Учебно-методическое пособие (1017796), страница 6
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Ïðè ýòîìlim f (x) = lim f (x) = lim f (x)x→ax→a−0x→a+0Çàìå÷àíèå. Îñíîâíûå òåîðåìû, êîòîðûå èñïîëüçóþòñÿ äëÿ âû÷èñëåíèÿ ïðåäåëîâ, ñïðàâåäëèâû è äëÿ îäíîñòîðîííèõ ïðåäåëîâ.2.3. Òî÷êè ðàçðûâàÎïðåäåëåíèå 2.4 Åñëè ñóùåñòâóåò lim f (x), íî ïðè ýòîìx→aôóíêöèÿ f (x) íå ÿâëÿåòñÿ íåïðåðûâíîé â òî÷êå a, òî òî÷êà aíàçûâàåòñÿ òî÷êîé óñòðàíèìîãî ðàçðûâà.72Çäåñü âîçìîæíû äâå ñèòóàöèè:1. lim f (x) ̸= f (a), åñëè f (a) îïðåäåëåíî;x→a2. f (a) íå îïðåäåëåíî, à lim f (x) ñóùåñòâóåò.{Ïðèìåð 2.1.
f (x) =x, x ̸= 01, x = 0.Çäåñü x = 0 òî÷êà óñòðàíè-ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀìîãî ðàçðûâà.x→aÏðèìåð 2.2. f (x) =sin x(íå îïðåäåëåíî ïðè x = 0,xsin x= 1).x→0 xÅñëè a òî÷êà óñòðàíèìîãî ðàçðûâà ôóíêöèè f (x), òî ôóíêöèþìîæíî äîîïðåäåëèòü äî íåïðåðûâíîé, ïîëàãàÿ f (a) = lim f (x).x→asinx, x ̸= 0Ïðèìåð 2.3. Ôóíêöèÿ f (x) =ÿâëÿåòñÿ íåïðåx1, x = 0.limðûâíîé.Îïðåäåëåíèå 2.5 Òî÷êà a íàçûâàåòñÿ òî÷êîé ðàçðûâà ïåðâîãîðîäà, åñëè ñóùåñòâóþò îáà îäíîñòîðîííèõ ïðåäåëà (êîíå÷íûõ),íî îíè íå ðàâíû äðóã äðóãó.(x − 1) 2· x . Çäåñü lim f (x) = −1,x→1−0|x − 1|lim f (x) = 1.
Ñëåäîâàòåëüíî, x = 1 òî÷êà ðàçðûâà ïåðâîãîx→1+0ðîäà.Ïðèìåð 2.4. f (x) =Îïðåäåëåíèå 2.6 Òî÷êà a íàçûâàåòñÿ òî÷êîé ðàçðûâà âòîðîãîðîäà, åñëè õîòÿ áû îäèí èç îäíîñòîðîííèõ ïðåäåëîâ íå ñóùåñòâóåò, èëè ðàâåí áåñêîíå÷íîñòè (ðàâåí +∞ èëè −∞).1Ïðèìåð 2.5. f (x) = . Çäåñü x = 0 òî÷êà ðàçðûâà âòîðîãîxðîäà.73Ïðèìåð 2.6. Óêàçàòü òî÷êè ðàçðûâà ôóíêöèè f (x) =|x + 2|·x+2ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx − 3 è îõàðàêòåðèçîâàòü èõ òèï.Ðåøåíèå: Ôóíêöèÿ ÿâëÿåòñÿ íåïðåðûâíîé âñþäó, êðîìå òî÷êèx = −2. Ðàññìîòðèì îäíîñòîðîííèå ïðåäåëû.()|x + 2|Ïðåäåë ñïðàâà lim·x−3 =x→−2+0x+2)(x+2· x − 3 = lim (x − 3) = −1= limx→−2+0 x + 2x→−2+0()|x + 2|Ïðåäåë ñëåâà lim·x−3 =x→−2−0x+2()x+2= lim−· x − 3 = lim (−x − 3) = −5x→−2−0x→−2−0x+2Îáà îäíîñòîðîííèõ ïðåäåëà ñóùåñòâóþò, íî íå ðàâíû ìåæäó ñîáîé, ñëåäîâàòåëüíî, x = −2 òî÷êà ðàçðûâà ïåðâîãî ðîäà.Ïðèìåð 2.7.
