Kalkulatory matematyczne krok po kroku

[britva]

\(\color{green}{\boxed{}}\)Całki - określone i nieokreślone

\(\color{blue}{\boxed{}}\)Równania różniczkowe zwyczajne

\(\color{red}{\boxed{}}\)Pochodne

\(\color{blown}{\boxed{}}\)Macierze (podświetlanie mnożników w algorytmach)

przykład:
https://mathdf.com/mat/pl/?expr=det(A)& ... 2!1~4~1%5D

Powiązanie:
\(\color{blue}{\rightarrow}\)https://mathdf.com/pl/

[britva]

B. P. Demidovich. Zbiór problemów i ćwiczeń z analizy matematycznej. Całki nieoznaczone
Reshebnik. Wygenerowane za pomocą kalkulatora:

\(\underline{\text{Dysk Google}}\)

[britva]

Przykłady:
a) \(\color{green}{\boxed{}}\)
1/(a12+b_13+c_n+tau_10+tau11+tau_z+sqrt(x)+x^(1/n)) \(\Rightarrow\int{\dfrac{1}{a_{12}+b_{13}+c_{n}+\tau_{10}+\tau_{11}+\tau_{z}+\sqrt{x}+x^{\frac{1}{n}}}}{\;\mathrm{d}\tau_{z}}\)
Po obliczeniach:
\(\int{\dfrac{1}{\sqrt[n]{x}+\sqrt{x}+\tau_{z}+\tau_{11}+\tau_{10}+c_{n}+b_{13}+a_{12}}}{\;\mathrm{d}\tau_{z}}=\ln\left(\tau_{z}+\tau_{11}+\tau_{10}+\sqrt[n]{x}+c_{n}+b_{13}+a_{12}+\sqrt{x}\right)+C\)

b) \(\color{red}{\boxed{}}\)
x^sinx \(\Rightarrow\left(x^{\sin\left(x\right)}\right)'_{x}\)
Po obliczeniach:

\(\left(x^{\sin\left(x\right)}\right)'_{x}=x^{\sin\left(x\right)-1}\,\sin\left(x\right)+x^{\sin\left(x\right)}\,\cos\left(x\right)\,\ln\left(x\right)\)

c) \(\color{blue}{\boxed{}}\)
x*y''=y'+x*(y'^2+x^2) \(\Rightarrow x\,y''=y'+x\,\left(\left(y'\right)^{2}+x^{2}\right)\)
Po obliczeniach:

\(x\,y''=x\,\left(y'^{2}+x^{2}\right)+y'\;\rightarrow\;y=\ln\left(\cos\left(\frac{x^{2}+C}{2}\right)\right)+C_{1}\)

d) \(\color{black}{\boxed{}}\)
detB*A^2
Po obliczeniach:

\(\left|\begin{matrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{matrix}\right|\cdot \left(\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}\right)^{2}=\left(\begin{matrix}a_{12}\,a_{21}\,b_{11}\,b_{22}+a_{11}^{2}\,b_{11}\,b_{22}-a_{12}\,a_{21}\,b_{12}\,b_{21}-a_{11}^{2}\,b_{12}\,b_{21}&a_{12}\,a_{22}\,b_{11}\,b_{22}+a_{11}\,a_{12}\,b_{11}\,b_{22}-a_{12}\,a_{22}\,b_{12}\,b_{21}-a_{11}\,a_{12}\,b_{12}\,b_{21}\\a_{21}\,a_{22}\,b_{11}\,b_{22}+a_{11}\,a_{21}\,b_{11}\,b_{22}-a_{21}\,a_{22}\,b_{12}\,b_{21}-a_{11}\,a_{21}\,b_{12}\,b_{21}&a_{22}^{2}\,b_{11}\,b_{22}+a_{12}\,a_{21}\,b_{11}\,b_{22}-a_{22}^{2}\,b_{12}\,b_{21}-a_{12}\,a_{21}\,b_{12}\,b_{21}\end{matrix}\right)\)

dodatkowo:
\(\boxed{{\small\text{\frac{x_{n}}{a_{b}}}}\Rightarrow\;\dfrac{x_{n}}{a_{b}}}\) pozwala ci pisać LaTeX

[britva]

Jak można napisać stopni i logarytm (do tej pory tylko w ten sposób):

2^(a/b)+2^(1/a)+sqrt(13)+3^(1/3)+2^(2/3)+3^(2/3)+4^(3/10):

\(\displaystyle\int{2^{\frac{a}{b}}+2^{\frac{1}{a}}+\sqrt{13}+3^{\frac{1}{3}}+2^{\frac{2}{3}}+3^{\frac{2}{3}}+4^{\frac{3}{10}}}{\;\mathrm{d}x}\)

