Учебно-методическое пособие (1017796), страница 10
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......... ...... ......... ..y=x+√xy=xxÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ0Ðèñ. 23.7. Îáùàÿ ñõåìà èññëåäîâàíèÿ ôóíêöèè è ïîñòðîåíèåãðàôèêàÎáùàÿ ñõåìà èññëåäîâàíèÿ ôóíêöèè ñîñòîèò èç òðåõ ýòàïîâ.Ýòà ñõåìà äàñò íàì ïðàêòè÷åñêèé ñïîñîá ïîñòðîåíèÿ ãðàôèêàôóíêöèè, îòðàæàþùåãî îñíîâíûå ÷åðòû å¼ ïîâåäåíèÿ.Ïåðâûé ýòàï - ýëåìåíòàðíîå èññëåäîâàíèå ôóíêöèè.Ïóñòü äàíà ôóíêöèÿ f (x). ż ýëåìåíòàðíîå èññëåäîâàíèå âêëþ÷àåò ñëåäóþùèå ïðîöåäóðû.1. Íàéòè å¼ îáëàñòü îïðåäåëåíèÿ D(f ).2.
Âûÿñíèòü îáùèå ñâîéñòâà ôóíêöèè, êîòîðûå ïîìîãóò â îïðåäåëåíèè å¼ ïîâåäåíèÿ:• ÿâëÿåòñÿ ëè ôóíêöèÿ ÷¼òíîé ëèáî íå÷åòíîé,• ÿâëÿåòñÿ ëè ôóíêöèÿ ïåðèîäè÷åñêîé.3. Íàéòè òî÷êè ïåðåñå÷åíèÿ ãðàôèêà ôóíêöèè ñ îñÿìèêîîðäèíàò.4. Íàéòè òî÷êè ðàçðûâà ôóíêöèè è âûÿñíèòü ïîâåäåíèå ôóíêöèè â îêðåñòíîñòè ýòèõ òî÷åê.1135. Âûÿñíèòü ïîâåäåíèå ôóíêöèè â îêðåñòíîñòè ãðàíè÷íûõ òî÷åê, âêëþ÷àÿ è íåñîáñòâåííûå òî÷êè −∞ è +∞.6. Íàéòè àñèìïòîòû.ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÐåçóëüòàòîì ýëåìåíòàðíîãî èññëåäîâàíèÿ ôóíêöèè ÿâëÿåòñÿïîñòðîåíèå ýñêèçà ãðàôèêà ôóíêöèè.Âòîðîé ýòàï èññëåäîâàíèå ôóíêöèè ñ ïîìîùüþ ïåðâîé ïðîèçâîäíîé.1. Íàéòè ïåðâóþ ïðîèçâîäíóþ çàäàííîé ôóíêöèè f (x).2.
Íàéòè êðèòè÷åñêèå òî÷êè ïåðâîãî ðîäà.3. Íàéòè ó÷àñòêè âîçðàñòàíèÿ è óáûâàíèÿ ôóíêöèè.4. Îïðåäåëèòü ëîêàëüíûå ýêñòðåìóìû.Òðåòèé ýòàï - èññëåäîâàíèå ôóíêöèè ñ ïîìîùüþ âòîðîé ïðîèçâîäíîé.1. Íàéòè âòîðóþ ïðîèçâîäíóþ çàäàííîé ôóíêöèè f (x).2. Íàéòè òî÷êè, ãäå âòîðàÿ ïðîèçâîäíàÿ ðàâíà íóëþ èëè íå ñóùåñòâóåò.3. Íàéòè ó÷àñòêè âûïóêëîñòè è âîãíóòîñòè ãðàôèêà ôóíêöèè.4. Íàéòè òî÷êè ïåðåãèáà.Ïîëó÷åííûå â êàæäîì ïóíêòå ðåçóëüòàòû ïîñëåäîâàòåëüíîôèêñèðóåì íà ðèñóíêå â êà÷åñòâå ýëåìåíòîâ èñêîìîãî ãðàôèêà èâ èòîãå ïîëó÷àåì ãðàôèê ôóíêöèè.(x + 1)3è ïîñòðîèòü ååÏðèìåð 7.1. Èññëåäîâàòü ôóíêöèþ y =(x − 2)2ãðàôèê.Ðåøåíèå: Ïåðâûé ýòàï - ýëåìåíòàðíîå èññëåäîâàíèå ôóíêöèè.1141.
