Qhov tseem ceeb ntawm kev hloov pauv hauv lej yog qhov zoo, vim tias thaum lub sijhawm nws muaj nyob, cov kws tshawb fawb tau tswj hwm ntau qhov kev tshawb pom hauv cheeb tsam no, thiab txhawm rau kom luv thiab meej qhia qhov no lossis qhov theorem, peb siv qhov sib txawv los sau cov qauv sib xws.. Piv txwv li, Pythagorean theorem ntawm ib daim duab peb sab yog: a2 =b2 + c2. Yuav sau li cas txhua zaus thaum daws qhov teeb meem: raws li Pythagorean theorem, lub square ntawm lub hypotenuse yog sib npaug rau cov lej ntawm cov squares ntawm ob txhais ceg - peb sau qhov no nrog ib tug qauv, thiab txhua yam tam sim ntawd ua kom meej.
Yog li, kab lus no yuav tham txog qhov hloov pauv yog dab tsi, lawv hom thiab cov khoom. Ntau yam kev qhia lej yuav raug txiav txim siab: qhov tsis sib xws, qauv, tshuab thiab cov txheej txheem rau lawv cov kev daws teeb meem.
Variable concept
Ua ntej ntawm tag nrho, dab tsi yog qhov sib txawv? Qhov no yog tus lej tus nqi uas tuaj yeem ua rau ntau tus nqi. Nws tsis tuaj yeem tsis tu ncua, txij li cov teeb meem sib txawv thiab sib npaug, kom yooj yim, peb muab cov kev daws teeb meem raws lisib txawv tus lej sib txawv, uas yog, piv txwv li, z yog ib qho kev xaiv dav dav rau txhua qhov uas nws tau noj. Feem ntau lawv tau sau los ntawm cov tsiaj ntawv Latin lossis Greek cov tsiaj ntawv (x, y, a, b, thiab lwm yam).
Muaj ntau yam sib txawv. Lawv tau teeb tsa ob qho tib si ntawm lub cev - txoj hauv kev (S), sijhawm (t), thiab tsis paub qhov tseem ceeb hauv kev sib npaug, kev ua haujlwm thiab lwm yam kev qhia.
Piv txwv, muaj ib tug formula: S=Vt. Ntawm no, cov kev hloov pauv qhia qee qhov muaj feem cuam tshuam nrog lub ntiaj teb tiag - txoj hauv kev, nrawm thiab sijhawm.
Thiab muaj qhov sib npaug ntawm daim ntawv: 3x - 16=12x. Ntawm no, x tau muab coj los ua tus lej paub daws teeb meem uas ua rau muaj kev nkag siab zoo hauv cov ntawv no.
Nyob ntau ntau
Tus nqi txhais tau tias ib yam dab tsi uas qhia txog cov khoom ntawm ib yam khoom, khoom siv lossis qhov tshwm sim. Piv txwv li, huab cua kub, qhov hnyav ntawm tus tsiaj, feem pua ntawm cov vitamins nyob rau hauv ib ntsiav tshuaj - cov no yog tag nrho cov khoom uas nws tus lej muaj nuj nqis suav tau.
Txhua qhov ntau muaj nws tus kheej ntsuas ntsuas, uas ua ke tsim ib qho system. Nws hu ua tus lej system (SI).
Dab tsi yog qhov sib txawv thiab qhov tsis tu ncua? Xav txog lawv nrog cov piv txwv tshwj xeeb.
Cia peb ua cov lus tsa suab rectilinear. Ib qho taw tes hauv qhov chaw txav ntawm tib lub nrawm txhua lub sijhawm. Ntawd yog, lub sijhawm thiab kev ncua deb hloov pauv, tab sis qhov ceev tseem zoo li qub. Hauv qhov piv txwv no, lub sijhawm thiab qhov deb yog qhov sib txawv, thiab qhov nrawm yog qhov tas li.
Or, piv txwv li, "pi". Qhov no yog tus lej irrational uas txuas ntxiv yam tsis tau rov ua duaib ntu ntawm tus lej thiab tsis tuaj yeem sau tag nrho, yog li hauv lej nws tau qhia los ntawm cov cim feem ntau lees paub uas tsuas yog siv tus nqi ntawm ib feem tsis kawg. Qhov ntawd yog, "pi" yog tus nqi tas li.
