca s bn tri, tip theo ta thc hin thao tc: -> Calc -> Evaluate, xut hin ca s ta nhp gi tr ca im xo cn tnh. Kt qu c thng bo 3 ca s bn di ln lt l : f(x), f(x), f(x). Ta c th thay i gi tr xo c c kt qu ti cc im khc nhau. 2.4.8. Tnh gi tr ca f(x) trong on [a,b] vi bc chia cch u CHC NNG CALCULATE TABLE CHO PHN HOCH ON [A,B] BI MT LI CC NT CCH U NHAU MT ON DX V TNH GI TR CA HM S F(X) TI CC IM CHIA. lp bng, trc tin ta chn hm ca s bn tri, v thao tc: >Calc ->Table, xut hin ca s Calculate table. Ta khai bo khong [a,b] v bc chia dx. Nhn nt Calc ta s c kt qu cn thit. 2.4.9. V cc im trn h trc to : s dng chc nng ny, ta thao tc nh sau:->Function ->Insert point series..., xut hin ca s: Insert point series. Ta cn khai bo to ca im cn v. Bn tri c cc la chn - Kiu v im: Style, - Mu v im: Color,-Kch thc im: Size, Hin to ca im Show coordinates... Khai bo song nhn OK, ta s nhn c hnh nh cc im trn mn hnh.
24 www.mathvn.com
www.MATHVN.com
2.4.10. In n kt qu in cc kt qu, ta chn: ->File -> Print. Xut hin ca s Page Setup ta xc nh cc thng s trc khi in. Nu cn la chn my in trong danh sch cc my in ci t, ta chn tip Printer... a ra my in, ta chn OK. 2.5. Th vin cc hm ca Graph TRONG PHN MM GRAPH, CC HM C THIT K CI T TRONG TH VIN TNG I PHONG PH, TUY NHIN CC HM SAU THNG C S DNG NHIU TRONG CHNG TRNH PH THNG: ABS - Hm ly gi tr tuyt i ca i s, SQR - Hm cho gi tr bnh phng ca i s, SQRT - Hm cho gi tr l cn bc hai ca i s, SIN - Hm cho gi tr hm s sin ca i s, COS - Hm cho gi tr hm s cosin ca i s, TAN - Hm cho gi tr hm s tang ca i s, ARCSIN - Hm cho gi tr ca hm s ngc ca hm s sin, ARCCOS - Hm cho gi tr ca hm s ngc ca hm cosin, ARCTAN - Hm cho gi tr ca hm s ngc ca hm tan, LN - Hm cho gi tr logarit c s e ca i s, LOG - Hm cho gi tr logarit c s thp phn ca i s, PI - Cho gi tr ca s pi,
25 www.mathvn.com
www.MATHVN.com
Ton t ^ : dng biu din lu tha, v d 10^3 l 1000, 2^8 l 256. bit thm chi tit, chn Help tra cu nhng thng tin cn thit. 2.6. Khai thc phn mm Graph Phng php ch yu l dng Graph minh ho v kim tra kt qu. Sau khi hc sinh hon thnh khi lng cng vic, gio vin c th s dng Graph hc sinh kim tra li kt qu tnh ton ca mnh v kho st chi tit thm nh vo cc cng c ca Graph chng hn ta c th s dng Graph v th sau lu tr th di dng nh a vo gio n son trn Word hoc PowerPoint... V d 1: S dng phn mm Graph biu din tp nghim ca bt phng trnh bc nht hai n ax+by < c (chng hn nh bt phng trnh 2x+y < 3). Bc 1: Chn chc nng Function/Insert Function v th hm s ca Graph. Ta nhp biu thc y=(c-ax)/b vo hp thoi Function Equation (v d y=3-2x), ta nhn c ng thng l th ca hm s y=(cax)/b. Bc 2: Ly mt im M0(x0, y0) khng thuc ng thng , chng hn ta ly gc to O(0; 0) v so snh gi tr biu thc ax0 + by0 vi c xc nh na mt phng b no s l min nghimHnh 1
ca ax + by < c
Bc 3: Chn chc nng Function/Insert Shade v la chn m hnh tng ng, ta thu c hnh nh biu din hnh hc tp nghim ca bt phng trnh bc nht hai n ax+by < c (hnh 1).
26 www.mathvn.com
www.MATHVN.com
minh ho tp nghim ca cc bt phng trnh khc, ta ch m hp thoi Insert Function thay i biu thc v t li (Shade) min nghim. V d 2: Biu din hnh hc tp nghim ca h bt phng trnh bc nht hai n:
x+ y+20 x y 1 0 2 x y + 1 0 Hon ton tng t nhHnh 2
v d 1, ta ln lt v v xc nh min nghim ca tng bt phng trnh bc nht hai n ca h. Kt qu ta c c biu din hnh hc tp nghim ca h (hnh 2). 2.7 Bi tp: 1) V th cc hm s : y1=ax3+bx2+cx+d; y2=ax4+bx3+cx2+dx+e.
ax + bx ax 2 + bx + c y4 = cx + dx dx + ex 2) Minh ho vic t th hm s f(x) suy ra th cc hm s: f(|x|). |f(x)|, |f(|x|)| y3 =cng nh tnh cht ca hm s m, hm s lgarit. 3) S dng cc chc nng ca Graph kim tra kt qu tnh ton cc bi tp tnh di, din tch v tch phn xc nh trong sch gio khoa THPT.
27 www.mathvn.com
www.MATHVN.com
Chng 3 S DNG PHN MM HNH HC NG CABRI GEOMETRY3.1. Tng quan v phn mm hnh hc ng Cabri Geometry 3.1.1. Khi ng Cabri Geometry Nu my tnh ca bn cha ci t phn mm Cabri Geometry th bn c th download Cabri Geometry trn Internet t. gi Cabri ra lm vic ta ln lt chn cc lnh: Start/Programs/Cabri Geometry II Plus/Cabri Geometry II Plus hoc bm chut vo logo ca Cabri Geometry trn mn hnh. 3.1.2. Giao din ca Cabri GeometryMenu bar
ci
H thng cng c
Vng v hnh
Hnh 3.1
Ca s lm vic ca Cabri Geometry bao gm cc thnh phn chnh nh: h thng menu bar, h thng cng c v vng lm vic dnh v, dng cc i tng hnh hc (hnh 3.1). 3.1.3. H thng menu bar ca Cabri Geometry H thng menu bar ca Cabri Geometry gm 5 nhm chc nng chnh, mi nhm ng vi mt h thng menu dc (PopUp). Nhm chc nng File: gm 11 chc nng (hnh 3.2)
28 www.mathvn.com
www.MATHVN.com
New (Ctrl+N): M mt tp mi. Open (Ctrl+O): M mt tp lu trn b nh ngoi. Khi xut hin ca s Open a File, ta phi chn a, th mc v tn tp tin cn m ri chn lnh Open. Close (Ctrl+F4): ng tp tin ang lm vic. Nu ta cha lu tr tp tin, xut hin thng bo (hnh 3.3). Khi nu chn Yes th Cabri Geometry s lu tr tp tin trc khiHnh 3.2
ng. Nu khng mun lu li thng tin ta chn No. Nu chn Cancel ta s tip tc lm vic vi tp tin hin thi.
Hnh 3.3
Save (Ctrl+S): Lu tr tp tin. Nu l ln lu tr u tin s xut hin ca s Save File As. Ta phi chn a, th mc v t tn cho tp tin ny. Nhng ln ghi sau, Cabri Geometry s ghi theo thng s chn (hnh 3.4). Save As: Lu tr tp tin c vi tn mi.
Hnh 3.4
Export figure for calcs...: Chuyn i tp tin theo nh dng ca cc my tnh in t c chc nng ho nh TI83; TI88; TI92... Revert: Chuyn giao din lm vic v tnh trng ban u. Show Page...: Xem ni dung trc khi in (c th chn vng in bng cch di chuyn khung ch nht n v tr cn thit).
