Учебно-методическое пособие (1017796), страница 7
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................ ..........0.... ...Ny=f(x)My0Mx-xÐèñ. 1.Åñëè ôóíêöèÿ f (x) íåïðåðûâíà â òî÷êå x0 è èìååò áåñêîíå÷íóþïðîèçâîäíóþ â ýòîé òî÷êå, òî êàñàòåëüíîé ê ãðàôèêó ÿâëÿåòñÿ√3âåðòèêàëüíàÿ ïðÿìàÿ x = x0 . Íàïðèìåð, äëÿ ôóíêöèè y = x283êàñàòåëüíîé â òî÷êå (0,0) ÿâëÿåòñÿ îñü OY (âåðòèêàëüíàÿ ïðÿìàÿx = 0).y6....... y =....................................................................................................................................................... ...... .............x2-xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ0√3Ðèñ. 2.Ïóñòü â òî÷êå (x0 , y0 ) ê êðèâîé y = f (x) ïðîâåäåíà êàñàòåëüíàÿ. Ïðÿìàÿ, ïðîõîäÿùàÿ ÷åðåç òî÷êó êàñàíèÿ ïåðïåíäèêóëÿðíîêàñàòåëüíîé, íàçûâàåòñÿ íîðìàëüþ ê êðèâîé y = f (x).y6..................................................................................................................................................................................................................................................................................
......................... ............ ................................................................. .......... ..0 ..........................s................. . .................. ................... ......0.. .êàñàòåëüíàÿíîðìàëüy=f(x)y0x-xÐèñ. 3.Óðàâíåíèå íîðìàëè ê êðèâîé y = f (x) â òî÷êå (x0 , y0 ) èìååòâèä:1(x − x0 )y = y0 − ′f (x0 )84Çäåñü ïðåäïîëàãàåòñÿ, ÷òî f ′ (x0 ) ̸= 0. Åñëè f ′ (x0 ) = 0, òî óðàâíåíèå íîðìàëè èìååò âèä: x = x0 .Ïðèìåð 4.1. Çàïèñàòü óðàâíåíèÿ êàñàòåëüíîé è íîðìàëè ê ãðàôèêó ôóíêöèè y = x2 − 3x + 5 â òî÷êå x0 = 2.Ðåøåíèå: Óðàâíåíèå êàñàòåëüíîé ê ãðàôèêó ôóíêöèè y = f (x)èìååò âèä: y = y0 + f ′ (x0 )(x − x0 ).
Íàéäåì çíà÷åíèå ôóíêöèè âòî÷êå x0 = 2: f (x0 ) = 22 − 3 · 2 + 5 = 3. Äëÿ îïðåäåëåíèÿ óãëîâî-ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀãî êîýôôèöèåíòà êàñàòåëüíîé íàõîäèì ïðîèçâîäíóþ îò çàäàííîéôóíêöèè: f ′ (x) = 2x − 3. Çíà÷åíèå ïðîèçâîäíîé â òî÷êå x0 = 2 èäàåò èñêîìûé óãëîâîé êîýôôèöèåíò: f ′ (x0 ) = 2 · 2 − 3 = 1.