Îáëàñòü îïðåäåëåíèÿ ôóíêöèè: x ̸= 2.2. Ôóíêöèÿ íå ÿâëÿåòñÿ íè ÷åòíîé, íè íå÷åòíîé; íå ÿâëÿåòñÿïåðèîäè÷åñêîé; íå ÿâëÿåòñÿ çíàêîïîñòîÿííîé.3. Ïåðåñå÷åíèå ñ îñüþ Ox: x = −1; y = 0; ñ îñüþ Oy : x = 0;ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ1y= .44. Òî÷êà ðàçðûâà ôóíêöèè x = 2.5. Ïîâåäåíèå ôóíêöèè â îêðåñòíîñòè ãðàíè÷íûõ òî÷åê, âêëþ÷àÿ è íåñîáñòâåííûå òî÷êè −∞ è +∞.(x + 1)3lim f (x) = lim= +∞x→+∞x→+∞ (x − 2)2(x + 1)3lim f (x) = lim= −∞x→−∞x→+∞ (x − 2)2(x + 1)3lim f (x) = lim= +∞x→2+x→2+ (x − 2)2(x + 1)3lim f (x) = lim= +∞x→2−x→2− (x − 2)26. Ïðÿìàÿ x = 2 âåðòèêàëüíàÿ àñèìïòîòà; ãîðèçîíòàëüíûõàñèìïòîò íå ñóùåñòâóåò, ïîñêîëüêó lim f (x) = ±∞.
Íàéäåìx→±∞ïðàâóþ íàêëîííóþ àñèìïòîòó:f (x)= 1, b+ = lim (f (x) − x) = 7.x→+∞ xx→+∞k+ = limÒàêèì îáðàçîì, ïðàâàÿ íàêëîííàÿ àñèìïòîòà: y = x + 7. Íàéäåì ëåâóþ íàêëîííóþ àñèìïòîòó:f (x)= 1, b− = lim (f (x) − x) = 7.x→−∞x→−∞ xk− = limÒàêèì îáðàçîì, ëåâàÿ íàêëîííàÿ àñèìïòîòà ñîâïàäàåò ñ ïðàâîé: y = x + 7.115Âòîðîé ýòàï èññëåäîâàíèå ôóíêöèè ñ ïîìîùüþ ïåðâîé ïðîèçâîäíîé.1. Íàéäåì ïåðâóþ ïðîèçâîäíóþ çàäàííîé ôóíêöèè f (x):3(x + 1)2 (x − 2)2 − (x + 1)3 2(x − 2)f (x) ==(x − 2)4(x + 1)2 (x − 8)=(x − 2)3ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ′2. Íàéäåì êðèòè÷åñêèå òî÷êè I ðîäà, ò.å.
òå òî÷êè, â êîòîðûõf (x) îïðåäåëåíà è íåïðåðûâíà, à ïåðâàÿ ïðîèçâîäíàÿ ðàâíàíóëþ f ′ (x0 ) = 0, ëèáî f ′ (x0 ) íå ñóùåñòâóåò. Ïåðâàÿ ïðîèçâîäíàÿ f ′ (x0 ) = 0 ïðè x = −1 è x = 8.  òî÷êå x = 2 ôóíêöèÿf (x) íå îïðåäåëåíà.3. Íàéäåì ó÷àñòêè âîçðàñòàíèÿ è óáûâàíèÿ ôóíêöèè è ýêñòðåìóìû. Äëÿ îïèñàíèÿ ïîâåäåíèÿ ôóíêöèè ñîñòàâèì ñëåäóþùóþ òàáëèöó:f ′ (x)f (x)x ∈ (−∞; −1)+âîçðàñòàåòx = −10x ∈ (−1; 2)+âîçðàñòàåòx=2@òî÷êà ðàçðûâàx ∈ (2; 8)−óáûâàåòx=80òî÷êà ëîêàëüíîãî ìèíèìóìà+814âîçðàñòàåòfmin =x ∈ (8; +∞)Òðåòèé ýòàï - èññëåäîâàíèå ôóíêöèè ñ ïîìîùüþ âòîðîé ïðî-116èçâîäíîé.1.