History
Keeb kwm ntawm qhov cim ntawm kev hloov pauv pib nyob rau xyoo pua 17th nrog tus kws tshawb fawb René Descartes.
Nws tau xaiv cov txiaj ntsig uas paub nrog thawj cov tsiaj ntawv: a, b thiab lwm yam, thiab rau cov tsis paub nws tau hais kom siv cov ntawv kawg: x, y, z. Nws yog ib qho tseem ceeb uas Descartes tau suav tias qhov hloov pauv no yog cov lej tsis zoo, thiab thaum ntsib nrog qhov tsis zoo, nws tau muab lub cim rho tawm nyob rau pem hauv ntej ntawm qhov sib txawv lossis, yog tias nws tsis paub tias tus lej twg yog, tus ellipsis. Tab sis dhau sij hawm, cov npe ntawm qhov sib txawv tau pib qhia cov lej ntawm ib qho kev kos npe, thiab qhov no pib nrog tus lej lej Johann Hudde.
Nrog rau qhov sib txawv, kev suav lej hauv lej yooj yim dua los daws, vim hais tias, piv txwv li, peb yuav daws qhov sib npaug biquadratic li cas tam sim no? Peb nkag mus rau qhov sib txawv. Piv txwv li:
x4 + 15x2 + 7=0
Rau x2 peb coj qee yam k, thiab qhov sib npaug ua kom pom tseeb:
x2=k, rau k ≧ 0
k2 + 15k + 7=0
Qhov ntawd yog qhov kev qhia txog kev hloov pauv coj mus rau lej.
Kev tsis sib xws, piv txwv ntawm kev daws
Ib qhov tsis sib xws yog cov ntaub ntawv uas ob qho kev qhia lej lossis ob tus lej txuas nrog cov cim sib piv:, ≦, ≧. Lawv nruj thiab raug qhia los ntawm cov paib lossis tsis nruj nrog cov paib ≦, ≧.
Thawj zaug cov paib no tau qhiaThomas Harriot. Tom qab Thomas tuag, nws phau ntawv nrog cov ntawv sau no tau luam tawm, cov lej nyiam lawv, thiab dhau sijhawm lawv tau siv dav hauv kev suav lej.
Muaj ntau txoj cai ua raws thaum daws qhov sib txawv tsis sib xws:
- Thaum hloov tus lej los ntawm ib feem ntawm qhov tsis sib xws mus rau lwm qhov, hloov nws lub cim rau qhov tsis sib xws.
- Thaum muab ntau lossis faib qhov tsis sib xws los ntawm tus lej tsis zoo, lawv cov cim tau thim rov qab.
- Yog tias koj muab ntau lossis faib ob sab ntawm qhov tsis sib xws los ntawm tus lej zoo, koj tau txais qhov tsis sib xws sib npaug rau qhov qub.
Kev daws qhov tsis sib xws txhais tau tias nrhiav txhua qhov tseem ceeb rau qhov sib txawv.
Ib qho piv txwv piv txwv:
10x - 50 > 150
Peb daws nws zoo li ib txwm kab sib npaug - peb txav cov ntsiab lus nrog qhov sib txawv mus rau sab laug, tsis muaj qhov sib txawv - mus rau sab xis thiab muab cov ntsiab lus zoo sib xws:
10x > 200
Peb faib ob sab ntawm qhov tsis sib xws los ntawm 10 thiab tau txais:
x > 20
Rau qhov tseeb, hauv qhov piv txwv ntawm kev daws qhov tsis sib xws nrog ib qho kev sib txawv, kos ib kab zauv, kos lub pierced point 20 rau nws, vim qhov tsis sib xws yog nruj, thiab tus lej no tsis suav nrog hauv cov txheej txheem ntawm nws cov kev daws teeb meem..
Kev daws qhov tsis sib xws no yog lub sijhawm (20; +∞).