Òàêèìîáðàçîì, çàïèñûâàåì óðàâíåíèå êàñàòåëüíîé: y = 3 + 1 · (x − 2)èëè y = x + 1. Óðàâíåíèå íîðìàëè ê ãðàôèêó ôóíêöèè y = f (x)11èìååò âèä: y = y0 − f ′ (x(x−x).Âíàøåìñëó÷àå:y=3−0)1 (x − 2)0èëè y = −x + 5.Îòâåò: óðàâíåíèå êàñàòåëüíîé y = x + 1; óðàâíåíèå íîðìàëèy = −x + 5.Ïðèìåð 4.2. Íàéòè óðàâíåíèå êàñàòåëüíîé è íîðìàëè ê êðèâîé,{çàäàííîé ïàðàìåòðè÷åñêè:x = et + 1,y = e2t − 1.ïðè t = 0Ðåøåíèå: Íàéä¼ì ñíà÷àëà ïðîèçâîäíûå îò x è y ïî ïåðåìåííîét: x′t = et , yt′ = e2t · 2 = 2e2t .Çàòåì íàéä¼ìyx′ïî ôîðìóëåyx′yt′= ′.xt2e2t äàííîì ñëó÷àå= t = 2et .eÏðè t = 0 ïðîèçâîäíàÿ yx′ = 2. Ýòîìó çíà÷åíèþ ïàðàìåòðàñîîòâåòñòâóåò òî÷êà x0 = 2, y0 = 0.Óðàâíåíèå êàñàòåëüíîé â òî÷êå (x0 , y0 ): y = 2(x − 2).1Óðàâíåíèå íîðìàëè â òî÷êå (x0 , y0 ): y = − (x − 2).2yx′85Ïðèìåð 4.3.
Íàéòè óðàâíåíèå êàñàòåëüíîé è íîðìàëè ê ýëëèïñóÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ()x2 y 23 √+= 1 â òî÷êàõ: à) M1, 3 ; á) M2 (3, 0).942Ðåøåíèå: Ïðåîáðàçóåì óðàâíåíèå ýëëèïñà ê âèäó 4x2 +9y 2 = 36.Âîñïîëüçóåìñÿ ïðàâèëîì äèôôåðåíöèðîâàíèÿ ôóíêöèè, çàäàííîé4xíåÿâíî: 8x + 18y · y ′ = 0. Îòñþäà íàéäåì ïðîèçâîäíóþ yx′ = − .9y√2 3.Çíà÷åíèå ïðîèçâîäíîé â òî÷êå M1 : y ′ (M1 ) = −9√ ()√2 33Óðàâíåíèå êàñàòåëüíîé â òî÷êå M1 : y = 3 −x−.92()√39Óðàâíåíèå íîðìàëè â òî÷êå M1 : y = 3 + √x−.22 3Çíà÷åíèå ïðîèçâîäíîé â òî÷êå M2 : y ′ (M2 ) = ∞.Óðàâíåíèå êàñàòåëüíîé â òî÷êå M2 : x = 3.Óðàâíåíèå íîðìàëè â òî÷êå M2 : y = 0.Çàìå÷àíèå. Äðóãîé ñïîñîá ðåøåíèÿ ýòîé çàäà÷è îñíîâàí íà èñïîëüçîâàíèè ïàðàìåòðè÷åñêîãî çàäàíèÿ ýëëèïñà{x = 3 cos t,y = 2 sin t.)(3 √πÒî÷êå M1, 3 ñîîòâåòñòâóåò çíà÷åíèå ïàðàìåòðà t = .
Òî÷23πêå M2 (3, 0) ñîîòâåòñòâóåò çíà÷åíèå ïàðàìåòðà t = . Íàéòè óðàâ2íåíèå êàñàòåëüíîé è íîðìàëè ñàìîñòîÿòåëüíî.4.2. Ïðèìåíåíèå äèôôåðåíöèàëà â ïðèáëèæåííûõâû÷èñëåíèÿõÅñëè ôóíêöèÿ f (x) äèôôåðåíöèðóåìà â òî÷êå x0 ,∆y = dy + α(∆x)∆x.ãäå α(∆x) → 0 ïðè ∆x → 0. Çäåñü ∆y = f (x0 + ∆x) − f (x0 ) ïðèðàùåíèå ôóíêöèè.86Ïðè ìàëûõ ∆x èìååì ïðèáëèæåííîå ðàâåíñòâî ∆y ≈ dy , ò.å.f (x0 + ∆x) ≈ f (x0 ) + f ′ (x0 )∆x.y6.............................. .....
