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        Ãëàâíàÿ ñòðàíèöà (Ñîäåðæàíèå)

   


Ïðàâîîáëàäàòåëÿì

Óðàâíåíèÿ è íåðàâåíñòâà. Íåñòàíäàðòíûå ìåòîäû ðåøåíèÿ: Ñïðàâî÷íèê.  Îëåõíèê Ñ.Í., Ïîòàïîâ Ì.Ê., Ïàñè÷åíêî Ï.È.

 Ì.: Èçä-âî Ôàêòîðèàë, 1997. - 219ñ. 

Ñïðàâî÷íèê ïîñâÿùåí çàäà÷àì, êîòîðûå äëÿ øêîëüíèêîâ ñ÷èòàþòñÿ çàäà÷àìè ïîâûøåííîé òðóäíîñòè, òðåáóþùèì íåñòàíäàðòíûõ ìåòîäîâ ðåøåíèé. Ïðèâîäÿòñÿ ìåòîäû ðåøåíèé óðàâíåíèé è íåðàâåíñòâ, îñíîâàííûå íà ãåîìåòðè÷åñêèõ ñîîáðàæåíèÿõ, ñâîéñòâàõ ôóíêöèé (ìîíîòîííîñòè, îãðàíè÷åííîñòè, ÷åòíîñòè), ïðèìåíåíèè ïðîèçâîäíîé. Êíèãà ñòàâèò ñâîåé öåëüþ ïîçíàêîìèòü øêîëüíèêîâ ñ ðàçëè÷íûìè, îñíîâàííûìè íà ìàòåðèàëå ïðîãðàììû îáùåîáðàçîâàòåëüíîé ñðåäíåé øêîëû, ìåòîäàìè ðåøåíèÿ, êàçàëîñü áû òðóäíûõ çàäà÷, ïðîèëëþñòðèðîâàòü øèðîêèå âîçìîæíîñòè èñïîëüçîâàíèÿ õîðîøî óñâîåííûõ øêîëüíûõ çíàíèé è ïðèâèòü ÷èòàòåëþ íàâûêè óïîòðåáëÿòü íåñòàíäàðòíûå ìåòîäû ðàññóæäåíèé ïðè ðåøåíèè çàäà÷. Äëÿ øêîëüíèêîâ, àáèòóðèåíòîâ, ðóêîâîäèòåëåé ìàòåìàòè÷åñêèõ êðóæêîâ, ó÷èòåëåé è âñåõ ëþáèòåëåé ðåøàòü çàäà÷è.

Ñïðàâî÷íîå èçäàíèå.

 

 

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Îãëàâëåíèå
Îò àâòîðîâ
                                                                                                                      7

Ãëàâà I. Àëãåáðàè÷åñêèå óðàâíåíèÿ è íåðàâåíñòâà                                                    8

1.1.  Ðàçëîæåíèå ìíîãî÷ëåíà íà ìíîæèòåëè                                                                8

1.1.1.  Âûíåñåíèå îáùåãî ìíîæèòåëÿ                                                                           8

1.1.2.  Ïðèìåíåíèå ôîðìóë ñîêðàùåííîãî óìíîæåíèÿ                                               9

1.1.3.  Âûäåëåíèå ïîëíîãî êâàäðàòà                                                                             10

1.1.4.  Ãðóïïèðîâêà                                                                                                        10

1.1.5.  Ìåòîä íåîïðåäåëåííûõ êîýôôèöèåíòîâ                                                          10

1.1.6.  Ïîäáîð êîðíÿ ìíîãî÷ëåíà ïî åãî ñòàðøåìó è ñâîáîäíîìó êîýôôèöèåíòàì  11

1.1.7.  Ìåòîä ââåäåíèÿ ïàðàìåòðà                                                                                13

1.1.8.  Ìåòîä ââåäåíèÿ íîâîé íåèçâåñòíîé                                                                 13

1.1.9.  Êîìáèíèðîâàíèå ðàçëè÷íûõ ìåòîäîâ                                                              14

 1.2.    Ïðîñòåéøèå ñïîñîáû ðåøåíèÿ àëãåáðàè÷åñêèõ óðàâíåíèé                               15

1.3.     Ñèììåòðè÷åñêèå è âîçâðàòíûå óðàâíåíèÿ                                                          19

 1.3.1.  Ñèììåòðè÷åñêèå óðàâíåíèÿ òðåòüåé ñòåïåíè                                                  19