Kev daws qhov tsis sib xws tsis nruj yog ua raws li txoj kev nruj:
6x - 12 ≧ 18
6x ≧ 30
x ≧ 5
Tab sis muaj ib qho kev zam. Cov ntaub ntawv ntawm daim ntawv x ≧ 5 yuav tsum to taub raws li hauv qab no: x yog ntau dua lossis sib npaug rau tsib, uas txhais tau tiastus naj npawb tsib suav nrog hauv cov txheej txheem ntawm txhua qhov kev daws teeb meem rau qhov tsis sib xws, uas yog, thaum sau cov lus teb, peb muab ib qho square bracket rau pem hauv ntej ntawm tus lej tsib.
x ∈ [5; +∞)
Square inequalities
Yog tias peb coj ib qho kev sib npaug ntawm daim ntawv ax2 + bx + c=0 thiab hloov qhov sib npaug ntawm qhov tsis sib xws hauv nws, ces peb yuav raws li tau txais quadratic tsis sib xws.
Yuav daws qhov tsis sib xws ntawm plaub, koj yuav tsum tuaj yeem daws qhov sib npaug plaub npaug.
y=ax2 + bx + c is a quadratic function. Peb tuaj yeem daws nws los ntawm kev sib cais, lossis siv Vieta theorem. Nco qab tias cov kev sib npaug no daws tau li cas:
1) y=x2 + 12x + 11 - txoj haujlwm yog parabola. Nws cov ceg raug coj mus rau sab saud, txij li qhov kos npe ntawm tus coefficient "a" yog qhov zoo.
2) x2 + 12x + 11=0 - sib npaug rau xoom thiab daws siv qhov kev ntxub ntxaug.
a=1, b=12, c=11
D=b2 - 4ac=144 - 44=100 > 0, 2 cag
Raws li cov qauv ntawm cov hauv paus hniav ntawm kev sib npaug plaub npaug, peb tau txais:
x1 =-1, x2=-11
Los yog koj tuaj yeem daws qhov sib npaug no siv Vieta theorem:
x1 + x2 =-b/a, x1 + x 2=-12
x1x2 =c/a, x1x2=11
Siv txoj kev xaiv, peb tau txais tib lub hauv paus ntawm kab zauv.
Parabola
Yog li, thawj txoj hauv kev los daws qhov tsis sib xws plaub ntug yog parabola. Lub algorithm rau kev daws nws yog raws li nram no:
1. Txiav txim siab qhov twg cov ceg ntawm parabola raug coj mus.
2. Sib npaug qhov muaj nuj nqi rau xoom thiab nrhiav cov hauv paus hniav ntawm kab zauv.
3. Peb tsim ib tug xov tooj kab, kos cov hauv paus hniav rau nws, kos ib tug parabola thiab nrhiav qhov sib txawv peb xav tau, nyob ntawm seb lub kos npe rau ntawm qhov tsis sib xws.
Tau qhov tsis sib xws x2 + x - 12 > 0
Sau tawm ua haujlwm:
1) y=x2 + x - 12 - parabola, ceg nce.
Set to zero.
2) x2 + x -12=0
Tom ntej, peb daws raws li qhov sib npaug plaub npaug thiab pom qhov xoom ntawm txoj haujlwm:
x1 =3, x2=-4
3) Kos tus lej kab nrog cov ntsiab lus 3 thiab -4 ntawm nws. Lub parabola yuav dhau los ntawm lawv, ceg nce thiab cov lus teb rau qhov tsis sib xws yuav yog cov txiaj ntsig zoo, uas yog, (-∞; -4), (3; +∞).
Interval method
Txoj kev thib ob yog txoj kev sib nrug. Algorithm rau kev daws nws:
1. Nrhiav cov hauv paus hniav ntawm qhov sib npaug uas qhov tsis sib xws yog sib npaug rau xoom.
2. Peb cim lawv ntawm kab zauv. Yog li, nws muab faib ua ob peb ntu.
3. Txiav txim siab lub cim ntawm ib ntus.
4. Peb tso cov paib rau ntawm qhov seem, hloov lawv tom qab ib qho.
daws qhov tsis sib xws (x - 4)(x - 5)(x + 7) ≦ 0
1) Inequality zeros: 4, 5 thiab -7.
2) Kos rau ntawm tus lej kab.
3) Txiav txim siab cov cim ntawm lub sijhawm.
teb: (-∞; -7]; [4; 5].
