Mga sunud-sunod na calculator:

Ang calculator na ito ay naglulutas ng \(F\left(x,\,y,\,y',\,y'',\dots,y^{\left(n\right)}\right)=0\) — mga ordinary differential equation (ODE) ng iba't ibang order, kabilang ang:

Mga separable na equation: \(p\left(x\right)\mathrm{d}x=q\left(y\right)\mathrm{d}y\)

Mga homogeneous na equation: \(y'=f\left(k\,x,\;k\,y\right)=f\left(x,\;y\right)\)

Mga first-order linear na equation: \(y'+a\left(x\right)\,y=b\left(x\right)\)

Mga equation na may anyo: \(y'=f\left(\frac{a_1\,x+b_1\,y+c_1}{a\,x+b\,y+c}\right)\)

Mga Bernoulli differential equation: \(y'+a\left(x\right)\,y=b\left(x\right)\,y^n\)

Mga Riccati equation: \(y'+a\left(x\right)\,y+b\left(x\right)\,y^2=c\left(x\right)\)

Mga exact differential equation: \(P\left(x,\;y\right)\,\mathrm{d}x+Q\left(x,\;y\right)\,\mathrm{d}y=0\)

Mga non-exact differential equation: \(\mu\cdot P\left(x,\;y\right)\,\mathrm{d}x+\mu\cdot Q\left(x,\;y\right)\,\mathrm{d}y=0\) — kung saan ang \(\mu\) ay isang integrating factor

Mga total differential equation: \(\mathrm{d}\left(F\left(x,\,y\right)\right)=0\)

Mga equation na hindi nalutas para sa derivative: \(F\left(x,\;y,\;y'\right)=0\)

Mga equation na may anyo: \(F\left(x,\,y^{\left(k\right)},\,y^{\left(k+1\right)},\dots,y^{\left(n\right)}\right)=0\) at \(F\left(y,\,y',\,y''\,\dots,y^{\left(n\right)}\right)=0\)

Mga linear differential equation na may constant coefficients: \(y^{\left(n\right)}+a_{n-1}\,y^{\left(n-1\right)}+\ldots+a_0\,y=f\left(x\right)\)

Mga Cauchy-Euler equation: \(x^n\,y^{\left(n\right)}+a_{n-1}\,x^{n-1}\,y^{\left(n-1\right)}+\ldots+a_{1}\,x\,y'+a_0\,y=0\)

Ang calculator ay naglulutas din ng mga sistema ng ordinary differential equations:

Mga linear homogeneous system na may constant coefficients: \(X'\left(t\right)=A\,X\left(t\right)\)

Mga linear nonhomogeneous system na may constant coefficients: \(X'\left(t\right)=A\,X\left(t\right)+f\left(t\right)\)

Naglulutas din ito ng mga equation at sistema na may initial conditions (initial value problems)

Nilulutas ng calculator na ito ang \(\displaystyle \int{f\left(x\right)\;\mathrm{d}x=F\left(x\right)+C}\) — mga indefinite integral nang hakbang-hakbang gamit ang mga sumusunod na pamamaraan at teknik:

Mga pangunahing formula ng integrasyon: \(\displaystyle\int{x^n}\;\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C,\;\left(n\neq-1\right)\), \(\displaystyle\int{a^x}\;\mathrm{d}x=\dfrac{a^x}{\ln\left(a\right)}+C\)\(\dots\)

Panuntunan sa suma at pagkakaiba: \(\displaystyle\int{\left(u\pm v\pm w\right)}\;\mathrm{d}x=\int{u}\;\mathrm{d}x\pm\int{v}\;\mathrm{d}x\pm\int{w}\;\mathrm{d}x\)

Panuntunan sa constant multiple: \(\displaystyle\int{c\,f\left(x\right)}\;\mathrm{d}x=c\int{f\left(x\right)}\;\mathrm{d}x\)

Panuntunan sa substitusyon (u-substitution): \(\displaystyle\int{f\left(x\right)}\;\mathrm{d}x=\left[\begin{array}{c}x=\varphi\left(t\right)\\\mathrm{d}x=\varphi'\left(t\right)\,\mathrm{d}t\end{array}\right]=\int{f\left(\varphi\left(t\right)\right)\,\varphi'\left(t\right)}\;\mathrm{d}t\)

