From 51302b581c43e649f72d50cb09b5d56971128e5e Mon Sep 17 00:00:00 2001 From: Szymon P Date: Sun, 12 Oct 2025 10:40:01 +0200 Subject: [PATCH] matematyka stosowana --- media/Pasted image 20251011092804.png | Bin 0 -> 77831 bytes media/Pasted image 20251011092828.png | Bin 0 -> 466568 bytes uczelnia/1_semestr/1 semestr.md | 1 + .../1_semestr/algorytmy/2 wykład/2 wykład.md | 7 +++++++ .../algorytmy/2 wykład/Algorytmy sortowania.md | 5 +++++ .../2 wykład/Algorytmy wyszukiwania.md | 2 ++ .../algorytmy/2 wykład/Cykl eulera.md | 1 + .../algorytmy/2 wykład/Najbliższy sąsiad.md | 3 +++ .../algorytmy/2 wykład/Najkrótsza ścieżka.md | 1 + .../Podstawy rozwiązywania problemów.md | 9 +++++++++ .../algorytmy/2 wykład/Problemy grafowe.md | 5 +++++ .../algorytmy/2 wykład/Przejście grafu.md | 1 + .../2 wykład/Przetwarzanie napisów.md | 1 + .../2 wykład/Przykłady algorytmów grafowych.md | 3 +++ .../algorytmy/2 wykład/Ramię robota.md | 3 +++ .../2 wykład/Schemat tworzenia algorytmu.md | 6 ++++++ .../2 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z+;Bc=lgiq#8a7O_I17Z9%lRDbwSU=pj~&=il2tR?sg}T=?G{anMHg&Lj+hDko-Z+U z0DRyVY@uyz1~0Vo^suVIMQ2%H^5{!%pHe<`O1k*c1Gbq+&fxE`VgEF^K==d>+{d4- zb5XCL525s+#Tk4nR0QpR(RU5~d6|gum=qute1-k^(Pt8AOVNCpMOY?j!fJ4|5xR8Y zokgh+iOLgT4;XR!fbnVDG=)h$4Q_?uAToZVdce%E?#eL5_+7iB@VM#~ 1, to szereg jest rozbieżny +Jeżeli granica z sqrt(an, n) = 1, to kryterium nie rozstrzyga zbieżności szeregu + +Dodatkowo, jeśli od pewnego momentu >= 1, to jest to szereg rozbieżny. + +Kryterium Cauchy'ego często wykorzystuje się podczas badania zbieżności szeregów, we wzorach których występują potęgi n-tego stopnia. + +Czyli intuicyjnie: +- Gdy granica pierwiastka stopnia n z każdego kolejnego elementu ciągu jest mniejsza niż 1, to te pierwiastki muszą sumować się do 1, bo będziemy sumować coraz to mniejsze ułamki (to tak jakbyśmy w nieskończoność zamalowywali coraz mniejszy element kwadratu- gdybyśmy w nieskończoność zamalowywali coraz to mniejsze części, to w końcu zamalujemy go całego) +- Gdy ta granica jest większa od 1, to dodajemy pierwiastki coraz to większych liczby, więc ucieka nam to w nieskończoność. +- Gdy ta granica jest równa 1, to nic nam to nie mówi \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium d'Alemberta (ułamkowe).md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium d'Alemberta (ułamkowe).md new file mode 100644 index 0000000..ca503de --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium d'Alemberta (ułamkowe).md @@ -0,0 +1,13 @@ +Dla ciągu o wyrazach (an + 1) / an: +- Jeżeli granica jest mniejsza niż 1, to szereg jest zbieżny +- Jeżeli granica jest większa od 1, to szereg jest rozbieżny +- Jeżeli jest równa 1, to nic nam to nie mówi + +Dodatkowo jeśli od pewnego momentu granica jest >= 1, to szereg jest rozbieżny. + +Na ludzki: +Gdy mamy taki szereg, że każdy kolejny element ciągu to element ciągu zwiększony o 1 i podzielony przez ten element, to +- przy granicy mniejszej od 1 będziemy dodawali ułamki, więc zbiegamy do 1 +- przy granicy większej od 1 będziemy uciekali do nieskończoności +- przy granicy równej 1 nic nam to nie mówi + diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium porównawcze.md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium porównawcze.md new file mode 100644 index 0000000..6ec8194 --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Kryterium porównawcze.md @@ -0,0 +1,17 @@ +Porównujemy szereg, którego zbieżność znamy z innym, który badamy. + +Dla dwóch dodatnich szeregów: +A i B + +Jeżeli dla prawie wszystkich n zachodzi a, <= bn to: +- ze zbieżności szeregu B wynika zbieżność szeregu A +- z rozbnieśności szeregu A wynika rozbieżność szeregu B + +Czyli tłumacząc to na język bardziej ludzki. Gdy kolejne elementy ciągu są mniejsze, niż elementy drugiego ciągu, to: +- ze zbieżności większego wynika zbieżność mniejszego +- z rozbnieżności mniejszego wynika rozbieżność większego + +Jeśli mniejszy ucieka w nieskończoność, to większy tym bardziej. +Jeśli większy zbiega do konkretnej wartości, to mniejszy tym bardziej. + +[Zadania](https://www.matemaks.pl/kryterium-porownawcze.html) \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szereg