..... ..... ..... ..... ..... ..... ..... .......r............................................................................. ... ................................ ..... ..... ..... ...... ..... ..... ..... ...............r......0. ...... ............ ................................. ................ ................ .........................................................0{y=f(x)ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀdf( x0 ; ∆x)y∆xz }| {x-xÐèñ.
4.√3,996.√Ðåøåíèå: Ðàññìîòðèì ôóíêöèþ f (x) = x. Òîãäà√√1f (x0 + ∆x) = x0 + ∆x, f (x0 ) = x0 , f ′ (x0 ) = √ .2 x0Ïðèìåð 4.4. Âû÷èñëèòü ïðèáëèæåííîÏðèáëèæåííîå ðàâåíñòâîf (x0 + ∆x) ≈ f (x0 ) + f ′ (x0 )∆xâ äàííîì ñëó÷àå ïðèìåò âèä√x0 + ∆x ≈√1x0 + √ · ∆x (x0 > 0)2 x0×èñëî 3,996 ìîæíî ïðåäñòàâèòü â âèäå 3,996 = 4 + (−0,004). Âýòîì ñëó÷àå x0 = 4, ∆x = −0,004. Òîãäà√√√13,996 = 4 + (−0,004) ≈ 4 + √ · (−0,004) = 1,9992 4Âîïðîñû òî÷íîñòè âû÷èñëåíèé áóäóò ðàññìîòðåíû ïîçäíåå.874.3. Ïðèêëàäíûå çàäà÷è íà èñïîëüçîâàíèå ïðîèçâîäíîé4.3.1. Ìãíîâåííàÿ ñêîðîñòü ïðè ïðÿìîëèíåéíîìäâèæåíèèÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÏóñòü ìàòåðèàëüíàÿ òî÷êà M äâèæåòñÿ ïî ïðÿìîé è å¼ ðàññòîÿíèå â ìîìåíò âðåìåíè t îò íåêîòîðîé ôèêñèðîâàííîé òî÷êèO íà ýòîé ïðÿìîé ðàâíî s = s(t).
Ñðåäíÿÿ ñêîðîñòü òî÷êè çàïðîèçâîëüíûé ïðîìåæóòîê âðåìåíè ∆t, çà êîòîðûé òî÷êà ïåðåìåùàåòñÿ èç ïîëîæåíèÿ s(t) â ïîëîæåíèå s(t + ∆t), îïðåäåëÿåòñÿêàê îòíîøåíèå∆s, ãäå∆s = s(t + ∆t) − s(t).∆tÌãíîâåííàÿ ñêîðîñòü òî÷êè â ìîìåíò t îïðåäåëÿåòñÿ êàê ïðåäåë ñðåäíåé ñêîðîñòè çà ïðîìåæóòîê âðåìåíè ∆t ïðè óñëîâèè∆t → 0. Òàêèì îáðàçîì, ïîëó÷àåì∆s= s′ (t),∆t→0 ∆tv(t) = limò.å. ìãíîâåííàÿ ñêîðîñòü åñòü ïåðâàÿ ïðîèçâîäíàÿ îò ïåðåìåùåíèÿ.Óñêîðåíèå äâèæóùåéñÿ òî÷êè â ìîìåíò âðåìåíè t ýòî ïåðâàÿ ïðîèçâîäíàÿ îò ñêîðîñòè v(t), ò.å. a(t) = v ′ (t). Òàêèì îáðàçîì, óñêîðåíèå a(t) åñòü âòîðàÿ ïðîèçâîäíàÿ îò ïåðåìåùåíèÿ s(t):a(t) = s′′ (t).Ïðèìåð 4.5. Òî÷êà äâèæåòñÿ ïðÿìîëèíåéíî ïî çàêîíó s = t2 .Íàéòè åå ñêîðîñòü è óñêîðåíèå â ìîìåíò âðåìåíè t = 3.