29 www.mathvn.com
www.MATHVN.com
Page Setup...: nh cc thng s trc khi in ni dung tp. Print (Ctrl+P): Thc hin lnh in. Exit (Ctrl+F4): Kt thc phin lm vic. Nhm chc nng Edit: gm 8 chc nng (hnh 3.5)
Hnh 3.5
Undo (Ctrl+Z): Hu b lnh va thc hin. Cut (Ctrl+X): Xo cc i tng c la chn trn mn hnh v lu tm chng vo b m Clipboard. Copy (Ctrl+C): Lu tr tm thi cc i tng c la chn trn mn hnh vo b m Clipboard. Paste (Ctrl+V): a cc i tng ang lu tr trong b m Clipboard ra vng lm vic. Clear (Del): Xo b cc i tng c la chn. Select All (Ctrl+A): nh du la chn tt c cc i tng. Replay Construction: Xem li ton b qu trnh dng hnh. Refresh Drawing (Ctrl+F): Ly li ho tit ca hnh dng. Nhm chc nng Options: gm 6 chc nng (hnh 3.6)
Hnh 3.6
30 www.mathvn.com
www.MATHVN.com
Show/Hide Attributes (F9): Hin hay n bng la chn thuc tnh cc i tng. Show Figure Description (F10): n hay hin bng lit k cc thao tc dng hnh thc hin. Preferences...: Khai bo la chn cc tham s h thng nh la chn mu i tng, ch hin th, font ch h thng, dng phng trnh... (hnh 3.7).
Hnh 3.7
Nu mun thay i cc thuc tnh ca i tng no th cn phi khai bo, la chn trong danh sch cc mu sn c, ri bm chut vo : [x] Keep as defaults. Nu mun lu tr cu hnh bm chn lnh Save to file. Language...: La chn ngn ng hin th. S c nhiu la chn nh ting Anh, Php, c, an Mch... ta cn bm chut vo ngn ng cn s dng. Font: La chn kiu ch cho i tng ang c la chn. Nhm chc nng Session: gm 5 chc nng (hnh 3.8)
Hnh 3.8
31 www.mathvn.com
www.MATHVN.com
Begin recording... (F2): Bt u ghi li chui cc thao tc v, dng hnh... v lu tr di dng tp tin trong th mc ring. Stop playing/ Read a session (F4): Kt thc qu trnh ghi hay c mt recording c (khi ta c th xem li cc bc dng hnh c ghi). Previous (F6): Chuyn v thao tc trc . Next (F7): Chuyn n thao tc tip theo. Print a session (F5): Ghi ni dung recording ra file. Nhm chc nng Window H thng gm cc lnh dng b tr sp xp cc ca s theo kiu dn ngang hay lp ngi, hoc ng cc ca s ang m. Chc nng Help
Hnh 3.9
Nu bt chc nng Help, khi ta ch chut vo cng c no th pha di ca s s hin ln chc nng ca cng c (hnh 3.9). 3.2. Thao tc vi cc cng c ca Cabri Geometry H thng cng c ca Cabri Geometry gm 11 nhm chc nng. Biu tng ca cng c ang c la chn s c mu sng. s dng mt cng c no , ta bm chut vo biu tng nhm chc nng ri di chuyn chut bm chn cng c cn s dng. Phn ny chng ti ch lit k cc cng c ca Cabri Geometry. thc hnh, bn c nn thao tc da theo cc v d chi tit phn 2.
32 www.mathvn.com
www.MATHVN.com
3.2.1. Nhm chc nng chn trng thi lm vic vi chut Khi bm chut vo nhm chc nng ny, xut hin danh sch 4 cng c: Pointer: S dng la chn, dch chuyn cc i
tng hnh hc. Sau khi chn cng c Pointer: chn mt i tng no , ta ch chut vo i tng v bm (click), khi chut s c dng hnh bn tay v hin ln ch thch kiu ca i tng. chn nhiu i tng mt lc, ta nhn phm Shift trong khi ln lt bm chut vo cc i tng cn chn. di chuyn mt i tng, sau khi chn i tng ta gi phm chut trong khi di chuyn chut (drag) thay i v tr hnh v. ca hnh. Sau khi chn cng c Rotate ta bm chut xc nh tm quay sau bm chut vo i tng v gi phm xoay hnh. Dilate: Thay i kch thc ca hnh bng mt php ng dng. Sau khi chn cng c Dilate ta cn bm chut xc nh mt im c chn lm tm ca php ng dng sau bm chut vo i tng v gi phm ko thay i kch thc. hnh. 3.2.2. Nhm chc nng chn cng c to im Khi bm chut vo nhm chc nng ny, xut hin 3 cng c: Point: To im. Point chut c hnh dng bt ch, chn li Rotale and Dilate: C th cng mt lc va xoay va thay i kch thc ca Rotate: S dng xoay hnh xung quanh mt im hay tm
Khi chn cng c khng cng c. cn
a u bt ch n v tr xc nh im, bm chut tri. C th xc nh nhiu im m
Point on Object: Ly im thuc mt i tng c.
33 www.mathvn.com
www.MATHVN.com
Sau khi chn cng c
Point on Object, ta a chut ch vo
i tng, xut hin cu thng bo, chng hnly im ny trn ng trn... cn chn im v tr no, ta bm chut ti v tr (hnh 3.10) Hnh 3.10
Intersection Points: Xc nh im l giao ca cc hnh xc nh giao ca hai i tng no , ta chn cng c Intersection Points ri a chut
hnh hc c.
ln lt bm vo hai i tng . Cng c th ch chut vo v tr l giao ca cc i tng, khi xut hin dng thng bo Ly ti giao im ta bm chut (hnh 3.11). 3.2.3. Nhm chc nng chn cng c v cc iHnh 3.11
tng hnh hc Khi bm chut chn nhm chc nng ny, xut hin bng 7 cng c dng cc i tng hnh hc c bn: Line: Dng mt ng thng.
Mt ng thng c xc nh bi hai im. dng mt ng thng, trc ht ta chn cng c Line sau
a chut bm chn v tr hai im trn mn hnh. Khi thay i v tr mt trong hai im th ng thng cng thay i v tr mt cch tng ng. Segment: Dng mt on thng.
Thao tc dng on thng tng t nh dng ng thng. Ta chn cng c Segment ri sau a chut bm vo v tr ca hai u mt on thng cn dng. Ray: Dng mt tia. Ray
dng mt tia ta phi xc nh im gc v hng ca tia. Chn cng c chut xc nh im tip theo, ta c tia cn dng.
sau bm chut xc nh im gc ca tia, di chuyn chut chn hng ca tia v bm
34 www.mathvn.com
www.MATHVN.com
Vector: Dng mt vect. Vector ri sau ln lt bm chut xc
dng mt vect ta chn cng c
nh im gc v im ngn ca vect cn dng. Triangle: Dng mt tam gic. Triangle ri sau di chuyn v bm
dng mt tam gic, ta chn cng c 3 im chn. Polygon: Dng a gic n cnh.
chut ln lt xc nh v tr 3 nh ca tam gic, ta s nhn c tam gic tng ng vi
Tng t nh dng tam gic, ta chn cng c vi cc im chn. Regular Polygon: Dng a gic u (n 1 - Cho t s k < 1 - Cho t s k = 1 - Cho t s k = -1 ... - M ' N ' = k MN - MN =|k|MN - M v M cng pha vi im O - M v M hai pha so vi im O - M v M trng nhau - M v M i xng nhau qua im O ... Hon ton tng t , qua qu trnh tng tc vi Cabri Geometry HS s khm ph c cc tnh cht khc ca php v t nh bin gc thnh gc bng n, bin tam gic thnh tam gic ng dng vi t s ng dng l |k|... V d 3.22: Cho ng trn (O) c ng knh AB. Gi C l im i xng vi A qua B v PQ l ng knh thay i ca (O) khc ng knh AB. ng thng CQ ct PA v PB ln lt ti M, N[1]. a). Chng minh rng Q l trung im ca CM, N l trung im ca CQ. Cho PQ thay i v tr, sau mt vi trng hp, HS nhn thy hnh nh BQ//PM! S dng Cabri Geometry kim tra, kt qu HS nhn c thng bo:khi PQ thay i thHnh 3.47
Kt qu
BQ lun lun song song vi BM (hnh 3.47). V B l trung
64 www.mathvn.com
www.MATHVN.com
im ca AC nn BQ l ng trung bnh ca tam gic CAM hay Q l trung im ca CM. n y HS i chng minh BQ l ng trung bnh ca tam gic CAM. Hon ton tng t, HS s tm cch chng minh BN l ng trung bnh ca tam gic CAQ nn N l trung im ca CQ. b). Tm qu tch cc im M, N khi ng knh PQ thay i. Xut pht t vic chng minh c Q l trung im ca CM nn khi ng knh PQ thay i, ta lun c: CM = 2CQ . Nh vy php v t V tm C, t s k=2 bin im Q thnh im M. Suy ra khi ng knh PQ thay i, Q chy trn ng trn (O) th qu tch M l nh ca ng trn (O) qua php v t V, tc l ng trn (O) mCO ' = 2CO .
