Matematická analýza II.
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- Cameron Fisher
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1 V. Diferenciálny počet (prezentácia k prednáške MANb/10) doc. RNDr., PhD. 1 1 ondrej.hutnik@upjs.sk umv.science.upjs.sk/analyza Prednáška 8 6. marca 2018
2 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Príklady: Vyšetrite rovnomernú spojitost funkcie f na množine M, ak f (x) = sin x, kde M = R; f (x) = 1 x na M = 1, + ); f (x) = 1 x na M = (0, + );
3 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Príklady: Vyšetrite rovnomernú spojitost funkcie f na množine M, ak f (x) = sin x, kde M = R; f (x) = 1 x na M = 1, + ); f (x) = 1 x na M = (0, + );
4 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Príklady: Vyšetrite rovnomernú spojitost funkcie f na množine M, ak f (x) = sin x, kde M = R; f (x) = 1 x na M = 1, + ); f (x) = 1 x na M = (0, + );
5 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Príklady: Vyšetrite rovnomernú spojitost funkcie f na množine M, ak f (x) = sin x, kde M = R; f (x) = 1 x na M = 1, + ); f (x) = 1 x na M = (0, + );
6 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Pozorovanie: f C u(m) ( x n, y n M) lim x n n y n = 0 pohrajte sa interaktívne: Veta V.10 C u (M) C (M) lim f (x n n) f (y n) = 0
7 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Pozorovanie: f C u(m) ( x n, y n M) lim x n n y n = 0 pohrajte sa interaktívne: Veta V.10 C u (M) C (M) lim f (x n n) f (y n) = 0
8 It has apparently not yet been observed, that... continuity at any single point... is not the continuity... which can be called uniform continuity, because it extends uniformly to all points and in all directions. Heine: Ueber trigonometrische Reihen (1870), p. 361 Definícia rovnomerná spojitost funkcie na množine Funkciu f nazývame rovnomerne spojitá na množine M D f, akk ( ε > 0) ( δ > 0) ( x, y M, x y < δ) f (x) f (y) < ε. Budeme písat f C u (M). Pozorovanie: f C u(m) ( x n, y n M) lim x n n y n = 0 pohrajte sa interaktívne: Veta V.10 C u (M) C (M) lim f (x n n) f (y n) = 0
9 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b EDUARD HEINE ( ) GEORG CANTOR ( )
10 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b Poznámky: Heineho-Cantorova veta neplatí na otvorenom intervale, napr. f (x) = cos 1 x na M = (0, 1 je spojitá, ale nie rovnomerne! Čo vieme povedat o natiahnutí spojitosti do krajných bodov intervalu? Tvrdenie: bod nespojitosti rovnomerne spojitej funkcie na otvorenom intervale je odstránitel ný Nech f C (a, b), kde a, b R. Potom f sa dá spojite rozšírit na a, b (t.j. existujú vlastné jednostranné limity v krajných bodoch) práve vtedy, ked f C u(a, b). Náčrt dôkazu: zrejmé, lebo ak sa dá spojite rozšírit na a, b, tak podl a Heineho-Cantorovej vety je f C u a, b, teda aj f C u(a, b) Ked že f C u(a, b), tak ( ε > 0)( δ > 0)( x, y (a, b)) ( x y < δ f (x) f (y) < ε), a teda to platí aj pre intervaly (a, a + δ) a (b δ, b), čiže podl a Dirichletovej vety existujú vlastné jednostranné limity v krajných bodoch. Predchádzajúci výsledok neplatí na neohraničenom intervale, napr. f (x) = sin x je rovnomerne spojitá na R, ale neexistujú jej limity v ±!