Ïóòüèçìåðÿåòñÿ â ìåòðàõ, âðåìÿ - â ñåêóíäàõ.Ðåøåíèå: Ñêîðîñòü v(t) = s′ (t) = 2t. Òîãäà ïðè t = 3 ñv(t) = 6 ì/ñ. Óñêîðåíèå a(t) = s′′ (t) = 2 ì/ñ2 .88Ìîæíî ðàññìîòðåòü îáðàòíóþ çàäà÷ó - ïî èçâåñòíîé ñêîðîñòè(èëè óñêîðåíèþ) íàéòè ïðîéäåííûé ïóòü.Ïðèìåð 4.6. Ìàòåðèàëüíàÿ òî÷êà ìàññû m ñâîáîäíî ïàäàåò ïîääåéñòâèåì ñèëû òÿæåñòè F = mg . Íàéòè çàêîí äâèæåíèÿ òî÷êè(áåç ó÷åòà ñîïðîòèâëåíèÿ âîçäóõà).Ðåøåíèå:  äàííîì ñëó÷àå óñêîðåíèå g = a(t) = s′′ (t).
Òàêèìîáðàçîì, çàäà÷à ñâîäèòñÿ ê íàõîæäåíèþ ôóíêöèè ïî çàäàííîéÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀâòîðîé ïðîèçâîäíîé. Ìîæíî ïðîâåðèòü, ÷òî ðåøåíèåì ÿâëÿåòñÿgt2ôóíêöèÿ s(t) =+ C1 t + C2 , ãäå C1 , C2 íåêîòîðûå ïðîèçâîëü2íûå ïîñòîÿííûå. Ýòè ïîñòîÿííûå èìåþò ñëåäóþùèé ôèçè÷åñêèéñìûñë: C2 = s(0) ïîëîæåíèå òî÷êè â íà÷àëüíûé ìîìåíò âðåìåíè, C1 = s′ (0) = v(0) íà÷àëüíàÿ ñêîðîñòü.Óðàâíåíèå âèäà s′′ (t) = g ÿâëÿåòñÿ äèôôåðåíöèàëüíûì óðàâíåíèåì âòîðîãî ïîðÿäêà.
Ìåòîäû ðåøåíèÿ òàêèõ óðàâíåíèé áóäóòèçó÷àòüñÿ ïîçäíåå.4.3.2. Ìîùíîñòü è íàïðÿæåíèåÏóñòü ÷åðåç ó÷àñòîê ýëåêòðè÷åñêîé öåïè ïðîõîäèò ýëåêòðè÷åñêèé çàðÿä. Ñêîðîñòü ïîñòóïëåíèÿ â öåïü ýëåêòðè÷åñêîé ýíåðãèèW (t) â ìîìåíò âðåìåíè t ïðåäñòàâëÿåò ñîáîé ìãíîâåííóþ ìîùíîñòüP (t) = W ′ (t).Ïðèìåð 4.7. Íàéòè ìãíîâåííóþ ìîùíîñòü, åñëè èçâåñòíà ýíåðãèÿ, ïîñòóïàþùàÿ â ïðèåìíèêW (t) = (A cos φ)t +AAsin(2ωt − φ) +sin φ,2ω2ωãäå A - àìïëèòóäà, φ - ôàçà, ω - ÷àñòîòà, êîòîðûå ïðåäïîëàãàþòñÿïîñòîÿííûìè.89Ðåøåíèå: Ìãíîâåííàÿ ìîùíîñòü ýòî ïðîèçâîäíàÿ ýíåðãèèW (t) ïî âðåìåíè:Acos(2ωt − φ)2ω =2ω= A cos φ + A cos(2ωt − φ).P (t) = W ′ (t) = A cos φ +ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÍàïðÿæåíèå íà ýëåìåíòå ýëåêòðè÷åñêîé öåïè èíäóêòèâíîñòèdi îïðåäåëÿåòñÿ ïî ñëåäóþùåé ôîðìóëå UL = L , ãäå i òîê âdtöåïè, L èíäóêòèâíîñòü.Ïðèìåð 4.8. Íàéòè íàïðÿæåíèå íà èíäóêòèâíîñòè, åñëè òîêi(t) = e−at , ãäå a ïîñòîÿííàÿ.diÐåøåíèå: UL = L = −aLe−at .dt4.3.3.