Integrasyon ng mga rational function: trigonometric \(\mathrm{R}\left(\sin\left(x\right),\;\cos\left(x\right)\right)\); hyperbolic \(\mathrm{R}\left(\sinh\left(x\right),\;\cosh\left(x\right)\right)\); partial fractions \(\dfrac{P_k\left(x\right)}{Q_n\left(x\right)}\)

Paraan ng undetermined coefficients: polynomial factorization, linear-fractional irrationalities \(\mathrm{R}\left(x,\,\left(\dfrac{a\,x+b}{c\,x+d}\right)^{r_1,\dots,\,r_n}\right)\), paraan ng Ostrogradsky–Hermite \(\displaystyle\int{\dfrac{P\left(x\right)}{Q\left(x\right)}}=\dfrac{P_2\left(x\right)}{Q_2\left(x\right)}+\int{\dfrac{P_1\left(x\right)}{Q_1\left(x\right)}}\), mga integral na may square roots ng quadratics \(\mathrm{R}\left(x, \sqrt{a\,x^2+b\,x+c}\right)\), direktang pamamaraan \(\displaystyle\int{\dfrac{P_n\left(x\right)}{\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\), \(\displaystyle\int{\dfrac{P_m\left(x\right)}{\left(x-\alpha\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\), \(\displaystyle\int{\dfrac{M\,x+N}{\left(x^2+p\,x+q\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\)

Integrasyon sa pamamagitan ng parts \(\displaystyle\int{u}{\;\mathrm{d}v}=u\,v-\int{v}{\;\mathrm{d}u}\), trigonometric at hyperbolic substitutions, Euler substitutions, mga integral ng binomial differentials \(\displaystyle\int{x^m\,\left(a\,x^n+b\right)^p}{\;\mathrm{d}x}\)

Mga produkto ng mga potensya ng \(\sin^n\left(x\right)\,\cos^m\left(x\right)\) at hyperbolic functions \(\sinh^n\left(x\right)\,\cosh^m\left(x\right)\)

Mga karaniwang formula ng integrasyon, integrasyon na may absolute values, special functions \(\Gamma\left(s,\,x\right)\), \(\operatorname{Ei}\left(x\right)\), \(\operatorname{li}\left(x\right)\), \(\operatorname{Si}\left(x\right)\), \(\operatorname{Ci}\left(x\right)\), \(\operatorname{Shi}\left(x\right)\), \(\operatorname{Chi}\left(x\right)\), \(\operatorname{Li_2}\left(x\right)\), \(\operatorname{S}\left(x\right)\), \(\operatorname{C}\left(x\right)\), \(\operatorname{erf}\left(x\right)\), \(\operatorname{erfi}\left(x\right)\), reverse chain rule \(\displaystyle\int{\mathrm{d}\left(\mathrm{F}\left(x\right)\right)}\), Weierstrass substitution (tangent half-angle), Euler's formula \(e^{i\,x}=\cos(x)+i\,\sin(x)\)

Mga exponential, logarithmic, trigonometric, at hyperbolic transformations

Mga algebraic substitutions at regrouping na may simplification

Nilulutas ng calculator na ito ang \(\displaystyle\int\limits_{a}^{b}{f\left(x\right)}{\;\mathrm{d}x}\) — mga definite integral sa pamamagitan ng pagkalkula ng antiderivative at paglalapat ng Fundamental Theorem of Calculus, gamit ang symmetry properties para sa even o odd functions sa mga symmetric intervals, at periodicity properties

Para sa improper integrals, kinakalkula ng calculator ang mga limit sa infinity at one-sided limits sa mga punto ng discontinuity sa loob ng integration interval

Mga sinusuportahang mathematical functions:

\(\ln\) \(\sin\) \(\cos\) \(\tan\) \(\cot\) \(\arctan\) \(\arcsin\) \(\arccos\) \(\operatorname{arccot}\) \(\sinh\) \(\cosh\) \(\tanh\) \(\coth\) \(\operatorname{sech}\) \(\operatorname{csch}\) \(\operatorname{arsinh}\) \(\operatorname{arcosh}\) \(\operatorname{artanh}\) \(\operatorname{arcoth}\) \(\operatorname{arcsec}\) \(\operatorname{arccsc}\) \(\operatorname{arsech}\) \(\operatorname{arcsch}\) \(\sec\) \(\csc\) \(\left|f\right|\)

Nilulutas ng calculator ang mga ekwasyon sa anyong \(f\left(x\right)=0\), kabilang ang:

Pagtukoy ng domain ng function \(\mathrm{dom}\left(f\right)\)

Mga linear na ekwasyon \(a\,x+b=0\)

Mga quadratic na ekwasyon na may real at complex na coefficients \(a\,x^2+b\,x+c=0\)

Mga cubic na ekwasyon sa anyong \(a\,x^3+b\,x^2+b\,x+a=0\)

Mga cubic na ekwasyon \(a\,x^3+b\,x^2+c\,x+d=0\)

Mga quartic na ekwasyon sa anyong \(a\,x^4+b\,x^3+c\,x^2\pm b\,x+a=0\) at \(a\,x^4+b\,x^3+c\,x^2+d\,x+\dfrac{a\,d^2}{b^2}=0\)

Mga produkto ng apat na termino sa arithmetic progression \(\left(a\,x+b\right)\,\left(a\,x+b+c\right)\,\left(a\,x+b+2\,c\right)\,\left(a\,x+b+3\,c\right)=d\)

Iba't ibang exponential, logarithmic, trigonometric, hyperbolic, at inverse na ekwasyon

Paggamit ng pamamaraan ni Ferrari upang malutas ang mga quartic na ekwasyon \(a\,x^4+b\,x^3+c\,x^2+d\,x+e=0\)

Paghahanap ng mga rational na ugat \(x=\dfrac{m}{n}\) at pag-factor \(f_1\left(x\right)\cdots f_n\left(x\right)=0\)

Mga kilalang solusyon ng mga pangunahing trigonometric, hyperbolic, at inverse na ekwasyon

Paghahanap ng mga ugat ng mga complex na numero \(\sqrt[n]{a+i\,b}\)

Half-angle tangent substitution \(\sin(x)=\dfrac{2\,t}{1+t^2}\) at \(\cos(x)=\dfrac{1-t^2}{1+t^2}\) kung saan \(t=\tan\left(\dfrac{x}{2}\right)\)

Ang binomial theorem \((a+b)^n=a^n+C^1_n\,a^{n-1}\,b+\ldots+C^{n-1}_n\,a\,b^{n-1}+b^n\)

Mga polynomial identity para sa mga suma at pagkakaiba \(x^n+y^n\), \(x^n-y^n\)

Pagsasama ng magkakatulad na termino at pag-factor ng common factors \(x^2+x\;\Rightarrow\; x\,(x+1)\)

Cross-multiplication ng mga fraction \(\dfrac{a}{b}=\dfrac{c}{d}\;\Rightarrow\;a\,d=b\,c\) at completing the square \((a+b)^2+c\)

Pag-exponentiate ng magkabilang panig upang alisin ang mga natural logarithm

Mga complex logarithm \(\ln\left(a+i\,b\right)\) at Euler's formula \(e^{i\,x}=\cos\left(x\right)+i\,\sin\left(x\right)\)

Mga pangunahing functional equation \(f\left(g\left(x\right)\right) = f\left(r\left(x\right)\right)\;\Rightarrow\;g\left(x\right)=r\left(x\right)\)

Kinakalkula ng calculator na ito ang derivative ng isang function na \(f\left(x\right)\) o \(f\left(x,\,y,\,y',\dots,\,z,\,z',\dots\right)\) at ipinapakita ang mga panuntunang ginamit upang kalkulahin ang derivative.