11 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b Poznámky: Heineho-Cantorova veta neplatí na otvorenom intervale, napr. f (x) = cos 1 x na M = (0, 1 je spojitá, ale nie rovnomerne! Čo vieme povedat o natiahnutí spojitosti do krajných bodov intervalu? Tvrdenie: bod nespojitosti rovnomerne spojitej funkcie na otvorenom intervale je odstránitel ný Nech f C (a, b), kde a, b R. Potom f sa dá spojite rozšírit na a, b (t.j. existujú vlastné jednostranné limity v krajných bodoch) práve vtedy, ked f C u(a, b). Náčrt dôkazu: zrejmé, lebo ak sa dá spojite rozšírit na a, b, tak podl a Heineho-Cantorovej vety je f C u a, b, teda aj f C u(a, b) Ked že f C u(a, b), tak ( ε > 0)( δ > 0)( x, y (a, b)) ( x y < δ f (x) f (y) < ε), a teda to platí aj pre intervaly (a, a + δ) a (b δ, b), čiže podl a Dirichletovej vety existujú vlastné jednostranné limity v krajných bodoch. Predchádzajúci výsledok neplatí na neohraničenom intervale, napr. f (x) = sin x je rovnomerne spojitá na R, ale neexistujú jej limity v ±!
12 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b Poznámky: Heineho-Cantorova veta neplatí na otvorenom intervale, napr. f (x) = cos 1 x na M = (0, 1 je spojitá, ale nie rovnomerne! Čo vieme povedat o natiahnutí spojitosti do krajných bodov intervalu? Tvrdenie: bod nespojitosti rovnomerne spojitej funkcie na otvorenom intervale je odstránitel ný Nech f C (a, b), kde a, b R. Potom f sa dá spojite rozšírit na a, b (t.j. existujú vlastné jednostranné limity v krajných bodoch) práve vtedy, ked f C u(a, b). Náčrt dôkazu: zrejmé, lebo ak sa dá spojite rozšírit na a, b, tak podl a Heineho-Cantorovej vety je f C u a, b, teda aj f C u(a, b) Ked že f C u(a, b), tak ( ε > 0)( δ > 0)( x, y (a, b)) ( x y < δ f (x) f (y) < ε), a teda to platí aj pre intervaly (a, a + δ) a (b δ, b), čiže podl a Dirichletovej vety existujú vlastné jednostranné limity v krajných bodoch. Predchádzajúci výsledok neplatí na neohraničenom intervale, napr. f (x) = sin x je rovnomerne spojitá na R, ale neexistujú jej limity v ±!
13 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b Poznámky: Heineho-Cantorova veta neplatí na otvorenom intervale, napr. f (x) = cos 1 x na M = (0, 1 je spojitá, ale nie rovnomerne! Čo vieme povedat o natiahnutí spojitosti do krajných bodov intervalu? Tvrdenie: bod nespojitosti rovnomerne spojitej funkcie na otvorenom intervale je odstránitel ný Nech f C (a, b), kde a, b R. Potom f sa dá spojite rozšírit na a, b (t.j. existujú vlastné jednostranné limity v krajných bodoch) práve vtedy, ked f C u(a, b). Náčrt dôkazu: zrejmé, lebo ak sa dá spojite rozšírit na a, b, tak podl a Heineho-Cantorovej vety je f C u a, b, teda aj f C u(a, b) Ked že f C u(a, b), tak ( ε > 0)( δ > 0)( x, y (a, b)) ( x y < δ f (x) f (y) < ε), a teda to platí aj pre intervaly (a, a + δ) a (b δ, b), čiže podl a Dirichletovej vety existujú vlastné jednostranné limity v krajných bodoch. Predchádzajúci výsledok neplatí na neohraničenom intervale, napr. f (x) = sin x je rovnomerne spojitá na R, ale neexistujú jej limity v ±!