Ïåðåõîäíûé ïðîöåññ â ëèíåéíîé ýëåêòðè÷åñêîéöåïèÏðèìåð 4.9. Ïóñòü ëèíåéíàÿ ýëåêòðè÷åñêàÿ öåïè, ñîñòîÿùàÿ èçïîñëåäîâàòåëüíî ñîåäèíåííûõ ýëåìåíòîâ ñîïðîòèâëåíèÿ R è èíäóêòèâíîñòè L ïðèñîåäèíÿåòñÿ ê èñòî÷íèêó ý.ä.ñ. ε = ε(t). Îïèñàòü ïåðåõîäíûé ïðîöåññ.Ðåøåíèå:Ïåðåõîäíûé ïðîöåññ â äàííîé ýëåêòðè÷åñêîé öåïè íà îñíîâàíèèâòîðîãî çàêîíà Êèðõãîôà îïèñûâàåòñÿ óðàâíåíèåìUR + UL = ε.diÏîñêîëüêó UR = R · i, à UL = L · , ïîëó÷àåì óðàâíåíèådtdi= ε.R·i+L·dtÌû ïðèøëè ê äèôôåðåíöèàëüíîìó óðàâíåíèþ ïåðâîãî ïîðÿäêà.Ìåòîäû ðåøåíèÿ òàêèõ óðàâíåíèé áóäóò èçó÷àòüñÿ ïîçäíåå. Ïðèâåäåì îòâåò â ñëó÷àå ïîñòîÿííîé ý.ä.ñ. ε(t) = E .()E ERti(t) = − exp −.R RL90Äàííûé ïðèìåð íàðÿäó ñ ïðèâåäåííûìè âûøå ïðèìåðàìè èëëþñòðèðóåò âàæíîñòü èñïîëüçîâàíèÿ ïîíÿòèÿ ïðîèçâîäíîé ïðèðåøåíèè ïðèêëàäíûõ çàäà÷.4.4. Ïðàâèëî ËîïèòàëÿÏðàâèëî Ëîïèòàëÿ - ýòî ñïîñîá ðàñêðûòèÿ íåîïðåäåëåííîñòåé,ò.å.
âû÷èñëåíèÿ ïðåäåëîâ îòíîøåíèé äâóõ áåñêîíå÷íî ìàëûõ èëèÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀäâóõ áåñêîíå÷íî áîëüøèõ âåëè÷èí.îòíîøåíèÿ áåñêîíå÷íî ìàëûõ( Ïðàâèëî Ëîïèòàëÿ äëÿ )0íåîïðåäåëåííîñòè âèäà0Ïóñòü ôóíêöèè f (x) è g(x) îïðåäåëåíû â íåêîòîðîé ïðîêîëîòîéîêðåñòíîñòè O(x0 ) òî÷êè x0 , g(x) ̸= 0 ïðè x èç O(x0 ), èlim f (x) = lim g(x) = 0.x→x0x→x0Ïðåäïîëîæèì, ÷òî ïðè x ̸= x0 ôóíêöèè f (x) è g(x) èìåþò ïðîèç-âîäíûå, g ′ (x) ̸= 0 ïðè x èç O(x0 ), è ñóùåñòâóåò ïðåäåë îòíîøåíèÿýòèõ ïðîèçâîäíûõ:f ′ (x).limx→x0 g ′ (x)Òîãäà ïðè x → x0 ñóùåñòâóåò ïðåäåë îòíîøåíèÿ ñàìèõ ôóíêöèéèf ′ (x)f (x)= lim ′ .limx→x0 g (x)x→x0 g(x)Çàìå÷àíèå.