S dng Cabri Geometry cho PQ thay i v quan st vt ca im M, ta nhn c hnh nh qu tch M. Vic tm qu tch im N hon ton tng t. V d 3.23: Tam gic ABC c hai nh B, C c nh cn nh A chy trn mt ng trn (O, R) c nh khng c im chung vi ng thng BC. Tm qu tch trng tm G ca tam gic ABC. Sau khi HS pht hin c php v t tm I, t sk= 1 bin im A thnh im G. V A chy trn 3
ng trn (O, R) nn qu tch G l nh ca ng trn (O, R) qua php v t ni trn. GV t vn : Trong trng hp ng trn (O, R) c im chung vi ng thng BC th qu tch im G nh th no?Hnh 3.48
Nu s dng phng php truyn thng HS cn
phi v rt nhiu trng hp mi c th a ra d on ca mnh (v i khi thi gian khng cho php). Nu ta s dng Cabri Geometry th cng vic tr nn n gin. HS ch vic thay i v tr ca ng thng BC v cho im A di chuyn trn ng trn (O, R) pht hin ra trng hp suy bin: Ba im A, B, C thng hng, im A trng vi B hoc C
HS cng c th s dng thm cc chc nng o c, kim tra ca Cabri Geometry i n xc nh qu tch im G (hnh 3.48).
65 www.mathvn.com
www.MATHVN.com
V d 3.24: Cho tam gic ABC ni tip ng trn (O). D l mt im chuyn ng trn cung BC khng cha nh A. Ni A vi D. H CH vung gc vi AD. Minh ho qu tch ca im H. S dng Cabri ta v hnh, sau cho im D di chuyn, ta pht hin c t nht c 3 im c nh thuc qu tch: - im E (chn ng cao h t nh C n cnh AB tng ng vi trng hp khi D chy n trng vi B). - im C (tng ng vi trng hp D trng vi C). - im F (chn ng cao h t nh A n cnh BC, ng vi trng hp AD trng vi ng cao h t A n BC). Nh vy, ta d on qu tch l cung cha gc. Dng chc nng li vt s c hnh nh qu tch im H. V d 3.25: Cho hnh thoi ABCD c cnh AB c nh. Minh ho qu tch giao im O ca hai ng cho ca hnh thoi . Bc 1: S dng chut cho hnh thoi ABCD thay i. - Hnh thoi ABCD tr thnh hnh vung ABC1D1 => Xc nh im O1 thuc qu tch. - Hnh thoi ABCD tr thnh hnh vung ABC2D2 => Xc nh im O2 thuc qu tch. - Hnh thoi ABCD c im C tin trng vi im B => im O trng vi im B. Nh vy, bng trc quan cng nh bng kim tra ta thy r 3 im khng thng hng, vy qu tch c kh nng l mt ng trn i qua B. V vai tr im A v B nh nhau nn khi cho im D tin trng vi im A, ta pht hin c im A cng thuc qu tch. Ta d on qu tch im O l ng trn nhn AB l ng knh. Bc 2: V mt trng hp bt k, ta kim tra im O c thuc ng trn nhn AB l ng knh hay khng. Kt qu cho thy im ny nm trn i tng.
66 www.mathvn.com
www.MATHVN.com
V d 3.26: Trong mt ng trn (O), AB l mt ng knh c nh, M l mt im chy trn ng trn. Ni MA, MB v trn tia i ca tia MA ta ly im I sao cho MI = 2MB. Tm tp hp cc im I ni trn. Vi Cabri ta cho v tr im M thay i, qua ba v tr c th ta c ngay d on: qu tch im I khng th l thng, nh vy c kh nng qu tch im I l mt cung cha gc. T y gi cho ta i tm yu t gc khng i. iu c bit bi ny l: Nu s dng tnh lun t ng dng ca tam gic MBI th ch dng vic a ra kt lun gc AIB khng i. Vy qu tch l cung cha gc dng trn on thng AB. Tuy nhin, vi Cabri ta c c kt lun tng i th v. Qu tch im I l na ng trn ng knh BIo. Trong Io nm trn tip tuyn vi ng trn ti im A sao cho AIo = 2AB. Ta m rng bi ton theo hai hng sau: + AB khng phi l ng knh m ch l mt dy cung ca (O). + MI = k.MB (vi k l s thc dng cho trc). Kt qu cng rt th v. Qu tch l mt phn ca cung cha gc i qua A, B.
67 www.mathvn.com
www.MATHVN.com
Chng 4 S DNG PHN MM MAPLE TRONG DY HC TON4.1. Tng quan chung v phn mm Maple Phn mm Maple l kt qu ca nhm cc nh khoa hc trng i hc Waterloo - Canada v l mt trong nhng b phn mm ton hc c s dng rng ri nht hin nay. MAPLE l phn mm c mt mi trng tnh ton kh phong ph, h tr hu ht cc lnh vc ca ton hc nh: Gii tch s, th, i s hnh thc... do ta d dng tnh c cc gi tr gn ng, rt gn biu thc, gii phng trnh, bt phng trnh, h phng trnh, tnh gii hn, o hm, tch phn ca hm s, v th, tnh din tch, th tch, bin i ma trn, khai trin cc chui, tnh ton thng k, x l s liu, s phc, phng trnh vi phn, phng trnh o hm ring... v lp trnh gii cc bi ton vi cu trc chng trnh n gin. Ngoi ra, vi phn mm ny ta d dng bin son cc sch gio khoa in t vi chc nng Hyperlink to cc siu vn bn rt n gin m khng cn n s h tr ca bt k mt phn mm no khc (chng hn PageText, Word, FrontPage...). Vi cc chc nng trn, MAPLE l cng c c lc h tr cho nhng ngi lm ton. Khi ng Maple: Nu Maple c ci t ng quy trnh, lm vic vi MAPLE ta chn: ->Start -> Programs -> Maple9 -> Classic Worksheet Maple 9 hoc bm chut vo biu tng ca Maple 9 trn mn hnh:
Giao din ca ca s lm vic ca MapleGiao din lm vic ca Maple gm cc thnh phn c bn nh sau:
68 www.mathvn.com
www.MATHVN.com
H thng Menu lnh
Thanh cng c
Lnh ca Maple
Kt qu thc hin lnh ca Maple
Cc thnh phn chnh ca ca s lm vic ca Maple: -Tittle Bar (Thanh tiu ): Dng cha tn chng trnh v tp ang m. - Menu Bar (Thc n ngang): Dng cha cc chc nng, ng vi mi chc nng l mt thc n dc tng ng. -Tool Bar (Thanh cng c): Cha mt s biu tng (Icon) th hin mt s lnh thng dng ngi s dng thao tc nhanh.
Ngoi ra trong ch vn bn Maple cn c thanh cng c Formatting Bar dng nh dng vn bn.