14 The general ideas of the proof of several theorems in 3 according to the principles of Mr. Weierstrass are known to me by oral communications from himself, from Mr. Schwarz and Mr. Cantor, so that... Heine: Die Elemente der Funktionenlehre (1872), p. 182 Veta (Heineho-Cantorova, 1872) C u a, b = C a, b Poznámky: Heineho-Cantorova veta neplatí na otvorenom intervale, napr. f (x) = cos 1 x na M = (0, 1 je spojitá, ale nie rovnomerne! Čo vieme povedat o natiahnutí spojitosti do krajných bodov intervalu? Tvrdenie: bod nespojitosti rovnomerne spojitej funkcie na otvorenom intervale je odstránitel ný Nech f C (a, b), kde a, b R. Potom f sa dá spojite rozšírit na a, b (t.j. existujú vlastné jednostranné limity v krajných bodoch) práve vtedy, ked f C u(a, b). Náčrt dôkazu: zrejmé, lebo ak sa dá spojite rozšírit na a, b, tak podl a Heineho-Cantorovej vety je f C u a, b, teda aj f C u(a, b) Ked že f C u(a, b), tak ( ε > 0)( δ > 0)( x, y (a, b)) ( x y < δ f (x) f (y) < ε), a teda to platí aj pre intervaly (a, a + δ) a (b δ, b), čiže podl a Dirichletovej vety existujú vlastné jednostranné limity v krajných bodoch. Predchádzajúci výsledok neplatí na neohraničenom intervale, napr. f (x) = sin x je rovnomerne spojitá na R, ale neexistujú jej limity v ±!
15 Spojitost a d alšie vlastnosti funkcií Zopakovanie Veta II.4 Ak f je rýdzomonotónna na M D f, tak f je injektívna na M. y x Čo je potrebné pridat k injektívnosti, aby sme dostali rýdzomonotónnost??? Veta V.11 Ak f je injektívna na M D f a f C (M), tak f je rýdzomonotónna na M.
16 Spojitost a d alšie vlastnosti funkcií Zopakovanie Veta II.4 Ak f je rýdzomonotónna na M D f, tak f je injektívna na M. y x Čo je potrebné pridat k injektívnosti, aby sme dostali rýdzomonotónnost??? Veta V.11 Ak f je injektívna na M D f a f C (M), tak f je rýdzomonotónna na M.
17 Spojitost a d alšie vlastnosti funkcií Zopakovanie Veta II.4 Ak f je rýdzomonotónna na M D f, tak f je injektívna na M. y x Čo je potrebné pridat k injektívnosti, aby sme dostali rýdzomonotónnost??? Veta V.11 Ak f je injektívna na M D f a f C (M), tak f je rýdzomonotónna na M.
18 Dôsledok = Veta II.4 + Veta V.11 Nech f C (M). Potom f je injektívna na M práve vtedy, ked f je rýdzomonotónna na M. Zopakovanie Dôsledok Darbouxovej vety Nech f C (M), kde M je interval. Potom N = {y = f (x); x M} je jednoprvková množina alebo interval. JEAN GASTON DARBOUX ( ) Za akých okolností sa dá tento výsledok obrátit?
19 Dôsledok = Veta II.4 + Veta V.11 Nech f C (M). Potom f je injektívna na M práve vtedy, ked f je rýdzomonotónna na M. Zopakovanie Dôsledok Darbouxovej vety Nech f C (M), kde M je interval. Potom N = {y = f (x); x M} je jednoprvková množina alebo interval. JEAN GASTON DARBOUX ( ) Za akých okolností sa dá tento výsledok obrátit?
