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b/uczelnia/1_semestr/matematyka_ze_statystyką/Matematyka ze statystyką.md @@ -1 +1,2 @@ -[Szeregi](Szeregi.md) \ No newline at end of file +[Szeregi](Szeregi.md) +[[Pierwsze kolokwium]] \ No newline at end of file diff --git a/uczelnia/1_semestr/matematyka_ze_statystyką/Pierwsze kolokwium.md b/uczelnia/1_semestr/matematyka_ze_statystyką/Pierwsze kolokwium.md new file mode 100644 index 0000000..e394a62 --- /dev/null +++ b/uczelnia/1_semestr/matematyka_ze_statystyką/Pierwsze kolokwium.md @@ -0,0 +1,60 @@ +Spis treści + +- [Obliczanie granicy ciągów](#Obliczanie%20granicy%20ciągów) +- [Zbieżność szeregów](#Zbie%C5%BCno%C5%9B%C4%87%20szereg%C3%B3w) +- [Ciągłość funkcji](#Ci%C4%85g%C5%82o%C5%9B%C4%87%20funkcji) +- [Pochodna z definicji](#Pochodna%20z%20definicji) +- [Pochodna funkcji](#Pochodna%20funkcji) +- [Granica funkcji lewo-prawo stronna](#Granica%20funkcji%20lewo-prawo%20stronna) +- [Przebieg zmienności funkcji](#Przebieg%20zmienno%C5%9Bci%20funkcji) +# Obliczanie granicy ciągów +![](../../../media/Pasted%20image%2020251119193520.png) +Na kolokwium będą takie same typy ciągów, ale inne liczby oczywiście + +1. Pierwiastek z funkcji +- coś: + - Takie coś mnożymy przez "sprzężenie"(zamieniamy znak pomiędzy wyrażeniami)/ sprzężenie + - Wtedy wychodzi wynik + - $\lim_{n \to \infty} \sqrt{9n^2 + 2n -5} -3n = \lim_{n \to \infty} \sqrt{9n^2 + 2n -5} -3n \times \frac{\sqrt{9n^2 + 2n -5} - 3n}{\sqrt{9n^2 + 2n -5} - 3n}$ + - $\lim_{n \to \infty} \left( \sqrt{9n^2 + 2n - 5} - 3n \right) = \lim_{n \to \infty} \frac{\left( \sqrt{9n^2 + 2n - 5} - 3n \right)\left( \sqrt{9n^2 + 2n - 5} + 3n \right)}{\sqrt{9n^2 + 2n - 5} + 3n}.$ +2. Twierdzenie o trzech ciągach: + - Bierzemy największy element pod pierwiastkiem + - Porównujemy do ciągu mniejszego i większego, czyli w mniejszym dajemy tylko największy element, a w większym zamieniamy wszystko na największy element + - Granica to powinien wyjść właśnie ten największy element + - Tak mówi chatgpt, ale to podobno *NiE jESt iStniEjąCa zalEŻnoŚć*, więc trzeba liczyć z tego +3. Operowanie na potęgach: + - Rozbijamy te potęgi + - Wyciągamy potęgę przed nawias + - Skracamy +# Zbieżność szeregów +[Szeregi](Szeregi.md) +![](../../../media/Pasted%20image%2020251119200743.png) +Jeden przykład na kryt. leibniza, drugie na d'alemberta/cauchiego. + + +# Ciągłość funkcji +![](../../../media/Pasted%20image%2020251119193335.png) +Dość proste zadanie (zwane również: łatwe punkty) +- Bierzemy równanie, gdzie funkcja nie przyjmuje wartości. +- Liczymy w tym miejscu granicę +- Te granicę przyrównujemy do drugiego gównania +- Mamy a +- włala + +# Pochodna z definicji +Tego typu +![](../../../media/Pasted%20image%2020251119193425.png) +[pochodne matemaks](https://www.matemaks.pl/pochodne.html) +## Śmieszna definicja +![](../../../media/Pasted%20image%2020251118195420.png) +$h=(x_0+h)−x_0$ +## Jak to liczyć? +Prosta sprawa, podstawiamy do wzoru. Liczymy granicę. Powinno wyjść... + +# Pochodna funkcji +Tu się przeważnie liczy pochodne funkcji złożonych, więc ze wzorów się da ogarnąć. + +# Granica funkcji lewo-prawo stronna +To musze poćwiczyć dla tych niektórych dziwnych funkcji… + +# Przebieg zmienności funkcji +Duuuuże zadanie. \ No newline at end of file