Qun l thng tin vi Maple Vi Maple, cc thao tc c bn nh: lu tr tp, m mt tp c, m mt tp mi,... hon ton tng t nh cc phn mm quen thuc trong mi trng Windows nh Winword, Excell,... nh dng cc i tng trong Maple
69 www.mathvn.com
www.MATHVN.com
nh dng cc i tng trong Maple, nh thay i kiu ch ca cc dng lnh, cc dng thng bo kt qu, l... ta tin hnh nh sau: Bc 1: La chn i tng. Bc 2: -> Format -> Paragraph. Khi xut hin bng chn cc tham s. thay i cc thng s ngm nh, ta chn: -> Format-> Styles. Xut hin bng ta khai bo cc thng s cn xc nh. Cc i tng c bn tch hp trong mt Maple Trang cng tc (Worksheet) l mi trng m ngi s dng c th tnh ton, thc hnh trn - cn c gi l trang cng tc. Khi ngi s dng lu tr cc kt qu ln a t, mi Worksheet c ghi thnh mt File vi phn m rng ngm nh l mws. Mt Worksheet ca Maple thng c nhng thnh phn sau: -Cm x l (Execution group) bao gm cc i tng c bn ca Maple nh: lnh, kt qu tnh ton ca Maple, th,... tp tin ca
70 www.mathvn.com
www.MATHVN.com
to mt cm x l mi, ta kch chut vo biu tng [> trn thanh cng c hoc chn: -> Insert -> Execution Group -> After cursor. -on (Paragraph): Khi nim Paragraph vi Maple c hiu nh khi nim Paragraph ca phn mm son tho vn bn Winword. to mt Paragraph mi, ta chn: Insert-> Paagraph -> After cursor. -Mc (Section): Mc c th coi nh l cc modul thnh phn cu thnh nn trang cng tc. Mt trang c th gm nhiu mc, mi mc c th cha nhng on v nhng mc con. Biu tng ca mc l du [+], nu ta nhy chut vo biu tng ny th ni dung ca mc c tri ra v biu tng ca mc s bin thnh [-], nu ta nhy chut vo biu tng [-] ny th ni dung ca mc s thu li. to mc mi, ta chn: -> Insert-> Section. -Siu lin kt (Hyperlink): Khi nim siu lin kt tr nn rt quen thuc vi chng ta trong thi i bng n ca Internet. Mt siu lin kt l i tng m nu ta kch hot vo th s dn ta n mt on, mt mc hay mt Worksheet no khc. to siu lin kt ta chn i tng mang siu lin kt sau chn: -> Format -> Convert to -> Hyperlink. Ti mc: Link Target c cc s la chn: -URL: Lin kt n mt a ch websize no . -Worksheet: Lin kt n mt tp no ca Maple. -Help Topic: Chuyn n mt ch trong ni dung Help ca Maple. -Bookmark: Chuyn n mt bookmark no c nh ngha trc . C th nhn Browse tm kim a ch ch ca mi lin kt. Khai bo xong nhn OK hon tt. - Vn bn (Text): l i tng c s dng rt nhiu trong Maple vi mc ch cung cp thng tin di dng vn bn. to on vn bn mi, ta kch chut vo biu tng ch [T] trn thanh Tool Bar hoc c th chn: -> Insert -> Text.
71 www.mathvn.com
www.MATHVN.com
Lnh v Kt qu ca Maple (Maple Input and Output). Lnh ca Maple (Maple Input) l nhng t ta ting Anh c s dng theo mt ngha nht nh v phi tun theo c php ca Maple. Lnh c nhp sau du nhc lnh "[>" v kt thc bi du : hoc ; , v d gii phng trnh 5x2 + 3x- 2 = 0, ta g lnh [> solve(5*x^2 + 3*x- 2,{x}); . Mi cu lnh ca Maple nu kt thc lnh bng du (;) kt qu s hin th ngay ra mn hnh, nu kt thc lnh bng du (:) th Maple vn tin hnh tnh ton bnh thng nhng kt qu khng hin th ra mn hnh. Lnh c thc hin khi con tr trong hoc cui dng lnh m ta nhn Enter (k hiu ). Lnh ca Maple c hai loi lnh tr v lnh trc tip: Lnh tr v lnh trc tip ch khc nhau ch ch ci u tin ca lnh tr vit in hoa. Lnh trc tip cho kt qu ngay, cn lnh tr ch cho ta biu thc tng trng. V d: Tnh2 x + 4 3 64 x ( thi tuyn sinh HTN - Khi D - 1999) Lim x x > 0
Nu ta s dng lnh tr Limit: [> Limit((2*sqrt(x+4)-(64-x)^(1/3))/x,x=0); kt qu nh sau:2 x + 4 ( 64 x ) lim x x 0( 1/3 )
Nu ta s dng lnh trc tip: [> limit((2*sqrt(x+4)-(64-x)^(1/3))/x,x=0); kt qu nh sau:( 1/3 ) 1 1 + 64 2 192
Tuy nhin kt qu trn cha gn, ta c th s dng lnh sau: [> simplify(limit((2*sqrt(x+4)-(64-x)^(1/3))/x,x=0)); kt qu nh sau:25 48
nh vy kt qu 2 + 192 64
1
1
( 1/3 )
sau khi rt gn l
25 . 48
Kt qu tnh ton (Maple Output) s c a ra mn hnh, thng l mu xanh c ban sau khi ta nhn phm enter thc hin cu lnh. Tuy nhin Maple cng c ch cho php thc hin nhm cc cu lnh (nh tp bat ca MS - DOS) ngi s dng thc hin mt nhm cc cu lnh nhm gii quyt mt
72 www.mathvn.com
www.MATHVN.com /2
vn no , v d tnh tch phn I2n= 1996). Ta nhp vo 2 dng lnh sau:
sin0
2n
xdx ( thi tuyn sinh HTN - khi A -
[> Int(sin(x)^(2*n),x=0..Pi/2); (nhn t hp Shift + Enter xung dng) T:=n->int(sin(x)^(2*n),x=0..Pi/2); mn hnh s hin kt qu nh sau: 01/2
sin( x )1/2
(2 n )
dx
T := n 0
sin( x )
(2 n )
dx
tnh gi tr tch phn vi mt n c th ta ch vic g lnh [>T(n) , chng hn vi n = 0, I0 = I100=2
, n=10, I10 =
46189 , vi n = 100, ta c: 524288
11318564332012910145675522134685520484313073709426667105165 401734511064747568885490523085290650630550748445698208825344 /2
v cn tnh I2009
sin0
2009
xdx , ta ch g [>T(2009) .