1.3.2.  Ñèììåòðè÷åñêèå óðàâíåíèÿ ÷åòâåðòîé ñòåïåíè                                             20

1.3.3.  Âîçâðàòíûå óðàâíåíèÿ                                                                                       22

1.3.4.  Óðàâíåíèÿ ÷åòâåðòîé ñòåïåíè ñ äîïîëíèòåëüíûìè óñëîâèÿìè íà êîýôôèöèåíòû  25

1.4.  Íåêîòîðûå èñêóññòâåííûå ñïîñîáû ðåøåíèÿ àëãåáðàè÷åñêèõ óðàâíåíèé     27

1.4.1. Óìíîæåíèå óðàâíåíèÿ íà ôóíêöèþ                                                                  27

1.4.2. Óãàäûâàíèå êîðíÿ óðàâíåíèÿ                                                                            29

1.4.3. Èñïîëüçîâàíèå ñèììåòðè÷íîñòè óðàâíåíèÿ                                                    32

1.4.4. Èñïîëüçîâàíèå ñóïåðïîçèöèè ôóíêöèé                                                            33

1.4.5. Èññëåäîâàíèå óðàâíåíèÿ íà ïðîìåæóòêàõ äåéñòâèòåëüíîé îñè                    34

1.5. Ðåøåíèå àëãåáðàè÷åñêèõ íåðàâåíñòâ                                                                   3 5

1.5.1.  Ïðîñòåéøèå ñïîñîáû ðåøåíèÿ àëãåáðàè÷åñêèõ íåðàâåíñòâ                           3 5

1.5.2.  Ìåòîä èíòåðâàëîâ                                                                                               38

Çàäà÷è

Ãëàâà Ï. Óðàâíåíèÿ è íåðàâåíñòâà, ñîäåðæàùèå ðàäèêàëû, ñòåïåíè, ëîãàðèôìû è ìîäóëè 48

1.5.3.  Îáîáùåííûé ìåòîä èíòåðâàëîâ                                                           41

2.1. Óðàâíåíèÿ è íåðàâåíñòâà, ñîäåðæàùèå íåèçâåñòíóþ ïîä çíàêîì êîðíÿ        48

2.1.1.  Âîçâåäåíèå â ñòåïåíü                                                                                     48

2.1.2.  Óðàâíåíèÿ âèäà -Jf(x) ± -\lg(x) =h(x)                                                                 51

2.1.3.  Óðàâíåíèÿ âèäà yf(x) ± \fg(x) = ô(õ)                                                                  53

2.1.4.  Óìíîæåíèå óðàâíåíèÿ èëè íåðàâåíñòâà íà ôóíêöèþ                                     56

2.2. Óðàâíåíèÿ è íåðàâåíñòâà, ñîäåðæàùèå íåèçâåñòíóþ â îñíîâàíèè                 59
ëîãàðèôìîâ

2.2.1.  Ïåðåõîä ê ÷èñëîâîìó îñíîâàíèþ                                                                      59

2.2.2.  Ïåðåõîä ê îñíîâàíèþ, ñîäåðæàùåìó íåèçâåñòíóþ                                          64

2.2.3.  Óðàâíåíèÿ âèäà log9(x)h(x) = log9(x) g(x), log/(x) ô(õ) = log^(x)ô(õ)                      65

2.2.4.  Óðàâíåíèÿ âèäà log/(x)g(x) = a                                                                           66

2.2.5.  Íåðàâåíñòâà âèäà log9(x)f(x) > log9(x)g(x)                                                           68

2.3.  Óðàâíåíèÿ è íåðàâåíñòâà, ñîäåðæàùèå íåèçâåñòíóþ â îñíîâàíèè è 70 ïîêàçàòåëå ñòåïåíè

2.4.  Óðàâíåíèÿ è íåðàâåíñòâà, ñîäåðæàùèå íåèçâåñòíóþ ïîä çíàêîì 75 àáñîëþòíîé âåëè÷èíû

2.4.1.  Ðàñêðûòèå çíàêîâ ìîäóëåé                                                                                 75

2.4.2.  Óðàâíåíèÿ âèäà |f(x)|=g(x)                                                                                  77

2.4.3.  Íåðàâåíñòâà âèäà |f(x)|<g(x)                                                                                78

2.4.4.  Íåðàâåíñòâà âèäà |f(x)|>g(x)                                                                                79

