Calculus
Section outline
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Predavanja: $\mathbb{N}$, matematična indukcija, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, sup, inf, max, min, absolutna vrednost v $\mathbb{R}$, kartezični zapis $\mathbb{C}$ in računanje v kartezičnem zapisu.
Lectures - pages in ($\cdot$) refer to this book: $\mathbb{N}$, mathematical induction (pages 39-42); $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ (p. 67-68, proof of Theorem 2.30 not required); sup, inf, max, min (p. 64-66, 69); absolute value in $\mathbb{R}$ (last paragraph of p. 61-63; proof of Theorem 2.25 not required); $\mathbb{C}$ in cartesian coordinates and basic operations in cartesian coordinates (last paragraph of p. 79-81).
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Uploaded 6/10/20, 17:49
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Predavanja: polarni zapis kompleksnega števila, računanje v polarni obliki, Eulerjeva formula, transformacije kompleksne ravnine. Algebraične enačbe, osnovni izrek algebra. n-ti koreni enote.
Lectures - pages in ($\cdot$) refer to this book: polar form of a complex number, basic operations in a polar form, Euler formula, transformations in a complex plane (from Theorem 2.15 on page 422-425). Algebraic equations and fundamental theorem of algebra (pages 428-432; proof of Theorem 15.5 not required). $n$-th roots of unity (pages 427-428).
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Uploaded 14/10/20, 20:35
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Uploaded 4/10/24, 18:08
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Predavanja: Zaporedja (definicija, eksplicitni, rekurzivni zapis). Lastnosti zaporedij (omejenost, natančna zgornja/spodnja meja, monotonost). Limita zaporedja. Naraščanje/padanje prek vseh meja.
Lecture: Sequences: definition, explicitly and recursively determined sequences, limit of a sequence, growing over all bounds (up to Theorem 1 in this notes). Properties of sequences: boundedness supremum/infimum, increasing sequences (link).
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Uploaded 21/10/20, 20:39
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Uploaded 9/10/24, 10:50
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Predavanja: Pravila za računanje z limitami, izrek o sendviču, izrek o monotoni konvergenci, $e$ kot limita zaporedja $a_n=(1+\frac{1}{n})^n$. Vrste - delne vsote, konvergenca, geometrijska vrsta.
Lectures (Pages 258-263): Rules for calculating with limits, sandwich theorem, theorem of monotone convergence, $e$ as a limit of the sequence $a_n=(1+\frac{1}{n})^n$. Series - partial sums, convergence, geometric series.
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Uploaded 29/10/20, 10:58
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Uploaded 21/10/24, 12:09
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Predavanja: Pravila za računanje z vrstami, konvergenčni kriteriji: primerjalni, kvocientni, korenski, Leibnizov. Ponovitev osnov funkcij ene spremenljivke.
Lectures: Rules for calculating with series (Theorem 11.2.2.), convergence tests: comparison test (Theorem 11.5.5), ratio test (Theorem 11.7.1, including the proof), root test (Theorem 11.7.3), alternating series and Leibniz test (Section 4.1, including the proof of Theorem 11.4.1). Basics of functions of one variable (Section 3.1 and 3.3).
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Uploaded 4/11/20, 19:12
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Uploaded 30/10/24, 16:52
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Posnetek predavanj
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Uploaded 23/10/24, 18:27
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Predavanja: Limita funkcije (leva in desna limita, neskončna limita, limita v neskončnosti), pravila za računanje z limitami, zveznost funkcije, lastnosti zveznosti, ničle zveznih funkcij, slika zaprtega intervala zvezne funkcije je zaprt interval (zvezna funkcija na zaprtem intervalu zavzame globalni minimum in globalni maksimum).
Lectures: Limit of a function (left and right limit, infinite limit, limit at infinity), rules for calculating limits (link), continuity of a function, properties of continuous functions (link), zeroes of continuous functions (Lemma 20.6), image of a closed interval w.r.t. the continuous function is a closed interval (Corollary 20.5 and Theorem 20.7). Section 3.2 gives an overview of elementary functions.
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Uploaded 11/11/20, 21:46
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Uploaded 31/10/24, 11:31
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Za radovedne: O reševanju kubične enačbe, iznajdbi kompleksnih števil in s tem povezanih življenjskih zgodbah.
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Zelo lepa geometrijska ideja dokaza. V angleščini.
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Kaj se lahko zgodi, če preuredimo vrstni red členov v vrsti?
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