- th (Graph): Maple cho php v v hin th th trong trang cng tc, tnh nng ny c gi l "Kh nng ho trc tip". V d: v th hm s y=1 3 2 x mx 2 x + m + vi m = 0 ( thi tuyn sinh vo 3 3
HTN - nm hc 1999 - 2000, khi A, B). Ta s dng lnh plot nh sau [> plot(x^3/3-x+2/3,x=-3..2); . Kt qu ta c th nh sau:
73 www.mathvn.com
www.MATHVN.com
4.2. S dng cc lnh n gin ca Maple Maple cung cp mt h thng cc hm hu nh ph khp cc lnh vc ca ton hc, bn c cn tham kho nhng ti liu ca nhm tc gi Phm Huy in, inh Th Lc, T Duy Phng, y chng ti ch m t mt s cu lnh n gin thng s dng. -Lnh xo i tt c cc bin nh ca vic tnh ton trc v khi ng mt quy trnh tnh ton mi: [> restart; - xc nh gi tr cho mt bin, mt hng, mt hm hoc khai bo mt th tc, Maple s dng cu lnh gn := , v d:Xc nh bin n nhn gi tr bng 5: [> n := 5;
n := 5
[> z := (x^2 + 1)/(x - y); #Khai bo dng tng qut cu Z z := x2 + 1 xy
-Lnh tm s nguyn t ng trc s nguyn a xc nh: prevprime(a); V d, vi a = 100, ta g lnh: [> prevprime(100); - Lnh tm s nguyn t ng sau s nguyn a: nextprime(a); V d a = 100 , ta g lnh: [> nextprime(100); -Lnh tm nghim nguyn ca phng trnh: isolve(f,{x,y...}); Trong f l biu thc ca phng trnh hoc h phng trnh, {x,y...} l danh sch cc n. V d, tm nghim nguyn ca bi ton c va g va ch b li cho trn 36 con 100 chn chn. Gi s g l x, s ch l y ta thc hin lnh: [> isolve({2*x+4*y=100,x+y=36},{x,y});{ y = 14, x = 22 }
Kt qu cho ta p s ca bi ton l: s g l 22, s ch l 14. -Lnh tm thng v phn d: iquo(a,b); v irem(a,b); trong a, b l cc biu thc. V d vi a=23, b = 4, ta g lnh: [> Thuong = iquo(23,4); Thuong = 5
[> Du = irem(23,4); Du = 3
74 www.mathvn.com
www.MATHVN.com
-Lnh tnh tng v hn v tng hu hn cc s hng sum(f,n=a..b); trong f l biu thc ca s hng tng qut, a..b N l cn di, cn trn ca gii hn tnh tng, v d: [> F = Sum((1+n^2)/(1+n+n^3),n=1..10); F=n=1 10
1 + n2 1 + n + n3
[>F = sum((1+n^2)/(1+n+n^3),n=1..10); F=
26427131228884246127 10434641980997032227
[>F = evalf(sum((1+n^2)/(1+n+n^3),n=1..10)); F = 2.532634208
-Lnh tnh tch hu hn v v hn cc s Product(f, n=a..b); trong f l biu thc ca s hng tng qut, a..b N l cn di, cn trn ca gii hn tnh tch, v d: [> F = Product((n^3+5*n+6)/(n^2+1),n=0..5); F=n=0 5
n3 + 5 n + 6 1 + n2
[>F = product((n^3+5*n+6)/(n^2+1),n=0..5); F= 2239488 85
-Xc nh chnh xc ca cc php tnh s hc: evalf(f,n); trong f l biu thc, n l s cc ch s sau du phy, v d: [> evalf(Pi,30); 3.14159265358979323846264338328
-Tnh ton vi cc s phc c Maple thc hin theo quy tc thng thng, v d: [> (1+3*I)*(3+8*I); -21 + 17 I
[> (1+3*I)/(3+8*I); 27 1 + I 73 73
-Chuyn s phc x v dng to cc: convert((x),polar), v d: [> convert((1+3*I)/(3+8*I),polar); 1 1 polar 73 730 , arctan 27
-Lnh khai trin biu thc i s: expand(f), v d: [> expand((2*x+y)^3); 8 x3 + 12 x2 y + 6 x y2 + y3
75 www.mathvn.com
www.MATHVN.com
-Lnh phn tch a thc thnh nhn t: factor(f), v d: [> factor(6*x^2+18*x-24); 6 (x + 4) (x 1)
-Lnh xc nh bc ca a thc: degree(f),v d: [> degree(x^12-x^10+x^15+1); 15
-Lnh vit a thc di dng bnh phng ca tng: completesquare() (lnh ny phi phi m gi cng c student), v d: [> with(student): completesquare(x^2 - 2*x*a + a^2 + y^2 -2*y*b + b^2 = 23, x); ( x a )2 + y2 2 y b + b2 = 23
-Lnh sp xp a thc theo bc: collect(f,x), trong f l biu thc, x l n chn xp theo th bc, v d: [> f := a^3*x-x+a^3+a; f := a 3 x x + a 3 + a
[> collect(f,x); ( a3 1 ) x + a3 + a
[> collect(f,x,factor); ( a 1 ) ( a2 + a + 1 ) x + a ( a2 + 1 )
-Lnh n gin (rt gn) biu thc: simplify(), v d: [> e := cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2*sin(x)^2 - cos(2*x): simplify(e); cos( x )4 ( cos( x ) + 1 )
-Lnh ti gin phn thc: normal(), v d: [> normal( (x^2-y^2)/(x-y)^3 ); x+y ( x y )2
-Lnh kh cn thc mu s: readlib(). Trc khi thc hin lnh ny cn m th vin readlib(rationalize), v d:[> readlib(rationalize):2/(2-sqrt(2)); 2 2 2
[> rationalize(%);2+ 2
-Khai bo hm s: nh ngha hm s ta dng du gn (:=). V d: [> f:=x->2*x^3+x^2/3+x-1;
76 www.mathvn.com
www.MATHVN.com1 f := x 2 x3 + x2 + x 1 3
Sau khi nh ngha hm s ta c th tnh gi tr ca hm s, v d tnh gi tr ca hm s ti x=0.12345: [> f(0.12345); -0.8677073006
-Gii phng trnh solve(f,{d/s bin}) Bc 1: nh ngha phng trnh bi lnh gn := , v d : [> eq := x^4-5*x^2+6*x=2; eq := x4 5 x2 + 6 x = 2
Bc 2: gii phng trnh bng lnh solve(); [> solve(eq,x); 1, 1, 3 1, 1 3
-Gii h phng trnh solve({d/s pt},{d/s n}). Bc 1: nh ngha cc phng trnh bng lnh gn :=, v d : [> Pt1:=x+y+z-3=0; Pt1 := x + y + z 3 = 0
[>Pt2:=2*x-3*y+z=2; Pt2 := 2 x 3 y + z = 2
[>Pt3:=x-y+5*z=5; Pt3 := x y + 5 z = 5
Bc 2: gii phng trnh bng lnh solve. [> solve({Pt1,Pt2,Pt3},{x,y,z});{z = 9 17 7 ,x= ,y= } 11 11 11
-Gii bt phng trnh solve() : Bc 1: nh ngha cc bt phng trnh bng lnh gn := [> Bpt:=sqrt(7*x+1)-sqrt(3*x-18) solve(Bpt,{x});{9 x}
Ta c th gii trc tip bt phng trnh trn nh sau :
77 www.mathvn.com
www.MATHVN.com
[> solve(sqrt(7*x+1)-sqrt(3*x-18) Bpt1:=x^3-11*x^2+10*x Bpt2:=x^3-12*x^2+32*x>0; Bpt2 := 0 < x3 12 x2 + 32 x
Bc 2: dng lnh :[> solve({Bpt1,Bpt2},x); { 1 < x, x < 4 }, { 8 < x, x < 10 }
Hoc ta c th a trc tip bt phng trnh vo trong cu lnh nh sau: [> solve({x^3-11*x^2+10*x0},x); { 1 < x, x < 4 }, { 8 < x, x < 10 }
V th trong khng gian hai chiu plot().
V d: v th hm s x4+2x3-x2+1[> restart: with(plots): [> plot(x^4+2*x^3-x^2+1,x=-3..3,-4..12);
V d: v th hm s |x4+2x3-x2+1| [> plot(abs(x^4+2*x^3-x^2+1),x=-3..3,-4..12);
78 www.mathvn.com
www.MATHVN.com
V d : v th hm s y= x4+2x3-x2+1v y=2x3-2*x+2 trn cng h trc to : [> plot([x^4+2*x^3-x^2+1,2*x^3-2*x+2],x=-3..3,-4..12);
V th hm n vi lnh implicitplot(). V d: v elip c phng trnh x2/9 + y2/4 = 1. [> implicitplot(x^2/9+y^2/4=1,x=-4..4,y=-2..2);
V th hm xc nh tng khc: Trc ht cn, khai bo hm tng khc vi cu lnh: piecewise() , [> piecewise(x*x>4 and x plot(piecewise(x*x>4 and x plot([sin(t),cos(t)],t=-Pi..Pi);
V th trong khng gian ba chiu Trc tin ta khi ng chng trnh v np th vin : [> restart: with(plots): with(plottools): Tip theo v mt hai chiu trong khng gian ba chiu bng lnh plot3d() [> f:=x*exp(-x^2-y^2); [> plot3d(sin(x+y), x=-1..1, y=-1..1);
[> c1:= [cos(x)-2*cos(0.4*y), sin(x)-2*sin(0.4*y), y]: c2:= [cos(x)+2*cos(0.4*y), sin(x)+2*sin(0.4*y), y]: c3:= [cos(x)+2*sin(0.4*y), sin(x)-2*cos(0.4*y), y]: c4:= [cos(x)-2*sin(0.4*y), sin(x)+2*cos(0.4*y), y]: plot3d({c1, c2, c3, c4}, x=0..2*Pi, y=0..10, grid=[25,15], style=patch, color=sin(x)); V th phc thuc tham s (ng)
80 www.mathvn.com
www.MATHVN.com
Maple c chc nng h tr s vn ng (animation) ca th hai chiu v th ba chiu vi c php [>animate (cho th hai chiu), v c php [>animate3d (cho th ba chiu). V d: V th hm y=tsin(xt). [> animate(t*sin(x*t),x=-Pi..Pi,t=-2..2);
Mun cho th chuyn ng th ti khung th ta nhn chut phi sau chn -> Animation -> Play
Mun cho th chuyn ng lin tc khng ngng trn thanh cng c: nhp chut vo nt th th s chuyn ng lin tc, nt Tng t nh vy ta c th cho th ba chiu vn ng. [> animate3d(cos(t*x)*sin(t*y),x=-Pi..Pi,y=-Pi..Pi,t=1..2); dng s vn ng.