Ang mga sumusunod na panuntunan ay tinukoy:

Mga karaniwang derivative ng \(x\), \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), \(\cot(x)\), \(e^x\), \(a^x\), \(\ln(x)\)\(\,\ldots\)

Panuntunan sa konstante: \((c)'=0\)

Panuntunan sa maramihang konstante: \(\left(c\,f(x)\right)'=c\,f'(x)\)

Panuntunan sa kabuuan: \(\left(f(x)+g(x)\right)'=f'(x)+g'(x)\)

Panuntunan sa pagkakaiba: \(\left(f(x)-g(x)\right)'=f'(x)-g'(x)\)

Panuntunan sa potensya: \(\left(x^n\right)'=n\,x^{n-1}\)

Panuntunan sa produkto: \(\left(f(x)\,g(x)\right)'=f(x)\,g'(x)+g(x)\,f'(x)\)

Panuntunan sa quotient: \(\left(\dfrac{f(x)}{g(x)}\right)'=\dfrac{g(x)\,f'(x)-f(x)\,g'(x)}{\left(g(x)\right)^2}\)

Panuntunan sa reciprocal: \(\left(\dfrac{1}{f(x)}\right)'=\dfrac{-f'(x)}{\left(f(x)\right)^2}\)

Panuntunan sa kadena: \(\left(f\left(g(x)\right)\right)'=f'_g\left(g\right)\,g'(x)\)

Absolute value: \(\left(\left|x\right|\right)'=\dfrac{x}{\left|x\right|}\)

Sign function: \(\left(\operatorname{sgn}\left(f\right)\right)'=2\,\delta\left(x\right)\), kung saan ang \(\delta\) ay ang Dirac delta function

Hinahanap ng calculator na ito ang limit ng isang function \(\displaystyle\lim_{x\to{a}}{f\left(x\right)}\) gamit ang mga sumusunod na katangian:

Limit ng isang constant \(\displaystyle\lim_{x\to{a}}C=C\)

Panuntunan ng constant multiple \(\displaystyle\lim_{x\to{a}}k\,f(x)=k\,\lim_{x\to{a}}f(x)\)

Panuntunan ng suma at pagkakaiba \(\displaystyle\lim_{x\to{a}}{f\left(x\right)\pm g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\pm\lim_{x\to{a}}{g\left(x\right)}\)

Panuntunan ng produkto \(\displaystyle\lim_{x\to{a}}{f\left(x\right)\,g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\,\lim_{x\to{a}}{g\left(x\right)}\)

Panuntunan ng quotient \(\displaystyle\lim_{x\to{a}}\dfrac{f(x)}{g(x)}=\dfrac{\displaystyle\lim_{x\to{a}}f(x)}{\displaystyle\lim_{x\to{a}}g(x)}\), kung \(\displaystyle\lim_{x\to{a}}g(x)\neq 0\)

Limit ng isang exponential function \(\displaystyle\lim_{x\to{a}}{a^{f\left(x\right)}}=a^{\displaystyle\lim_{x\to{a}}{f\left(x\right)}}\)

Mga karaniwang limit \(\displaystyle\lim_{x\to{0}}{\dfrac{\sin\left(x\right)}{x}}=1\) at \(\displaystyle\lim_{x\to{\infty}}{(1+\dfrac{1}{x})^x}=e\)

Squeeze theorem: kung \(g\left(x\right)\leq f\left(x\right)\leq h\left(x\right)\) at \(\displaystyle\lim_{x\to{a}}g(x)=\lim_{x\to{a}}h(x)=L\;\Rightarrow\;\lim_{x\to{a}}f(x)=L\)

Panuntunan ni L'Hôpital: kung \(\displaystyle\lim_{x\to{a}}f(x)=0\) at \(\displaystyle\lim_{x\to{a}}g(x)=0\) (o parehong limit ay katumbas ng \(\infty\)), kung gayon \(\displaystyle\lim_{x\to{a}}{\dfrac{f\left(x\right)}{g\left(x\right)}}=\lim_{x\to{a}}{\dfrac{f'\left(x\right)}{g'\left(x\right)}}\)

Taylor series \(f(x)=\sum\limits_{n=0}^{\infty}\dfrac{f^{\left(n\right)}\left(a\right)}{n!}\,\left(x-a\right)^n\)

Ginagamit ang pagpaparami sa conjugate, mga substitution, at formula ni Euler

Sinusuri ang parehong two-sided limits \(x\to{a}\) at one-sided limits \(x\to{a^+}\)