Íàéäåì âòîðóþ ïðîèçâîäíóþ çàäàííîé ôóíêöèè:f ′′ (x) =54(x + 1)(x − 2)42. Âòîðàÿ ïðîèçâîäíàÿ ðàâíà íóëþ ïðè x = −1.ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3. Ó÷àñòêè âûïóêëîñòè è âîãíóòîñòè ãðàôèêà ôóíêöèè, òî÷êèïåðåãèáà îïðåäåëÿþòñÿ íà îñíîâàíèè ñëåäóþùåé òàáëèöû.f ′′ (x)f (x)x ∈ (−∞; −1)−âûïóêëà ââåðõx = −10òî÷êà ïåðåãèáàx ∈ (−1; 2)+âûïóêëà âíèçx=2@òî÷êà ðàçðûâàx ∈ (2; +∞)+âûïóêëà âíèçÏî ðåçóëüòàòàì èññëåäîâàíèÿ ñòðîèì ãðàôèê.y6......
........ ................. ... .............. ...... ........... ................................ ........ .... .....3...... .....1.... ......... ......2 2 ..... ... ...... . ..... .. ................ ......... ...... . ..... ...... ......... .....2... ................ ..... .................... ....... ...... ............... ........... ......... ...........y=(x + )(x − )y=x + 7x=Ðèñ. 24.-x117Ïðèìåð 7.2. Èññëåäîâàòü ôóíêöèþ y = xex è ïîñòðîèòü åå ãðàôèê.Ðåøåíèå: Èññëåäîâàíèå ôóíêöèè áóäåì ïðîâîäèòü ïî ñõåìå,îïèñàííîé â ïðåäûäóùåì ïðèìåðå.1. Îáëàñòü îïðåäåëåíèÿ ôóíêöèè: x ∈ (−∞; +∞).2.
Ôóíêöèÿ íå ÿâëÿåòñÿ íè ÷åòíîé, íè íå÷åòíîé; íå ÿâëÿåòñÿÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀïåðèîäè÷åñêîé; ïðè x > 0 f (x) > 0, ïðè x < 0 f (x) < 0.3. Ãðàôèê ïåðåñåêàåò îñè êîîðäèíàò â òî÷êå (0; 0).4.lim f (x) = lim xex = +∞; lim f (x) = lim xex = 0.x→+∞x→+∞x→−∞x→−∞Òàêèì îáðàçîì, y = 0 ëåâàÿ ãîðèçîíòàëüíàÿ àñèìïòîòà.Âåðòèêàëüíûõ àñèìïòîò íåò.5. Íàéäåì ïðàâóþ íàêëîííóþ àñèìïòîòó:f (x)= lim ex = +∞.x→+∞ xx→+∞k+ = limÒàêèì îáðàçîì, íàêëîííûõ àñèìïòîò íåò.6.  äàííîì ñëó÷àå ôóíêöèÿ áåñêîíå÷íî äèôôåðåíöèðóåìàÿ.Äëÿ îïðåäåëåíèÿ òî÷åê ëîêàëüíîãî ýêñòðåìóìà íàéäåì ñòàöèîíàðíûå òî÷êè, ò.å.
òå òî÷êè, â êîòîðûõ ïåðâàÿ ïðîèçâîä-íàÿ ðàâíà íóëþ. f (x) = xex ⇒ f ′ (x) = ex + xex = (x + 1)ex ,f ′ (x) = 0 ïðè x = −1. Ýòî è åñòü ñòàöèîíàðíàÿ òî÷êà. Ïðîèçâîäíàÿ f ′ (x) < 0 ïðè x < −1 è f ′ (x) > 0 ïðè x > −1,ñëåäîâàòåëüíî, x = −1 òî÷êà ëîêàëüíîãî ìèíèìóìà.7.