Óêàçàòü òî÷êè ðàçðûâà ôóíêöèè f (x) = e1/(x−1) èîõàðàêòåðèçîâàòü èõ òèï.Ðåøåíèå: Ôóíêöèÿ ÿâëÿåòñÿ íåïðåðûâíîé âñþäó, êðîìå òî÷êèx = 1. Ðàññìîòðèì îäíîñòîðîííèå ïðåäåëû.Ïðåäåë ñïðàâà lim e1/(x−1) = +∞x→1+0Ïðåäåë ñëåâà lim e1/(x−1) = 0x→1−0 ýòîì ñëó÷àå x = 1 ÿâëÿåòñÿ òî÷êîé ðàçðûâà âòîðîãî ðîäà.tg xÏðèìåð 2.8. Óêàçàòü òî÷êè ðàçðûâà ôóíêöèè f (x) =xîõàðàêòåðèçîâàòü èõ òèï.πÐåøåíèå: Òî÷êè ðàçðûâà: x = 0, x = + πk , ãäå k öåëîå.è2tg x= 1 (ïî òàáëèöå ýêâèâàëåíòíîñòåé), òîÏîñêîëüêó limx→0 xx = 0 òî÷êà óñòðàíèìîãî ðàçðûâà.  ýòîì ñëó÷àå ôóíêöèþ ìîæíî äîîïðåäåëèòü äî íåïðåðûâíîé ôóíêöèè ïðè x = 0 tg x , x ̸= 0,f (x) =x1, x = 0.74π+ πk , ãäå k öåëîå, ÿâëÿþòñÿ òî÷êàìè2ðàçðûâà âòîðîãî ðîäà (ðàññìîòðåòü ñàìîñòîÿòåëüíî).Òî÷êè ðàçðûâà x =3.
Äèôôåðåíöèðîâàíèå ôóíêöèè îäíîé ïåðåìåííîéÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3.1. Ïðîèçâîäíàÿ ôóíêöèèÎïðåäåëåíèå 3.1 Ïðîèçâîäíîé ôóíêöèè f (x) â òî÷êå a íàçûâà-þò ïðåäåë (åñëè îí ñóùåñòâóåò) îòíîøåíèÿ ïðèðàùåíèÿ ôóíêöèè ê ïðèðàùåíèþ àðãóìåíòà:f (a + ∆x) − f (a)∆f= lim∆x→0∆x→0 ∆x∆xf ′ (a) = limÏðàâèëà äèôôåðåíöèðîâàíèÿÏóñòü ôóíêöèè u(x), v(x) äèôôåðåíöèðóåìûå ôóíêöèè. Òî′′′ãäà 1.
(u ± v) = u ± v ,′′′2. (uv) = u v + uv , 3.3.2. Òàáëèöà ïðîèçâîäíûõ123456789(c)′ = 0, c ïîñòîÿííàÿ(xn )′ = nxn−1(sin x)′ = cos x(cos x)′ = − sin x1(tg x)′ =cos2 x1(ctg x)′ = − 2sin x(ex )′ = ex(ax )′ = ax ln a1(ln x)′ =x( u )′vu′ v − uv ′=v27510111211ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ121(loga x)′ =x ln a1(arcsin x)′ = √, |x| < 11 − x21(arccos x)′ = − √, |x| < 11 − x21(arctg x)′ =1 + x21(arcctg x)′ = −1 + x2 ñëåäóþùèõ ïðèìåðàõ ðàññìàòðèâàåòñÿ âû÷èñëåíèå ïðîèçâîäíîé y ′ çàäàííîé ôóíêöèè y .Ïðèìåð 3.1. y = x4 − 6x2 + 8Íà îñíîâàíèè ïðàâèëà äèôôåðåíöèðîâàíèÿñóììû,ðàçíîñòè è( 4)′′2òàáëèöû ïðîèçâîäíûõ èìååì y = x − 6x + 8 =( )′( )= x4 − 6 x2 + (8)′ = 4x3 − 6 · 2x + 0 = 4x3 − 12x√3Ïðèìåð 3.2. y = 2 3 x + 2xÏî ôîðìóëå äèôôåðåíöèðîâàíèÿ ñòåïåííîé ôóíêöèè èìååì()′()′( −2 )′ 2 −2/3√3′1/33+3 x= xy = 2 x+ 2 =2 x− 6x−3 =x326= √−33 x2 x3Ïðèìåð 3.3. y = x2 sin xÈñïîëüçóåì ïðàâèëî äèôôåðåíöèðîâàíèÿ ïðîèçâåäåíèÿ:)′ ( 2 )′y = x sin x = x sin x + x2 (sin x)′ = 2x sin x + x2 cos xcos xÏðèìåð 3.4.