20 Dôsledok = Veta II.4 + Veta V.11 Nech f C (M). Potom f je injektívna na M práve vtedy, ked f je rýdzomonotónna na M. Veta V.12 Nech f je monotónna na intervale M D f a N = {y = f (x); x M} je jednoprvková množina alebo interval. Potom f C (M). Veta (o spojitosti inverznej funkcie) Nech f C (M) je injektívna na intervale M D f a N = {y = f (x); x M}. Potom f C (N) je rýdzomonotónna. Veta V.9 Všetky základné elementárne funkcie sú spojité v každom bode svojho definičného oboru. všeobecná mocninná funkcia, logaritmická funkcia, cyklometrické a hyperbolometrické funkcie
21 Dôsledok = Veta II.4 + Veta V.11 Nech f C (M). Potom f je injektívna na M práve vtedy, ked f je rýdzomonotónna na M. Veta V.12 Nech f je monotónna na intervale M D f a N = {y = f (x); x M} je jednoprvková množina alebo interval. Potom f C (M). Veta (o spojitosti inverznej funkcie) Nech f C (M) je injektívna na intervale M D f a N = {y = f (x); x M}. Potom f C (N) je rýdzomonotónna. Veta V.9 Všetky základné elementárne funkcie sú spojité v každom bode svojho definičného oboru. všeobecná mocninná funkcia, logaritmická funkcia, cyklometrické a hyperbolometrické funkcie
22 Dôsledok = Veta II.4 + Veta V.11 Nech f C (M). Potom f je injektívna na M práve vtedy, ked f je rýdzomonotónna na M. Veta V.12 Nech f je monotónna na intervale M D f a N = {y = f (x); x M} je jednoprvková množina alebo interval. Potom f C (M). Veta (o spojitosti inverznej funkcie) Nech f C (M) je injektívna na intervale M D f a N = {y = f (x); x M}. Potom f C (N) je rýdzomonotónna. Veta V.9 Všetky základné elementárne funkcie sú spojité v každom bode svojho definičného oboru. všeobecná mocninná funkcia, logaritmická funkcia, cyklometrické a hyperbolometrické funkcie
23 The extent of this calculus is immense: it applies to curves both mechanical and geometrical; radical signs cause it no difficulty, and even are often convenient; it extends to as many variables as one wishes; the comparison of infinitely small quantities of all sorts is easy. And it gives rise to an infinity of surprising discoveries concerning curved or straight tangents, questions De maximis & minimis, inflexion points and cusps of curves, envelopes, caustics from reflexion or refraction, &c. as we shall see in this work. Marquis de L Hospital: Analyse des infiniment petits (1696)
24 Infinitezimálny počet motivácia And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I ever desired to know. Descartes: La Geometrie, Appendix to the Discours de la methode (1637) What contempt for the non-english! We have found these methods, without any help from the English. Joh. Bernoulli: Opera, vol. IV (1735), p. 70 Isaac Newton was not a pleasant man. His relations with other academics were notorious, with most of his later life spent embroiled in heated disputes... A serious dispute arose with the German philosopher Gottfried Leibniz. Both Leibniz and Newton had independently developed a branch of mathematics called calculus, which underlies most of modern physics... Hawking: A Brief History of Time (1988) Problém: Nech y = f (x) je daná krivka. V každom bode x chceme poznat sklon krivky, dotyčnicu ku krivke a normálu ku krivke. Motivácie: výpočet uhla, pod ktorým sa dve krivky pretínajú (Descartes); konštrukcia d alekohl adov (Galilei) a hodín (Huygens 1673); nájdenie maxima a minima funkcie (Fermat 1638); rýchlost a zrýchlenie pohybu (Galilei 1638, Newton 1686); astronómia, overenie gravitačného zákona (Kepler, Newton).
25 Infinitezimálny počet okolnosti vzniku a historické súvislosti NEWTON (1671, publikované až v roku 1736): Uvažuje premenné v, x, y, z ako as gradually and indefinitely increasing,... And the velocities by which every Fluent is increased by its general motion, (which I may call Fluxions) I shall represent by the same Letters pointed thus v, ẋ, ẏ, ż. Hodnoty dostal spôsobom rejecting the Terms... as being equal to nothing.
26 Infinitezimálny počet okolnosti vzniku a historické súvislosti LEIBNIZ (1684): Nech y = x 2. Ak x narastie o x, tak y narastie na y + y = (x + x) 2 = x 2 + 2x x + ( x) 2 y = 2x x + ( x) 2. Podl a Leibniza x a y predstavujú nekonečne malé veličiny, ktoré označil dx a dy. Zanedbaním výrazu (dx) 2, ktorý je nekonečne menší ako 2xdx dostal dy = 2xdx alebo dy dx = 2x.