2.4.5.  Óðàâíåíèÿ è íåðàâåíñòâà âèäà |f(x)|=|g(x)|, |f(x)|<g(x)                                       81

2.4.6.  Èñïîëüçîâàíèå ñâîéñòâ àáñîëþòíîé âåëè÷èíû                                  82

Çàäà÷è

Ãëàâà III. Ñïîñîá çàìåíû íåèçâåñòíûõ ïðè ðåøåíèè óðàâíåíèé                                  87

3.1. Àëãåáðàè÷åñêèå óðàâíåíèÿ                                                                                    87

3.1.1.  Ïîíèæåíèå ñòåïåíè óðàâíåíèÿ                                                                         87

3.1.2.  Óðàâíåíèÿ âèäà (õ + îñ)4 + (õ +13)4 = ñ                                                            88

3.1.3.  Óðàâíåíèÿ âèäà (õ- à)(õ-ð)(õ- f)(x- <5)=À                                                             89

3.1.4.  Óðàâíåíèÿ âèäà (àõ2 + Üõõ + ñ)(àõ2 + Ü2õ + ñ) = Àõ2                                      90

3.1.5.  Óðàâíåíèÿ âèäà (õ- à)(õ-ð)(õ- f)(x- S)=Ax^                                                        91

3.1.6.  Óðàâíåíèÿ âèäà à(ñõ2 + ðõõ + q)2 + b(cx2 + p2x + q) = Ax2                             92

3.1.7.  Óðàâíåíèÿ âèäà Ð(õ)=0, Ð(õ)=Ð(à-õ)                                                                93

3.2. Ðàöèîíàëüíûå óðàâíåíèÿ                                                                                     95

3.2.1.   Óïðîùåíèå óðàâíåíèÿ                                                                                      95

3.2.2.   Óðàâíåíèÿ âèäà-------- 1— +----- — +... + —=— = À

x + pj     õ + ð2              õ + ðò

. _ . ,.                                 îñ,õ + à,    à2õ + à2                  àïõ + àï     ï                               99

3.2.3.   Óðàâíåíèÿ âèäà —---------- - + —------ - + ... + —------- - = D

x + bx        x + b2                      Õ + Üï

. _ . ,.                                      a,x + h                   a2x + b2                         ax + bn                .    100

3.2.4.   Óðàâíåíèÿ âèäà-------- ^----- !--- +----- ^---------- + - +------- Ã---------- =À

_ _ _ ..                               a,x2+hx+c,    a2x2 + b2x + c2                          a„x2+b„x + c„      .    Þ2

3.2.5.   Óðàâíåíèÿ âèäà —---------- ;----- L + -------------- +... + --------------- = À

à1õ + ^>1                  à2õ + |32                        à„õ + |3„

^ „ ^ ÷ò                                 Àõ                    À2õ                           Àïõ                          103

3.2.6.   Óðàâíåíèÿ âèäà —;—;------------- 1--- -2—---------- Ó... ë------ —---------- = Â

àõ +Üãõ + ñ    àõ +b2x + c                       ax +bnx + c

3.3.   Èððàöèîíàëüíûå óðàâíåíèÿ                                                                                          104

3.3.1.       Óðàâíåíèÿ âèäà -Jax + b ± yjcx + d = f(x)                                                                    104

3.3.2.       Óðàâíåíèÿ âèäà Ìà-õ ±Ìõ-Ü = d                                                                                  107

3.3.3.       Ñâåäåíèå ðåøåíèÿ èððàöèîíàëüíîãî óðàâíåíèÿ ê ðåøåíèþ òðèãîíîìåòðè÷åñêîãî óðàâíåíèÿ  111

3.4.   Óðàâíåíèÿ âèäà                                                                                                           114

a0f"(x) + aif"-i(x)g(x) + ... + an_1f(x)g"-1(x) + ang"(x) = 0

3.5.   Ðåøåíèå íåêîòîðûõ óðàâíåíèé ñâåäåíèåì èõ ê ðåøåíèþ ñèñòåì                                120
óðàâíåíèé îòíîñèòåëüíî íîâûõ íåèçâåñòíûõ

Çàäà÷è                                                                                                                              127

Ãëàâà IV. Ðåøåíèå óðàâíåíèé è íåðàâåíñòâ ñ èñïîëüçîâàíèåì ñâîéñòâ                                 131
âõîäÿùèõ â íèõ ôóíêöèé