Podczas obliczania:

\(\displaystyle\int{2^{\frac{a}{b}}+\sqrt[a]{2}+\sqrt{13}+\sqrt[3]{9}+\sqrt[3]{3}+\sqrt[3]{4}+\sqrt[5]{8}}{\;\mathrm{d}x}=2^{\frac{a}{b}}\,x+\sqrt[a]{2}\,x+\sqrt{13}\,x+\sqrt[3]{9}\,x+\sqrt[3]{3}\,x+\sqrt[3]{4}\,x+\sqrt[5]{8}\,x+C\)

\(\log\left(x\right)=\log_e\left(x\right)=\ln\left(x\right)\)

\(\log_a\left(x\right)=\dfrac{\log_e\left(x\right)}{\log_e\left(a\right)}=\dfrac{\ln\left(x\right)}{\ln\left(a\right)}\)

[britva]

Zaktualizowano sqrt = root, log, ln:
sqrt5(32) + sqrt123(9) + sqrt(3,x) + sqrt(a,3) + root(2,x) + sqrt(x) + log3(x) + log_4(x) + log(5,x) + log(x,x) + log(e,x):

\(\int{\sqrt[{5}]{32}+\sqrt[{123}]{9}+\sqrt[{3}]{x}+\sqrt[{a}]{3}+\sqrt{x}+\sqrt{x}+\log_{3}\left(x\right)+\log_{4}\left(x\right)+\log_{5}\left(x\right)+\log_{x}\left(x\right)+\ln\left(x\right)}{\;\mathrm{d}x}\)

\(\left(\mathrm{\begin{matrix}\sqrt{x}&\sqrt[{\sqrt{x}}]{x}\\\sqrt[{\log_{\sqrt{x}}\left(x\right)}]{x}\,x&\sqrt[{\log^{x}_{\sqrt{x}}\left(x\right)}]{x}\,x\end{matrix}}\right)^{-1}=\left(\begin{matrix}\dfrac{x^{\frac{1}{2^{x}}+1}}{x^{\frac{1}{2^{x}}+\frac{3}{2}}-x^{\frac{1}{\sqrt{x}}+\frac{3}{2}}}&-\dfrac{\sqrt[{\sqrt{x}}]{x}}{x^{\frac{1}{2^{x}}+\frac{3}{2}}-x^{\frac{1}{\sqrt{x}}+\frac{3}{2}}}\\-\dfrac{x^{\frac{3}{2}}}{x^{\frac{1}{2^{x}}+\frac{3}{2}}-x^{\frac{1}{\sqrt{x}}+\frac{3}{2}}}&\dfrac{\sqrt{x}}{x^{\frac{1}{2^{x}}+\frac{3}{2}}-x^{\frac{1}{\sqrt{x}}+\frac{3}{2}}}\end{matrix}\right)\)

[britva]

Zaktualizowano:

Gdyby \({\small\quad y=y\left(x\right),\;z=z\left(x\right)}\)
wtedy:

\(\left(y\,z\right)'_{x}=y\,z'+y'\,z\)

\(\left(y\,z'+y'\,z\right)'_{x}=y\,z''+2\,y'\,z'+y''\,z\)

\(\left(y\,z''+2\,y'\,z'+y''\,z\right)'_{x}=y\,z'''+3\,y'\,z''+3\,y''\,z'+y'''\,z\)

lub

\(\left(t\,\cos\left(y'\right)\right)'_{t}{\small\quad y=y\left(t\right)} \Rightarrow \left(t\,\cos\left(y'\right)\right)'_{t}=\cos\left(y'\right)-t\,y''\,\sin\left(y'\right)\)

https://mathdf.com/der/pl/?expr=yz&arg=x&funcs=y%2Cz

[britva]

Dodano kalkulator numeryczny:

https://mathdf.com/calc/pl/

Spoiler

Spoiler

Spoiler

[britva]

Dodano kalkulator granic funkcji:

\(\displaystyle\lim_{x\to{\infty}}{\frac{1+\sin\left(x\right)}{x}}\)

\(\displaystyle\lim_{x\to{\infty}}{\frac{\ln\left(1+3^{x}\right)}{\ln\left(1+2^{x}\right)}}\)

\(\displaystyle\lim_{x\to{\infty}}{\sin\left(\sqrt{x+1}\right)-\sin\left(\sqrt{x}\right)}\)

\(\displaystyle\lim_{x\to{0}}{\frac{1-\ln\left(x\right)}{x}}\)

\(x\to{a}\)
\(x\to{a-0}\) lub \(x\to{a+0}\)

https://mathdf.com/lim/pl/