y =1 + 4x3Èñïîëüçóåì ïðàâèëî äèôôåðåíöèðîâàíèÿ äðîáåé:′(276y′ =(cos x1 + 4x3)′=())(33 ′(cos x) 1 + 4x − cos x · 1 + 4x′=4x3 )2(1 +()− sin x · 1 + 4x3 − 12x2 cos x=(1 + 4x3 )2ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3.3. Äèôôåðåíöèðîâàíèå ñëîæíîé ôóíêöèèÒåîðåìà 3.1 Ïóñòü ôóíêöèÿ u = g(x) äèôôåðåíöèðóåìà â òî÷-êå x0 , ôóíêöèÿ y = f (u) äèôôåðåíöèðóåìà â òî÷êå u0 = g(x0 ).Òîãäà ñëîæíàÿ ôóíêöèÿ y = f (g(x)) äèôôåðåíöèðóåìà â òî÷êåx0 , ïðè÷åì [f (g(x))]′x0 = f ′ (u0 ) · g ′ (x0 ). ñëåäóþùèõ ïðèìåðàõ ðàññìàòðèâàåòñÿ âû÷èñëåíèå ïðîèçâîäíîé y ′ çàäàííîé ôóíêöèè y .Ïðèìåð 3.5. y = sin 6xÄàííàÿ ôóíêöèÿ ÿâëÿåòñÿ ñëîæíîé, îíà ñîñòîèò èç äâóõ çâå”íüåâ“ : u = 6x, y = sin u.
Ïî òåîðåìå î äèôôåðåíöèðîâàíèè ñëîæíîé ôóíêöèè èìååì: y ′ = (sin 6x)′ = cos 6x · (6x)′ = 6 cos 6x.Ïðèìåð 3.6. y = (tg 7x)5Ýòà ôóíêöèÿ ñëîæíàÿ: u = 7x, a = tg u, y = a5 . Çäåñü òðèçâåíà“ .”[]′1Òîãäà y ′ = (tg 7x)5 = 5(tg 7x)4 ··72 7xcos√Ïðèìåð 3.7. y = ln [ln (1 + 2 x)]111√ ·√ ·√y′ =ln (1 + 2 x) 1 + 2 xxÏðèìåð 3.8.
y = cos5 (arcsin 8x)1·8y ′ = 5 cos4 (arcsin 8x) · (− sin(arcsin 8x)) · √1 − 64x2e−5x arctg 7xÏðèìåð 3.9. y =ln 4x + cos3 x77 äàííîé çàäà÷å ñíà÷àëà èñïîëüçóåì ïðàâèëî äèôôåðåíöèðîâàíèÿ äðîáåé, çàòåì äèôôåðåíöèðóåì ñëîæíûå ôóíêöèè è ïîëüçóåìñÿ ïðàâèëàìè äèôôåðåíöèðîâàíèÿ ïðîèçâåäåíèÿ è ñóììû.)′−5xearctg7x=y′ =ln 4x + cos3 x( −5x)′ ()3e arctg 7x · ln 4x + cos x=−(ln 4x + cos3 x)2( −5x) ()′3e arctg 7x · ln 4x + cos x−=23(ln 4x + cos x)()()73−5e−5x arctg 7x + e−5x·ln4x+cosx1 + 49x2−=(ln 4x + cos3 x)2()( −5x)1e arctg 7x ·· 4 + 3 cos2 x(− sin x)4x−(ln 4x + cos3 x)2ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ(3.4. Âû÷èñëåíèå ëîãàðèôìè÷åñêîé ïðîèçâîäíîéÏðèìåð 3.10.
Âû÷èñëèòü ïðîèçâîäíóþ ôóíêöèè y = xxxÏðåäñòàâèì ôóíêöèþ xx â âèäå xx = eln x = ex ln x .Èñïîëüçóåì äèôôåðåíöèðîâàíèå ñëîæíîé ôóíêöèèI ñïîñîá.()′y ′ = (xx )′ = ex ln x = ex ln x (x ln x)′ = ex ln x (ln x + 1) = xx (ln x + 1)II ñïîñîá.Ïðîëîãàðèôìèðóåì ôóíêöèþ y = xxln y = ln xxÏî ñâîéñòâó ëîãàðèôìîâ èìååì ln y = x ln x. Ïðîäèôôåðåíöèðóåìïîëó÷åííîå âûðàæåíèå (ln y)′ = (x ln x)′ .