81 www.mathvn.com
www.MATHVN.com
Khai bo hm bng ton t -> khai bo mt hm (f) c xc nh bi biu thc (bt(x)), trong Maple s dng ton t -> vi c php nh sau: [> tn hm := x-> biu thc xc nh hm f (i s).V d [> f:=x-> x^3+2*x-5*x+4; f := x x3 3 x + 4
Sau khi khai bo hm, tnh gi tr ca hm ti mt gi tr no , ta ch vic thay gi tr c th vo li gi hm [> f(value);, v d: [> f(3); 22
Khai bo hm t to bng proc()..... end Gia proc(d/s tham s)... end: l cc cu lnh ca hm. v d: [> Max:=proc(a,b,c) if a < b then if b < c then c else b fi; elif a < c then c else a fi; end: Sau nu gi hm Max vi cc s c th s c kt qu v d: [> Max(23,5,87); Cc cu trc c bn c s dng trong lp trnh ca Maple - Cu trc lp iu kin trc While Do < danh sch cc cu lnh> od; Vng lp While s thc hin lp i lp li cc cu lnh gia do v od nu iu kin sau t kho While cn ng. Nu mun thot sm khi vng lp cn phi s dng cc lnh Return, Break, Quit.
82 www.mathvn.com
www.MATHVN.com
V d: thut ton clit tm c s chung ln nht ca hai s t nhin: [> restart; [> a:=126:b:=34: # khai bao hai so tu nhien a=126, b=34 [> while b 0 do d:=irem(a,b); a:=b; b:=d; od; print(` USCLN cua hai so la:`); value(a); V d: vit ra mn hnh n s hng u ca dy Fibonacci [>restart; f(0):=1; f(1):=1; n:=2; while n in Do ; od; V d: tnh tng bnh phng cc s chn trong mng: [> restart; mang:=[2,5,7,8,9,23,45,67,89,24,36,42]; tong:=0; for i in mang do if irem(i,2)=0 then tong:=tong+i^2; fi; od; print(` tong can tim la:`,tong); Khi thc hin ta c kt qu:mang := [ 2, 5, 7, 8, 9, 23, 45, 67, 89, 24, 36, 42 ] tong := 0 tong can tim la:, 3704
- Cu trc r nhnh
83 www.mathvn.com
www.MATHVN.com
if then < d/s cc cu lnh 1>; elif < iu kin 2> then < d/s cc cu lnh 2>; else < d/s cc cu lnh 3>; fi; V d gii phng trnh bc 2, trc tin ta khai bo mt proc() : [> ptb2:=proc(a,b,c) local delta,x1,x2; delta:=b*b-4*a*c; if delta < 0 then print(` phuong trinh da cho vo nghiem`); elif delta=0 then x1:=-b/(2*a); print(` phuong trinh co nghiem kep:x1=`,x1); else x1:=(-b-sqrt(delta))/(2*a); x2:=(-b+sqrt(delta))/(2*a); print(` phuong trinh da cho co 2 nghiem phan biet :`); print(x1); print(x2); fi; end; gii phng trnh bc hai, ta ch cn gi tn proc() vi cc h s thc s, v d: [> ptb2(1,2,1); phuong trinh co nghiem kep:x1=, -1
[> ptb2(1,2,-1); phuong trinh da cho co 2 nghiem phan biet : 1 2 1 + 2
[> ptb2(1,2,3); phuong trinh da cho vo nghiem
tm hiu v lp trnh vi Maple, bn c s tm thy nhng hng dn chi tit, chuyn su trong cc ti liu [1],[2],[3],[4] 4.3. S dng cc cu lnh ca Maple h tr dy hc kho st hm s 4.3.1. Nhng cu lnh ca Male h tr dy hc kho st hm s Ta c th s dng cc hm ca Maple khi kho st hm s, chng hn nh: xc nh min gi tr, khong n iu, min li, cc tr v im un, v th,... Xc nh min xc nh ca hm s f(x):
84 www.mathvn.com
www.MATHVN.com
xc nh min gi tr ca cc hm phn thc c trnh by trong sch gio khoa gii tch lp 12, trc tin ta dng lnh denom() tch ly mu s. Min xc nh ca hm s chnh l tp cc gi tr lm cho mu s c ngha. V d tm min xc nh ca hm s:Y=
x2 + x + 1 2x+2
Ta dng nhm cc lnh sau: [> restart; Y:=(x^2+x+1)/(2*x+2); Y:=simplify(Y): print(`Tap xac dinh cua ham so la:`); a:=solve(denom(Y)=0,x): if(type(denom(Y),realcons)=true)or(coeff(denom(Y),x^2)0 and type(a[1],realcons) =falsse) then D=R;fi; if coeff(denom(Y),x^2)=0 and coeff(denom(Y),x)0 then D={xa};fi; Kt qu thc hin chng trnh:Y := x2 + x + 1 2x+2
Tap xac dinh cua ham so la: D = { x -1 }
Tm khong n iu ca hm s:
Bc 1: Tm o hm ca hm s vi lnh: [> diff(f(x),x); Bc 2: Xc nh chiu bin thin: Xc nh khong ng bin ca hm s (tc l tm nhng khong m o hm ca hm s khng m), ta s dng lnh: [> dhbn := bieuthuc f'(x) >=0; Bc 3: Gii phng trnh bng lnh [> solve(dhbn,{x}); Xc nh khong nghch bin ca hm s, tng t nh trn, ta dng lnh: dhbn := bieuthuc f'(x) solve(dhbn,{x});3 2 Th d: Tm khong n iu ca hm s: y = x 6 x + 4 x 8
[>
Bc 1: Tnh o hm: [> diff(x^3-6*x^2+4*x-8,x); 3 x2 12 x + 4
Bc 2: Thit lp bt phng trnh [> dhbn:=(3*x^2-12*x+4>=0);
85 www.mathvn.com
www.MATHVN.comdhbn := 0 3 x 2 12 x + 4
Bc 3: Gii bt phng trnh: [> solve(dhbn,{x});{x 2 2 2 6 }, { 2 + 6 x} 3 3
Tm min li, min lm ca hm s
Bc 1: Tnh o hm bc nht: [> dhb1:=diff(f(x),x); Bc 2: Tnh o hm bc hai: [> dhb2:=diff(dhb1,x); Bc 3: Gii bt phng trnh f"(x) 0 tm min li ca hm s, bng lnh: [> solve(dhb2>=0,x); V d xt hm sy = x4 2 x2
Bc 1: Tm o hm bc nht: [> a:=diff(x^4-x^2,x); a := 4 x3 2 x
Bc 2: Tm o hm bc 2 : [> b:=diff(a,x); b := 12 x2 2
Bc 3: Gii phng trnh tm min dng ca o hm bc 2 (min li ca hm s) [> solve(b>=0,x); 1 1 RealRange , 6 , RealRange 6 6, 6
Tm cc i, cc tiu: xc nh cc i, cc tiu ca hm s ta xt o hm bc nht v tnh n iu
ca hm s hoc dng tnh li thng qua o hm bc hai, c th: Bc 1: Tm o hm ca hm s: [> diff(f(x), x); Bc 2: Gii phng trnh f'(x)=0 tm cc im nghi ng l cc tr. [> solve(f'(x)=0, x); Bc 3: Tm khong ng bin v nghch bin ca hm s: [> solve(f'(x)>=, x); Bc 4: Xt xem ti x0 : 1) Nu o hm i du t dng sang m th x0 l im cc i. 2) Nu o hm i du t m sang dng th x0 l im cc tiu. 3) Nu qua x0 o hm khng i du th x0 khng phi l im cc tr.3 2 V d tm cc tr ca hm s y = x 6 x + 4 x 8
86 www.mathvn.com
www.MATHVN.com
Bc 1: [> a:=diff(x^3-6*x^2+4*x-8,x); a := 3 x2 12 x + 4
Bc 2: [> b:=solve(a=0,{x});b := { x = 2 + 2 2 6 }, { x = 2 6} 3 3
Bc 3: [> c:=solve(a>=0,{x});c := { x 2 2 2 6 }, { 2 + 6 x} 3 3
Bc 4: [> c:=solve(a dhb1:=diff(f(x),x); Bc 2: Gii phng trnh f'(x)=0 tm cc im nghi ng l cc tr. [> solve(dhb1=0,x); Bc 3: Tm o hm bc hai: [> dhb2:=diff(dhb1,x); Th d Tm cc tr cay = x3 6 x2 + 4 x 8 .