Äëÿ îïðåäåëåíèÿ ïðîìåæóòêîâ âûïóêëîñòè è âîãíóòîñòè íàéäåì òî÷êè, â êîòîðûõ âòîðàÿ ïðîèçâîäíàÿ ðàâíà íóëþ.118f (x) = xex ⇒ f ′ (x) = (x+1)ex ⇒ f ′′ (x) = (x+2)ex , f ′′ (x) = 0ïðè x = −2.Ðåçóëüòàòû èññëåäîâàíèé çàíåñåì â òàáëèöó.f ′ (x) f ′′ (x)x ∈ (−∞; −2)−f (x)−óáûâàåòÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀâûïóêëà ââåðõx = −2−0òî÷êà ïåðåãèáàx ∈ (−2; −1)−+óáûâàåòâûïóêëà âíèçx = −10+ëîêàëüíûé ìèíèìóìfmin = −e−1x ∈ (−1; +∞)++âîçðàñòàåòâûïóêëà âíèçÏî ðåçóëüòàòàì èññëåäîâàíèÿ ñòðîèì ãðàôèê.y6..................................x..................................-2.-1.....................................................................................................................
0...................................................................................................................................................y= x e-xÐèñ. 25.Ïðèìåð 7.3. Èññëåäîâàòü ôóíêöèþ y = 2x3 − 3x2 + x + 5 è119ïîñòðîèòü åå ãðàôèê.Ðåøåíèå:1. Îáëàñòü îïðåäåëåíèÿ ôóíêöèè: x ∈ (−∞; +∞).2. Ôóíêöèÿ íå ÿâëÿåòñÿ íè ÷åòíîé, íè íå÷åòíîé; íå ÿâëÿåòñÿïåðèîäè÷åñêîéÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3. Ïåðåñå÷åíèå ñ îñÿìè: ãðàôèê ïåðåñåêàåò îñü Oy â òî÷êå x = 0,y = 5.
Äëÿ íàõîæäåíèÿ ïåðåñå÷åíèé ãðàôèêà ñ îñüþ Ox ñëåäóåò ðåøèòü óðàâíåíèå 2x3 −3x2 +x+5 = 0. Ðåøåíèåì äàííîãîóðàâíåíèÿ ÿâëÿåòñÿ åäèíñòâåííûé êîðåíü x0 ≈ −0,919.f (x)= +∞, ñëåäîâàòåëüíî, íàêëîííûõ àñèìïòîò íåò.x→±∞ xÂåðòèêàëüíûõ àñèìïòîò íåò (ïî÷åìó?).4. limÄàëüíåéøåå èññëåäîâàíèå è ïîñòðîåíèå ãðàôèêà ïðåäëàãàåòñÿïðîâåñòè ñàìîñòîÿòåëüíî. Äëÿ ïðîâåðêè ðåøåíèÿ ïðèâåäåì îòâåò.y6...............2........5.......................................... ........................................................................................................................................................................................................................................................................................0123........y = 2x 3 − 3x + x +xxxxxÐèñ. 26.√13Çäåñü x0 ≈ −0,919, x1 = −≈ 0,211 òî÷êà ëîêàëüíîãî26120√31ìàêñèìóìà, x3 = +≈ 0, 789 òî÷êà ëîêàëüíîãî ìèíèìóìà,261òî÷êà x2 = òî÷êà ïåðåãèáà.28.
Ôîðìóëà ÒåéëîðàÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ8.1. Ìíîãî÷ëåí ÒåéëîðàÏóñòü ôóíêöèÿ f (x) îïðåäåëåíà â íåêîòîðîé îêðåñòíîñòè(x0 − δ; x0 + δ) íåêîòîðîé òî÷êè x0 ∈ R è èìååò âñþäó â ýòîéîêðåñòíîñòè ïðîèçâîäíûå f (k) (x) ïðè k = 1, 2, . . . , n. Ìíîãî÷ëå-íîì Òåéëîðà ñòåïåíè n â òî÷êå x0 íàçûâàåòñÿ ìíîãî÷ëåí P (x)ñòåïåíè n òàêîé, ÷òî åãî çíà÷åíèå è çíà÷åíèå âñåõ åãî ïðîèçâîä-íûõ, âû÷èñëåííûå â òî÷êå x0 , ðàâíû ñîîòâåòñòâóþùèì çíà÷åíèÿìôóíêöèè f (x) è å¼ ïðîèçâîäíûõ f (k) (x) äî ïîðÿäêà n:P (k) (x0 ) = f (k) (x0 ); k = 0, 1, 2, .
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