Ang calculator na ito ay nagko-convert ng complex na expression \(f(z)\) sa algebraic na anyo nito \(z=a+i\,b\), trigonometric na anyo \(z=r\cdot(\cos(\varphi)+i\,\sin(\varphi))\), at exponential na anyo \(z=r\,e^{i\,\varphi}\) gamit ang:

Modulus ng complex number: \(r=\left|a+i\,b\right|=\sqrt{a^2+b^2}\)

Ugat ng complex number: \(\sqrt[n]{z}=\sqrt[n]{r}\,\left(\cos\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)+i\,\sin\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)\right)\)

Kapangyarihan ng complex number: \(z^n=r^n\,\left(\cos\left(n\,\varphi\right)+i\,\sin\left(n\,\varphi\right)\right)\)

Pag-rationalize ng fraction sa pamamagitan ng conjugate nito: \(\dfrac{z}{a+i\,b}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{\left(a+i\,b\right)\cdot\left(a-i\,b\right)}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{a^2+b^2}\)

Complex logarithm: \(\operatorname{Log}\left(z\right)=\ln\left(r\right)+i\,(\varphi+2\,\pi\,\mathrm{k})\)

Principal value ng complex logarithm: \(\mathrm{Im}\operatorname{Log}\in(-\pi,\,\pi]\)

Mga trigonometric at hyperbolic na pagkakakilanlan tulad ng \(\sin\left(\alpha\pm\beta\right)=\sin\left(\alpha\right)\,\cos\left(\beta\right)\pm\cos\left(\alpha\right)\,\sin\left(\beta\right)\) o \(\sinh\left(i\,b\right)=i\,\sin\left(b\right)\), at Euler's formula \(e^{i\,\varphi}=\cos\left(\varphi\right)+i\,\sin\left(\varphi\right)\)

Ang calculator na ito ay nagsusuri ng mga ibinigay na matrix expression na may mga matrix na \(\mathrm{A}\), \(\mathrm{B}\), at \(\mathrm{C}\)

Kasama sa mga kakayahan nito ang mga operasyon ng matrix tulad ng: pagdaragdag \(\mathrm{A}+\mathrm{B}\), pagbabawas \(\mathrm{A}-\mathrm{B}\), pagpaparami \(\mathrm{C}\cdot\mathrm{B}\), determinant \(\left|\mathrm{A}\right|\), transpose \(\mathrm{B}^{\mathrm{T}}\), ranggo \(\operatorname{rank}\mathrm{C}\), kabaligtaran \(\mathrm{A}^{-1}\), pagpaparami sa scalar \(a\cdot\mathrm{B}\), o pagdaragdag ng scalar \(c+\mathrm{A}\)

Kinakalkula ang derivative ng mga elemento ng matrix \(\left(\mathrm{C}\right)'_x={\scriptsize\left(\begin{gathered}\left(\mathrm{a_{11}}\right)'_x&\left(\mathrm{a_{12}}\right)'_x\\\left(\mathrm{a_{21}}\right)'_x&\left(\mathrm{a_{22}}\right)'_x\end{gathered}\right)}\) o ang integral ng mga elemento ng matrix \(\int{\mathrm{A}}{\;\mathrm{d}x}={\scriptsize\left(\begin{gathered}\int{\mathrm{a_{11}}}{\;\mathrm{d}x}&\int{\mathrm{a_{12}}}{\;\mathrm{d}x}\\\int{\mathrm{a_{21}}}{\;\mathrm{d}x}&\int{\mathrm{a_{22}}}{\;\mathrm{d}x}\end{gathered}\right)}\)

Inilalapat ang mga mathematical function na \(\sin\), \(\cos\)\(\,\ldots\) sa bawat elemento ng matrix, halimbawa \(\ln\left(\mathrm{A}\right)={\scriptsize\left(\begin{gathered}\ln\left(\mathrm{a_{11}}\right)&\ln\left(\mathrm{a_{12}}\right)\\\ln\left(\mathrm{a_{21}}\right)&\ln\left(\mathrm{a_{22}}\right)\end{gathered}\right)}\)

Nagsusuri ng parehong mga numeric na halaga at mga kombinasyon ng mga operasyong aritmetika at mga function