Bc 1: [> a:=diff(x^3-6*x^2+4*x-8,x); a := 3 x2 12 x + 4
Bc 2: Tm nhng im m o hm bc nht bng 0: [> solve(a=0,{x});{x = 2 + 2 2 6 }, { x = 2 6} 3 3
Bc 3: Tm o hm bc hai: [> b:=diff(a,x); b := 6 x 12
Bc 4: Tnh gi tr ca o hm bc hai ti nhng im m ti o hm bc nht bng khng: [> subs(x=2+2/3*sqrt(6),b);
87 www.mathvn.com
www.MATHVN.com4 6
[> subs(x=2-2/3*sqrt(6),b); 4 6
Bc 5: Xt gi tr ca o hm bc hai v kt lun, chng hn v d ny, ta c: y"( 2 + (2 2 6 )= 4 6 >0 3
nn
x2 = 2 +
2 6 3
l im cc tiu, cn
y"
2 6 2 6 )= 4 6 < 0 nn x1 = 2 3 3
l im cc i ca hm s.
Tm im un: im un l im m ti o hm bc hai i du. xc nh im un ca
hm s, ta ln lt thc hin cc cu lnh sau: Bc 1: Tnh o hm bc nht: [> dhb1:=diff(f(x),x); Trong f( x ) l hm s m ta cn kho st Bc 2: Tnh o hm bc hai: [> dhb2:=diff(dhb1,x); Bc 3: im x0 l im un ca hm s nu x0 l nghim chung ca hai bt phng trnh: [> solve(dhb2>=0); v [> solve(dhb2 a:=diff(x^4-2*x^2,x); a := 4 x3 4 x
Bc 2: [> b:=diff(a,x); b := 12 x2 4
Bc 3: [> solve(b >=0); solve(b minimize(exp(tan(x)), x=0..10);
88 www.mathvn.com
www.MATHVN.com0
[> minimize(x^2-3*x+y^2+3*y+3); -3 2
[> minimize(x^2-3*x+y^2+3*y+3, location); -3 3 -3 -3 , { { x = , y = }, } 2 2 2 2
[> minimize(x^2-3*x+y^2+3*y+3, x=2..4, y=-4..-2, location); -1, { [ { y = -2, x = 2 }, -1 ] }
[> minimize(abs(exp(-x^2)-1/2), x=-4..4); 0
[> minimize(x^4 - x^2, x=-3..3, location=true); -1 1 -1 1 -1 , {{x = 2 }, , { x = 2 }, } 4 2 4 2 4
[> minimize(x^2 + cos(x), [x=0..3]); 1
Xc nh cc tr a phng ca hm s: extrema(f,{},x). V d xc nh gi tr cc tiu, cc i ca hm s x4- 2x2 [> y:=x^4-2*x^2 ;# khai bo hm sy := x4 2 x2
[> extrema(y,{},x);# xc nh cc i, cc tiu ca hm s{ -1, 0 }
Xc nh cc ng tim cn: Ta s dng lnh tch mu s ca f(x) bi lnh denom(), dng lnh solve() tm nghim ca mu s ta c tim cn ng. Ln lt tnh cc gii hn a= lim(f(x)/x) v b=lim(f(x)-ax) khi x tin ti v cng, nu cc gii hn ny tn ti s cho ta tim cn xin y=ax+b. V d xc nh tim cn ca hm s :Y= x2 + x + 1 2x+2
[> Y:=(x^2+x+1)/(2*x+2); a:=limit(Y/x,x=infinity); b:=limit(Y-a*x,x=infinity); ms:=solve(denom(Y)=0,x);
89 www.mathvn.com
www.MATHVN.com
if ainfinity or a -infinity then print(`tiem can dung:`,x=ms); print(` tiem can xien y=`,x*a+b); fi; Kt qu thc hin chng trnh:Y= x2 + x + 1 2x+2a := 1 2
b := 0 ms := -1 tiem can dung:, x = -1 tiem can xien y=, x 2
Xc nh giao im ca th hm s Y=f(x )vi cc trc to . S dng gi cng c student, sau dng cc lnh: Tm giao im vi trc tung intercept(y=Y,x=0,{x,y}), tm giao im vi trc honh: intercept(y=Y,y=0,{x,y}), v d xc nh giao im vi cc trc ton ca hm s: Y = [> restart:with(student): Y:=(x^2+x+1)/(2*x+2); # m gi cng c v khai bo hm [> intercept(y=Y,y=0,{x,y}); # giao im vi trc honh intercept(y=Y,x=0,{x,y}); # giao im vi trc tung Kt qu thc hin chng trnh:{ y = 0, x = RootOf( 1 + _Z 2 + _Z, label = _L1 ) } 1 {y = , x = 0} 2
x2 + x + 1 2x+2
Nh vy th hm s khng ct trc honh m ch c mt giao im duy nht vi trc tung ti im c to x = 0, y = 1/2. V th hm s: V th l mt trong nhng chc nng mnh ca Maple. v
90 www.mathvn.com
www.MATHVN.com
th hm s f(x) trn on [a,b], ta s dng lnh: [>plot(f(x),x=a..b); v d vi hm s x4-2x2 trn on [-2, 2]: [> plot(x^4-2*x^2,x=-2..2); Kt qu ta c th nh hnhv. Qua hnh dng th, mt ln na chng t vic xc nh chiu bin thin, im un, chiu li, lm, cc i, cc tiu .. vi cc cu lnh ca Maple 4.3.2. Mt s v d minh ho V d 1: Kho st hm s y=1 3 2 x mx 2 x + m + vi m = 0 ( thi tuyn sinh vo 3 3
H Thi Nguyn - Nm hc 1999-2000, khi A, B). Ta s dng chng trnh con sau: [> Restart: print(` Khao sat ham so y=x^3/3-mx^2-x+m+2/3 voi gia tri m=0`); Y:=(x^3/3-x+2/3); print(`Tap xac dinh cua ham so la D=R:`); print(`Tinh dao ham bac nhat cua ham so`); dy/dx=factor(simplify(diff(Y,x))); print(`giai phuong trinh f' = 0:`); solve(diff(Y,x)=0,{x}); print(` Ham so se dong bien tren khoang`); solve(diff(Y,x)>0); print(` Ham so nghich bien tren khoang`); solve(diff(Y,x)0); print(` Ham so nghich bien tren khoang`); solve(diff(Y,x) with(Student[Calculus1]): NewtonsMethodTutor(f, a) NewtonsMethodTutor(f, var=a) Trong : f - (tu chn) biu thc i s cha mt bin, a - (tu chn) im, var - (tu chn) bin s. V d.Tm nghim gn ng ca phng trnh x5 + x2-1=0 trong khong (-2; 2). Ta t f(x) = x5 + x2 - 1. Khi , d thy f(0).f(1) = - 1 < 0 f(x) = 0 c nghim trong khong (0; 1): [> with(Student[Calculus1]): NewtonsMethodTutor(x^5 + x^ 2 - 1, x = 0..2);
Minh ho kt qu ca nh l Lagrng Cu lnh:[> > with(Student[Calculus1]): MeanValueTheoremTutor(f, var=a..b) Trong : f - (tu chn) biu thc i s cha mt bin, a..b - (tu chn) min v th, var - (tu chn) bin s. V d.[> with(Student[Calculus1]):
132 www.mathvn.com
www.MATHVN.com
MeanValueTheoremTutor(x*sin(x), -2*Pi..2*Pi );