4.1.   Ïðèìåíåíèå îñíîâíûõ ñâîéñòâ ôóíêöèé                                                                         131

4.1.1.       Èñïîëüçîâàíèå ÎÄÇ                                                                                                 131

4.1.2.       Èñïîëüçîâàíèå îãðàíè÷åííîñòè ôóíêöèé                                                                     134

4.1.3.       Èñïîëüçîâàíèå ìîíîòîííîñòè                                                                                    138

4.1.4.       Èñïîëüçîâàíèå ãðàôèêîâ                                                                                           141

4.1.5.       Ìåòîä èíòåðâàëîâ äëÿ íåïðåðûâíûõ ôóíêöèé                                                               147

4.2.   Ðåøåíèå íåêîòîðûõ óðàâíåíèé è íåðàâåíñòâ ñâåäåíèåì èõ ê ðåøåíèþ        149
ñèñòåì óðàâíåíèé èëè íåðàâåíñòâ îòíîñèòåëüíî òîé æå íåèçâåñòíîé

4.2.1.       Óðàâíåíèÿ âèäà       /2(õ) + /22(õ) + ... + /ê2(õ) = 0,\Ìõ)\ + \/2(õ)\+...+ \/ê(õ)\=0                       150

4.2.2.       Íåðàâåíñòâà âèäà   f2(x) + f2(x) + ... + f2(x)>Q,\Mx)\ + \f2(x)\+...+ \fk(x)\>0                           151

4.2.3.       Èñïîëüçîâàíèå îãðàíè÷åííîñòè ôóíêöèé                                                                    153

4.2.4.       Èñïîëüçîâàíèå ñâîéñòâ ñèíóñà è êîñèíóñà                                                                  155

4.2.5.       Èñïîëüçîâàíèå ÷èñëîâûõ íåðàâåíñòâ                                                                          158

4.3.   Ïðèìåíåíèå ïðîèçâîäíîé                                                                                            160

4.3.1.  Èñïîëüçîâàíèå ìîíîòîííîñòè                                                                         160

4.3.2.  Èñïîëüçîâàíèå íàèáîëüøåãî è íàèìåíüøåãî çíà÷åíèé ôóíêöèè                162

4.3.3.  Ïðèìåíåíèå òåîðåìû Ëàãðàíæà                                                                      166

Çàäà÷è 166

Îòâåòû     172

Äîïîëíåíèå 1

Íåêîòîðûå çàäà÷è èç âàðèàíòîâ âñòóïèòåëüíûõ ýêçàìåíîâ ïî ìàòåìàòèêå â      176

ÌÃÓ èì. Ì. Â. Ëîìîíîñîâà

Äîïîëíåíèå 2

Îáðàçöû âàðèàíòîâ ïèñüìåííûõ ðàáîò, ïðåäëàãàâøèõñÿ íà âñòóïèòåëüíûõ       184

ýêçàìåíàõ ïî ìàòåìàòèêå â ÌÃÓ èì. Ì. Â. Ëîìîíîñîâà â 1992—1994 ãã.

Îòâåòû ê äîïîëíåíèþ 2                                                                                             212

 

 

 

Ïðèìå÷àíèå: Ñëó÷àéíî ïðî÷èòàë íà ôîðóìå Çàî÷íîé ôèçèêî-òåõíè÷åñêîé øêîëû ïðè ÌÔÒÈ òàêîå ìíåíèå:

"Îëåõíèê, Ïîòàïîâ, Ïàñè÷åíêî. Óðàâíåíèÿ è íåðàâåíñòâà. Åñòü ó ýòîé êíèæêè åñòü ïîäçàãîëîâîê "íåñòàíäàðòíûå ìåòîäû ðåøåíèÿ", òî ýòî òîæå êíèæêà âåëèêàÿ, íî ñèëüíî åþ íå óâëåêàéòåñü. Òàì îíè ëåçóò â íåìåðÿííûå äåáðè, êîòîðûå âñòðå÷àþòñÿ äîâîëüíî ðåäêî, íî ÏÐÎ×ÈÒÀÒÜ åå ñòîèò è íåìíîæêî ïîñìîòðåòü íà çàäà÷è, ÷òîáû ïîòîì íå áûòü â øîêå è íå ãîâîðèòü, ÷òî âû òàêîãî íå âèäåëè."

 

 


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