harmoniczny.md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szereg harmoniczny.md new file mode 100644 index 0000000..12199f7 --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szereg harmoniczny.md @@ -0,0 +1,6 @@ +Tak wygląda szereg harmoniczny (to szereg ciągu 1/n\[Czyli innymi słowy: mamy ciąg 1/n, dodajemy wszystkie jego elementy(to jest szereg), wtedy jego granica(suma szeregu) wynosi nieskończoność(bo wszystkie jego wyrazy można uprościć do 1 + 1/2 + 1/2 + 1/2 ... + 1/2, jeśli zgrupujemy kolejne wyrazy w nawiasy, które zawierają w sobie 2^n elementów) więc nigdy nie przestaje on rosnąć.)]) +![[../../../../media/Pasted image 20251011092804.png]] + + +Dowód zbieżności (a raczej rozbieżności) szeregu harmonicznego. +![[../../../../media/Pasted image 20251011092828.png]] \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szeregi liczbowe.md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szeregi liczbowe.md new file mode 100644 index 0000000..bf30de8 --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Szeregi liczbowe.md @@ -0,0 +1,11 @@ +[Bardzo fajne źródło](https://www.matemaks.pl/szeregi-liczbowe.html) +Dla ciągu Xn określamy ciąg Sn wzorem Sn = (sigma)^n i=1 Xi +wtedy +S1 = X1 +S2 = X1 + X2 +S3 = X1 + X2 + X3 +itd. +[[Zbieżność]] +[[Granica-Suma szeregu]] +[[Szereg harmoniczny]] +[[Badanie zbieżności]] \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Zbieżność.md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Zbieżność.md new file mode 100644 index 0000000..353aa7d --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Zbieżność.md @@ -0,0 +1,5 @@ +Szereg jest zbieżny, gdy Sn sum częściowych jest zbieżny. +Zbieżność oznacza dążenie do konkretnej liczby. +Rozbieżność oznaczy dążenie nieokreślonej wartości (nieskończoność, minus nieskończoność\[to jest - szereg rośnie, rośnie i rośnie, nie wiemy kiedy przestanie rosnąść]) + +Jeżeli szereg jest zbieżny, to granica ciągu jest równa 0. \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Złożenie funkcji.md b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Złożenie funkcji.md new file mode 100644 index 0000000..8d6c45d --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/2_wykład/Złożenie funkcji.md @@ -0,0 +1,3 @@ +f o g(x) = f(g(x)) | Funkcja f jest złożona z funkcją g(x) tzn, że jako argumentu funkcji f możemy użyć g(x) + +**To nie jest przemienne** diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/Matematyka ze statystyką.md b/uczelnia/1_semestr/matematyka_ze_statystyką/Matematyka ze statystyką.md index 4d05e15..81a2620 100644 --- a/uczelnia/1_semestr/matematyka_ze_statystyką/Matematyka ze statystyką.md +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/Matematyka ze statystyką.md @@ -1,2 +1,3 @@ [[uczelnia/1_semestr/matematyka_ze_statystyką/1_wykład/1 wykład|1 wykład]] +[[2_wykład/2 wykład|2 wykład]] Na 7 i 14 spotkaniu: kolokwia \ No newline at end of file diff --git a/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/2 wykład.md b/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/2 wykład.md new file mode 100644 index 0000000..ab041e9 --- /dev/null +++ b/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/2 wykład.md @@ -0,0 +1 @@ +[[Zapis liczb rzeczywistych]] \ No newline at end of file diff --git a/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/Zapis liczb rzeczywistych.md b/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/Zapis liczb rzeczywistych.md new file mode 100644 index 0000000..294f31f --- /dev/null +++ b/uczelnia/1_semestr/wprowadzenie_do_informatyki/2 sykład/Zapis liczb rzeczywistych.md @@ -0,0 +1,6 @@ +Rejestr - elektroniczny element pamięciowy, w którym jestesmy w stanie przechowywać zera i jedynki. Jest zbudowany z przerzutników. + +Notacja wykładnicza liczby. + +float - mniejsza precyzja +double - większa precyzja \ No newline at end of file diff --git a/uczelnia/1_semestr/wprowadzenie_do_informatyki/Wprowadzenie do informatyki.md b/uczelnia/1_semestr/wprowadzenie_do_informatyki/Wprowadzenie do informatyki.md new file mode 100644 index 0000000..796d21e --- /dev/null +++ b/uczelnia/1_semestr/wprowadzenie_do_informatyki/Wprowadzenie do informatyki.md @@ -0,0 +1 @@ +[[2 sykład/2 wykład|2 wykład]] \ No newline at end of file