Kho st ng cong Cu lnh:[> with(Student[Calculus1]): CurveAnalysisTutor(f, a..b); CurveAnalysisTutor(f, var=a..b); Trong : f - (tu chn) biu thc i s cha mt bin, a..b - (tu chn) min v th, var - (tu chn) bin s. V d. Xc nh cc im cc i, cc tiu; min tng, gim; khong li, lm ca hm s y = xsinx trn on [ 2 ;2 ] . [> with(Student[Calculus1]): CurveAnalysisTutor(x*sin(x));
Tnh gi tr ring ca ma trn Cu lnh:[> with(Student[LinearAlgebra]): EigenvaluesTutor(M)
133 www.mathvn.com
www.MATHVN.com
Trong : M - ma trn vung. V d. Tnh cc gi tr ring ca ma trn vung sau:
1 2 0 2 3 2 0 2 1 [> with(Student[LinearAlgebra]): M := ; 1 2 0 M := 2 3 2 0 2 1 [> EigenvaluesTutor( M );
Tip tc ta c kt qu sau:
Tnh vct ring ca ma trn Cu lnh:[ > with(Student[LinearAlgebra]): EigenvectorsTutor(M) Trong : M - ma trn vung. V d. [> with(Student[LinearAlgebra]): M := ;
134 www.mathvn.com
www.MATHVN.com
1 M := 2 0 [> EigenvectorsTutor( M );
2 3 2
0 2 1
Tip tc ta c: + Vi t = 1 vct ring l (-1; 0; 1). + Vi t = 5 vct ring l (1; 2; 1). a ma trn v dng Gaux Cu lnh:[ > with(Student[LinearAlgebra]): GaussianEliminationTutor(M) GaussianEliminationTutor(M, v) Trong : M - Ma trn, v - Vct. Ch . Trong Section ny s chiu ca ma trn khng c ln hn 5x5. V d. [> with(Student[LinearAlgebra]): [> M := ;
1 M := 2 0
2 3 2
0 2 1
3 5 5
[> v := ;
5 v := 4 2
[> GaussianEliminationTutor(M); GaussianEliminationTutor(M, v);
135 www.mathvn.com
www.MATHVN.com
Chng ta c th tm c khng gian cc vct ct, khng gian cc vct dng, hng ca ma trn... a ma trn v dng Gaux - Jordan Cu lnh:[ > with(Student[LinearAlgebra]): GaussJordanEliminationTutor(M) GaussJordanEliminationTutor(M, v) Trong : M - Ma trn, v - Vct. Ch . Trong Section ny s chiu ca ma trn khng c ln hn 5x5. V d: [> with(Student[LinearAlgebra]): M := ;
136 www.mathvn.com
www.MATHVN.com
1 M := 2 0 [> v := ;
2 3 2
0 2 1
3 5 5
5 v := 4 2
[> GaussJordanEliminationTutor( M ); GaussJordanEliminationTutor( M, v );
Tnh ma trn nghch o
137 www.mathvn.com
www.MATHVN.com
Cu lnh:[ > with(Student[LinearAlgebra]): InverseTutor(M); Trong : M - Ma trn vung. V d. Tnh ma trn nghch o ca ma trn :
1 2 0 2 3 2 0 2 1 [> with(Student[LinearAlgebra]): M := ;
1 M := 2 0 1 5 2 5 -4 5
2 3 2
0 2 1
[> M^(-1);
[> InverseTutor( M );
2 5 -1 5 2 5
-4 5 2 5 1 5
138 www.mathvn.com
www.MATHVN.com
Chng ta c th thc hin tng bc v th tnh ton trn tng dng, tng ct. a ra kt qu cui cng ta nhn nt Hint. Gii h phng trnh tuyn tnh Cu lnh:[ > with(Student[LinearAlgebra]): LinearSolveTutor(M) LinearSolveTutor(M, v) Trong : M - Ma trn, v - Vct. V d:[> with(Student[LinearAlgebra]): M := ;
1 M := 2 0
2 3 2
0 2 1
3 5 5
[> v := ;
5 v := 4 2
[> LinearSolveTutor( M ); LinearSolveTutor( M, v );
139 www.mathvn.com
www.MATHVN.com
S dng nt Hint a ra kt qu cui cng ca bi ton. Trong tt c cc Section c trnh by trn u c mc Help nhm hng dn cch s dng v khai thc c hiu qu cc chng trnh ny. Cc v d tng t cn rt nhiu, do khun kh trong phm vi mt mc nh khng th a ra ht c, chng ti coi nh bi tp bn c tip tc tm hiu, khm ph.
140 www.mathvn.com
www.MATHVN.com
Ngun ti liu gio trnh trch dn, tham kho[1]. Phm Huy in, inh Th Lc T Duy Phng -Hng dn thc hnh tnh ton trn chng trnh Maple V. NXB Gio dc 1998. [2]. Phm Huy in (ch bin)-Tnh ton, lp trnh v ging dy ton hc trn Maple. NXB KH&KT 2002. [3] . Nguyn B Kim -Phng php dy hc mn ton. NXB HSP 2002. [4]. Mai Cng Mn -S dng Maple trong ging dy mn hnh hc phng Lun vn Thc s ton hc 2000. [5]. o Thi Lai ng dng CNTT v vn i mi PPDH mn Ton Tp ch Nghin cu Gio dc , s 9/2002. [6] . Nguyn Vn Qu, Nguyn Tin Dng, Nguyn Vit H -Gii ton trn my vi tnh NXB Nng,1998. [7]. L Cng Trim, Nguyn Quang lc Mt s quan im v c s l lun dy hc ca vic s dng MTT .Tp ch NCGD 1992 [8] . Sue Johnston Wilder, David Pimm The free NCET (1995) leaflet, Mathematics ang IT - apupil's entitlement [9] .Sue Johnston Wilder, David Pimm The free NCET (1995) leaflet, Mathematics ang IT - apupil's entitlement [10]. Investigating transformation usng Geometers sketchpad through coopeerative learning SM-106 SEAMEO. [11]. Tran Vui Investigating Geometry with the Geometers Sketchpad A Conjecturing Approach. SEAMEO RECSAM, Penang, Malaysia. [12]. Asst.Prof.Krongthong Khairiee Teaching and Learning Mathematies Using The Geometers Sketchpad SEAMEO RECSAM, Penang Malaysia, 2002. [13]. Technology for Teaching Priscilia Norton, Debra Sprague George Mason University 2001 [14. Leone Burton and barbara Jaworski -Technology in mathematies Teaching and Learning A bridge between teaching ang learning. Chartwell Bratt, England, 1995.
141 www.mathvn.com
![[Www.mathVN.com] - Dap an 40 de Thi Thu Mon Toan](https://static.fdocuments.net/doc/165x107/5486e15bb4af9f964f8b47c1/wwwmathvncom-dap-an-40-de-thi-thu-mon-toan.jpg)