Ïîñêîëüêó y = y(x), òîy′y′⇒= ln x + 1. Âûðàçèì èç ýòîãî ðàâåíñòâà y ′ è(ln y) =yyxïîäñòàâèì y = x : y ′ = y(ln x + 1) = xx (ln x + 1).′783.5. Âû÷èñëåíèå ïðîèçâîäíîé ôóíêöèè, çàäàííîéïàðàìåòðè÷åñêèÏóñòü ôóíêöèÿ y(x) çàäàíà ñëåäóþùèì îáðàçîì{x = x(t),y = y(t).ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀïðè÷åì x(t) è y(t) äèôôåðåíöèðóåìûå ôóíêöèè àðãóìåíòà t,x′ (t) ̸= 0. Òîãäàyx′yt′= ′.xt{x = sin 3t,Ïðèìåð 3.11. Íàéòè ïðîèçâîäíóþ ôóíêöèèy = te2t .Ðåøåíèå: Íàéäåì îòäåëüíî ïðîèçâîäíûå ôóíêöèé x(t) è y(t)( )′x′ (t) = (sin 3t)′ = 3 cos 3t, y ′ (t) = te2t = e2t + 2te2tyt′e2t + 2te2t′′Ïî ôîðìóëå yx = ′ èìååì yx =xt3 cos 3t3.6. Âû÷èñëåíèå ïðîèçâîäíîé ôóíêöèè, çàäàííîé íåÿâíîÅñëè y = y(x) çàäàíà ñ ïîìîùüþ ñîîòíîøåíèÿ F (x, y) = 0,òî ãîâîðÿò, ÷òî ôóíêöèÿ y = y(x) çàäàíà íåÿâíî.
 ýòîì ñëó÷àåïðîèçâîäíàÿ ìîæåò áûòü íàéäåíà ñëåäóþùèì îáðàçîì:1. íàõîäèì ïðîèçâîäíóþ îò âûðàæåíèÿ F (x, y) (ðàññìàòðèâàÿ yêàê ôóíêöèþ îò x), ïðèðàâíèâàåì åå ê íóëþ;2. èç ïîëó÷åííîãî óðàâíåíèÿ âûðàæàåì y ′ (x).Ïðèìåð 3.12. Íàéòè ïðîèçâîäíóþ ôóíêöèè y(x), çàäàííîéíåÿâíî x3 y 4 + 5xy + 4y + 5 = 0.79Ðåøåíèå:  äàííîì ñëó÷àå F (x, y) = x3 y 4 +5xy+4y+5. Íàõîäèìïðîèçâîäíóþ( 3 4)′x y + 5xy + 4y + 5 x = 3x2 y 4 + x3 · 4y 3 y ′ + 5y + 5xy ′ + 4y ′Äàëåå, ïðèðàâíÿåì íàéäåííóþ ïðîèçâîäíóþ ê íóëþ è âûðàçèì èçïîëó÷åííîãî ñîîòíîøåíèÿ y ′ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3x2 y 4 + x3 · 4y 3 y ′ + 5y + 5xy ′ + 4y ′ = 0()3x2 y 4 + y ′ 4x3 y 3 + 5x + 4 + 5y = 03x2 y 4 + 5yy =− 3 34x y + 5x + 4′3.7. Ïðîèçâîäíûå âûñøèõ ïîðÿäêîâÎïðåäåëåíèå 3.2 Ïðîèçâîäíîé n-îãî ïîðÿäêà îò ôóíêöèè y =y(x) íàçûâàþò ïðîèçâîäíóþ îò åå (n − 1)-îé ïðîèçâîäíîé()′(n)(n−1)y = yÏðèìåð 3.13.