27 Infinitezimálny počet okolnosti vzniku a historické súvislosti JACOB A JOHANN BERNOULLI: tretie znovuobjavenie diferenciálneho kalkulu na základe Leibnizovho článku z roku Johann dáva v roku 1691/92 súkromné hodiny nového kalkulu markízovi de L Hospitalovi, pričom nekonečne malé veličiny chápal ako veličiny, ktoré sa môžu pripočítat ku konečným veličinám bez zmeny ich hodnôt kritika tejto novej matematiky od B. NIEVENTIJTA z roku 1694
28 Infinitezimálny počet okolnosti vzniku a historické súvislosti GUILLAUME DE L HOSPITAL (1696): vydal knihu Analyse des infiniment petits, ktorá sa stala prvou učebnicou diferenciálneho počtu a znamenala prelom v používaní nového kalkulu dokonca aj vo Francúzsku, kde veda bola ovplyvnená dlhé desat ročia karteziánmi (abbé Catelan, Papin, Rolle, atd.)
29 Derivácia rodiaca sa: derivácia umožňuje popis fyzikálnych dejov a geometrické správanie sa funkcií ISAAC NEWTON ( ) GOTTFRIED WILHELM LEIBNIZ ( )
30 Derivácia uháňajúca: derivácia udáva rýchlost zmien fyzikálnych veličín derivácia umožňuje merat rýchlost dejov najrôznejšieho druhu
31 Derivácia tvarujúca: derivácia funkcie v bode je smernica jej dotyčnice v danom bode, t.j. udáva ako rýchlo funkcia rastie alebo klesá druhá derivácia udává mieru konvexnosti či konkávnosti derivácia je vhodný nástroj pre popis kriviek najrôznejšieho druhu
32 Derivácia zjednodušujúca: derivácia slúži na aproximáciu funkcie, ak ju nahradíme jej dotyčnicou namiesto všeobecne komplikovaných závislostí medzi veličinami pracujeme s lineárnymi funkciami a rovnicami
33 Derivácia všeobklopujúca: ked že väčšina fyzikálnych dejov prebieha tak, že zmena jednej veličiny vyvoláva zmenu či prítomnost inej veličiny, je derivácia ideálnym prostriedkom pre formulovanie fyzikálnych zákonov popisuje všetko, čo nás obklopuje
34 ... tangentem invenire, esse rectam ducere, quæ duo curvæ puncta distantiam infinite parvam habentia, jungat,... Leibniz: Nova methodus pro maximis et minimis... (1684) Definícia derivácia funkcie v bode Nech x 0 D f je hromadný bod D f. Deriváciou funkcie f v bode x 0 nazývame f (x) f (x 0 ) vlastnú limitu lim a označujeme ju f (x 0 ). x x0 x x 0
35 ... tangentem invenire, esse rectam ducere, quæ duo curvæ puncta distantiam infinite parvam habentia, jungat,... Leibniz: Nova methodus pro maximis et minimis... (1684) Definícia derivácia funkcie v bode Nech x 0 D f je hromadný bod D f. Deriváciou funkcie f v bode x 0 nazývame f (x) f (x 0 ) vlastnú limitu lim a označujeme ju f (x 0 ). x x0 x x 0
36 f f (x) f (x 0 ) f (x 0 + h) f (x 0 ) (x 0 ) = lim = lim x x0 x x 0 h 0 h Príklady: Nájdite deriváciu funkcie f v l ubovol nom bode x 0 R, ak f (x) = c, kde c R; f (x) = x n pre n N; f (x) = a x pre a > 0, a 1;
37 f f (x) f (x 0 ) f (x 0 + h) f (x 0 ) (x 0 ) = lim = lim x x0 x x 0 h 0 h Príklady: Nájdite deriváciu funkcie f v l ubovol nom bode x 0 R, ak f (x) = c, kde c R; f (x) = x n pre n N; f (x) = a x pre a > 0, a 1;
38 f f (x) f (x 0 ) f (x 0 + h) f (x 0 ) (x 0 ) = lim = lim x x0 x x 0 h 0 h Príklady: Nájdite deriváciu funkcie f v l ubovol nom bode x 0 R, ak f (x) = c, kde c R; f (x) = x n pre n N; f (x) = a x pre a > 0, a 1;
Matematická analýza II.
V. Diferenciálny počet (prezentácia k prednáške MANb/10) doc. RNDr., PhD. 1 1 ondrej.hutnik@upjs.sk umv.science.upjs.sk/analyza Prezentácie k prednáškam čast II 21. februára 2018 The extent of this calculus
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