Ïðàâèëî Ëîïèòàëÿ ñïðàâåäëèâî è äëÿ îäíîñòîðîííèõ ïðåäåëîâx → x0 −, x → x0 +è äëÿ ñëó÷àåâ, êîãäàx → ∞, x → −∞, x → +∞.91ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx3 − 1Ïðèìåð 4.10. Âû÷èñëèòü ïðåäåë ôóíêöèè lim, èñïîëüx→1 ln xçóÿ ïðàâèëî Ëîïèòàëÿ.Ðåøåíèå: Ïîäñòàíîâêà ïðåäåëüíîãî çíà÷åíèÿ x = 1 ïðèâîäèò ê( 3)0íåîïðåäåëåííîñòè âèäà , òàê êàê lim x − 1 = 0 è lim ln x = 0.x→1x→10Ïðåäåë îòíîøåíèÿ ïðîèçâîäíûõ ñóùåñòâóåò:( 3)′x −13x2lim= 3,= limx→1 (ln x)′x→1 1xïîýòîìó( 3)′x −1x3 − 13x2lim= lim= lim=3x→1 ln xx→1 (ln x)′x→1 1xxe −1−xÏðèìåð 4.11.
Âû÷èñëèòü ïðåäåë limx→0sin2 xÐåøåíèå: Ïîäñòàíîâêà ïðåäåëüíîãî çíà÷åíèÿ x = 0 ïðèâîäèò ê0íåîïðåäåëåííîñòè âèäà . Âû÷èñëåíèå ïðåäåëà ìîæíî óïðîñòèòü,0çàìåíèâ çíàìåíàòåëü sin2 x íà ýêâèâàëåíòíóþ áåñêîíå÷íî ìàëóþsin2 x ∼ x2 ïðè x → 0:ex − 1 − xex − 1 − xlim= limx→0x→0x2sin2 xÏî ïðàâèëó Ëîïèòàëÿex − 1 − x(ex − 1 − x)′ex − 1lim= lim= limx→0x→0x→0 2xx2(x2 )′0. Ýòîò ïðåäåë ìîæíî0âû÷èñëèòü ëèáî ïî ïðàâèëó Ëîïèòàëÿ, ëèáî, çàìåíèâ ÷èñëèòåëüíà ýêâèâàëåíòíóþ áåñêîíå÷íî ìàëóþ: ex − 1 ∼ x ïðè x → 0. ÒîãäàÌû ñíîâà èìååì íåîïðåäåëåííîñòü âèäàex − 1x1lim= lim=x→0 2xx→0 2x292Ïðàâèëî( Ëîïèòàëÿ äëÿ îòíîøåíèÿ) áåñêîíå÷íî∞∞Ïóñòü ôóíêöèè f (x) è g(x) îïðåäåëåíû â íåêîòîðîé ïðîêîëîòîéîêðåñòíîñòè O(x0 ) òî÷êè x0 èáîëüøèõíåîïðåäåëåííîñòè âèäàlim f (x) = lim g(x) = ∞.x→x0x→x0ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÏðåäïîëîæèì, ÷òî ïðè x ̸= x0 ôóíêöèè f (x) è g(x) èìåþò ïðîèçâîäíûå, g ′ (x) ̸= 0 ïðè x èç O(x0 ), è ñóùåñòâóåò ïðåäåë îòíîøåíèÿýòèõ ïðîèçâîäíûõ:f ′ (x)lim.x→x0 g ′ (x)Òîãäà ïðè x → x0 ñóùåñòâóåò ïðåäåë îòíîøåíèÿ ñàìèõ ôóíêöèéèf (x)f ′ (x)lim= lim ′ .x→x0 g(x)x→x0 g (x)∞ïðàâèëîÇàìå÷àíèå.
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