Íàéòè âòîðóþ ïðîèçâîäíóþ ôóíêöèè y = tg x.Ðåøåíèå: Íàõîäèì ïåðâóþ ïðîèçâîäíóþ1cos2 xÎò ïîëó÷åííîãî âûðàæåíèÿ ñíîâà âû÷èñëÿåì ïðîèçâîäíóþ()′( −2 )′1y ′′ = (tg x)′′ ==cos x =cos2 x()2 sin x= −2 cos−3 x (− sin x) =cos3 xÏðèìåð 3.14. Íàéòè y ′′′ (x), åñëè y(x) = (x + 4)5 .y ′ = (tg x)′ =Ðåøåíèå: Íàõîäèì ïåðâóþ ïðîèçâîäíóþ y ′ = 5(x + 4)4 , çàòåìâòîðóþ y ′′ = 5 · 4(x + 4)3 = 20(x + 4)3 è, íàêîíåö, òðåòüþy ′′′ = 60(x + 4)2 .803.8. Äèôôåðåíöèàë ôóíêöèèÎïðåäåëåíèå 3.3 Äèôôåðåíöèàëîì dy ôóíêöèè y = f (x) íàçû-âàåòñÿ ëèíåéíàÿ ÷àñòü ïðèðàùåíèÿ ∆y :dy = f ′ (x)∆x.ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÄèôôåðåíöèàë íåçàâèñèìîé ïåðåìåííîé dx ïî îïðåäåëåíèþðàâåí ïðèðàùåíèþ íåçàâèñèìîé ïåðåìåííîé ∆x , ò.å.dx = ∆x.Òàêèì îáðàçîì, äèôôåðåíöèàë ôóíêöèè ðàâåí ïðîèçâåäåíèþïðîèçâîäíîé ýòîé ôóíêöèè íà äèôôåðåíöèàë íåçàâèñèìîé ïåðåìåííîédy = f ′ (x)dx.Îòñþäà âûòåêàåò ïðåäñòàâëåíèå ïðîèçâîäíîé ôóíêöèè â âèäå÷àñòíîãî äâóõ äèôôåðåíöèàëîâf ′ (x) =df.dxÎñíîâíûå òåîðåìû î äèôôåðåíöèàëàõÒåîðåìà 3.2 d(c) = 0, ãäå c const.Òåîðåìà 3.3 d(u ± v) = du ± dv .Òåîðåìà 3.4 d(c · u) = c · du, ãäå c const.Òåîðåìà 3.5 d(u · v) = u · dv + v · du.Òåîðåìà 3.6 d(u)v=v · du − u · dv.v281Òåîðåìà 3.7 Ïóñòü y = f (u), u = u(x).
Òîãäà äèôôåðåíöèàëñëîæíîé ôóíêöèèdy = f ′ (u(x)) · u′ (x)dx = f (u)du.(èíâàðèàíòíîñòü ôîðìû ïåðâîãî äèôôåðåíöèàëà).Äèôôåðåíöèàëû âûñøèõ ïîðÿäêîâ îïðåäåëÿþòñÿ ðåêóððåíò-()dn y = d dn−1 y .ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀíîé ôîðìóëîéÎòñþäà ïîëó÷àåì ôîðìóëódn y = y (n) (dx)n .Ïðèìåð 3.15. Íàéòè äèôôåðåíöèàë ôóíêöèè y = sin x + x3 + 1.()′()dy = sin x + x3 + 1 dx = cos x + 3x2 dxÏðèìåð 3.16. Íàéòè äèôôåðåíöèàë ôóíêöèè y = ln(tg 6x).dy =11·· 6dxtg 6x cos2 6xÏðèìåð 3.17. Íàéòè d2 y , åñëè y = tg 3x.Âû÷èñëèì ñíà÷àëà ïåðâóþ è âòîðóþ ïðîèçâîäíûå()′( −2 )′13′′′y =· 3, y == 3 cos 3x =cos2 3xcos2 3x()sin 3x= −6 cos−3 3x · (− sin 3x) · 3 = 18 3cos 3xÑëåäîâàòåëüíî,d2 y = 18sin 3x2(dx)cos3 3x824. Ïðèëîæåíèÿ ïðîèçâîäíîé4.1. Óðàâíåíèå êàñàòåëüíîé è íîðìàëè ê êðèâîéÊàñàòåëüíîé ê êðèâîé y = f (x) â òî÷êå M0 íàçûâàåòñÿ ïðÿìàÿM0 N , ÿâëÿþùàÿñÿ ïðåäåëüíûì ïîëîæåíèåì ñåêóùèõ ïðè óñëî-ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀâèè, ÷òî òî÷êà M1 ïðèáëèæàåòñÿ ê òî÷êå êàñàíèÿ M0 .Óðàâíåíèå êàñàòåëüíîé ê ãðàôèêó y = f (x) â òî÷êå (x0 , y0 ) ,ãäå y0 = f (x0 ) :y = y0 + f ′ (x0 )(x − x0 )y6..............................................................................................................................................
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