Tau ib qhov tsis sib xws: x2(3x - 6)(x + 2)(x - 1) > 0
1. Inequality zeros: 0, 2, -2 thiab 1.
2. Kos rau ntawm tus lej kab.
3. Txiav txim siab lub caij nyoog.
Txoj kab muab faib ua ntu ntu - ntawm -2 txog 0, ntawm 0 txog 1, ntawm 1 txog 2.
Tau tus nqi ntawm thawj ntu - (-1). Hloov hauv kev tsis sib xws. Nrog rau tus nqi no, qhov tsis sib xws dhau los ua qhov zoo, uas txhais tau hais tias lub cim ntawm lub sijhawm no yuav yog +.
Ntxiv mus, pib ntawm thawj qhov sib txawv, peb npaj cov paib, hloov lawv tom qab ib qho.
Qhov tsis sib xws ntau dua xoom, uas yog, koj yuav tsum nrhiav cov txheej txheem zoo ntawm kab.
Teb: (-2; 0), (1; 2).
Systems of equations
Ib qho kev sib npaug nrog ob qhov sib txawv yog ob qhov sib npaug sib npaug los ntawm curly brace uas yuav tsum tau nrhiav kev daws teeb meem.
Systems yuav sib npaug yog hais tias qhov kev daws teeb meem ntawm ib qho yog kev daws ntawm lwm tus, los yog ob qho tib si tsis muaj kev daws teeb meem.
Peb yuav kawm txog kev daws teeb meem ntawm cov kab ke sib npaug nrog ob qhov sib txawv. Muaj ob txoj hauv kev los daws lawv - txoj kev hloov pauv lossis cov txheej txheem algebraic.
Algebraic method
Yuav kom daws tau qhov system qhia hauv daim duab uas siv cov qauv no, koj yuav tsum xub muab ib qho ntawm nws qhov chaw los ntawm tus lej, yog li tom qab ntawd koj tuaj yeem sib koom ua ke ib qho sib txawv los ntawm ob qho tib si ntawm qhov sib npaug. Ntawm no peb muab peb los ntawm peb, kos ib kab hauv qab qhov system thiab ntxiv nws qhov chaw. Raws li qhov tshwm sim, x's ua zoo tib yam hauv modulus, tab sis opposite hauv kos npe, thiab peb txo lawv. Tom ntej no, peb tau txais ib kab zauv sib npaug nrog ib qho sib txawv thiab daws nws.
Peb pom Y, tab sis peb nres tsis tau, vim peb tseem tsis tau pom X. HloovY rau qhov uas nws yuav yooj yim rau rho tawm X, piv txwv li:
-x + 5y=8, nrog y=1
-x + 5=8
daws qhov sib npaug thiab nrhiav x.
-x=-5 + 8
-x=3
x=-3
Qhov tseem ceeb hauv kev daws teeb meem yog sau cov lus teb kom raug. Ntau tus tub ntxhais kawm ua yuam kev ntawm kev sau ntawv:
Teb: -3, 1.
Tab sis qhov no yog qhov nkag tsis raug. Tom qab tag nrho, raws li tau hais los saum toj no, thaum daws qhov system ntawm kev sib npaug, peb tab tom nrhiav kev daws teeb meem rau nws qhov chaw. Cov lus teb raug yuav yog:
(-3; 1)
Kev hloov pauv
Qhov no yog txoj kev yooj yim tshaj plaws thiab nws nyuaj rau ua yuam kev. Wb coj txoj kev sib npaug 1 ntawm daim duab no.
Hauv nws thawj ntu, x tau raug txo qis rau hauv daim ntawv peb xav tau, yog li peb tsuas yog yuav tsum hloov pauv mus rau lwm qhov sib npaug:
5y + 3y - 25=47
Txav tus lej yam tsis muaj qhov hloov pauv mus rau sab xis, nqa cov ntsiab lus zoo li rau tus nqi sib xws thiab nrhiav tus y:
8y=72
y=9
Tom qab ntawd, raws li txoj hauv kev algebraic, peb hloov tus nqi ntawm y hauv ib qho kev sib npaug thiab nrhiav x:
x=3y - 25, nrog y=9
x=27 - 25
x=2
Teb: (2; 9).
