\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

wxMaxima, predavanja

slajd 8, maxima, komandna linija

(%i1) %pi;
\[\tag{%o1} \ensuremath{\pi} \]
(%i2) float(%pi);
\[\tag{%o2} 3.141592653589793\]
(%i3) ev(%pi, numer);
\[\tag{%o3} 3.141592653589793\]
(%i4) %e;
\[\tag{%o4} \% e\]
(%i5) float(%e);
\[\tag{%o5} 2.718281828459045\]
(%i6) float(%i);
\[\tag{%o6} \% i\]
(%i7) %i^2;
\[\tag{%o7} -1\]
(%i8) sqrt(−1);
\[\tag{%o8} \% i\]
(%i9) sin(%pi/4);
\[\tag{%o9} \frac{1}{\sqrt{2}}\]

slajd 9, simboli, brojevi, razlomci

(%i10) 3/4;
\[\tag{%o10} \frac{3}{4}\]
(%i11) 3.0/4;
\[\tag{%o11} 0.75\]
(%i12) 3/4.;
\[\tag{%o12} \frac{3}{4}\]
(%i13) 3/4.0;
\[\tag{%o13} 0.75\]
(%i14) float(3/4);
\[\tag{%o14} 0.75\]
(%i15) ev(3/4, numer);
\[\tag{%o15} 0.75\]
(%i16) 1/2−1/3;
\[\tag{%o16} \frac{1}{6}\]
(%i17) float(%);
\[\tag{%o17} 0.1666666666666666\]
(%i18) sin(2);
\[\tag{%o18} \sin{(2)}\]
(%i19) sin(2.0);
\[\tag{%o19} 0.9092974268256817\]
(%i20) 63^3;
\[\tag{%o20} 250047\]
(%i21) 3^63;
\[\tag{%o21} 1144561273430837494885949696427\]
(%i22) 10!;
\[\tag{%o22} 3628800\]
(%i23) 100!;
\[\tag{%o23} 933262154439441526816992388562[98 digits]916864000000000000000000000000\]
(%i24) 1000!;
\[\tag{%o24} 402387260077093773543702433923[2508 digits]000000000000000000000000000000\]

slajd 10, problemi sa =, osnovna prepreka primeni

(%i25) solve(2·x−8 = 2, x);
\[\tag{%o25} [x=5]\]
(%i26) solve(x^2−5·x=−6, x);
\[\tag{%o26} [x=3,x=2]\]
(%i27) solve(x^2−2, x);
\[\tag{%o27} [x=-\sqrt{2},x=\sqrt{2}]\]
(%i28) a;
\[\tag{%o28} a\]
(%i29) a: 4;
\[\tag{a}4\]
(%i30) a;
\[\tag{%o30} 4\]
(%i31) a$
(%i32) a^3;
\[\tag{%o32} 64\]
(%i33) sqrt(a);
\[\tag{%o33} 2\]
(%i34) f(x):=x^2;
\[\tag{%o34} \operatorname{f}(x):={{x}^{2}}\]
(%i35) f(3);
\[\tag{%o35} 9\]
(%i36) f(a);
\[\tag{%o36} 16\]
(%i37) f(b);
\[\tag{%o37} {{b}^{2}}\]

slajd 11, =, % i solve

(%i38) %;
\[\tag{%o38} {{b}^{2}}\]
(%i39) %i245;
\[\tag{%o39} \mathit{\% i245}\]
(%i40) %o245;
\[\tag{%o40} \mathit{\% o245}\]
(%i41) 3·x+2=8;
\[\tag{%o41} 3 x+2=8\]
(%i42) solve(%, x);
\[\tag{%o42} [x=2]\]
(%i43) solve(y^3=27, y);
\[\tag{%o43} [y=\frac{{{3}^{\frac{3}{2}}} \% i-3}{2},y=-\frac{{{3}^{\frac{3}{2}}} \% i+3}{2},y=3]\]
(%i44) solve(f(t)=64, t);
\[\tag{%o44} [t=-8,t=8]\]

slajd 12, undefinisanje

(%i45) fundef(f);
\[\tag{%o45} \operatorname{f}(x):={{x}^{2}}\]
(%i46) remfunction(f);
\[\tag{%o46} [f]\]
(%i47) remfunction(all);
\[\tag{%o47} []\]
(%i48) values;
\[\tag{%o48} [a]\]
(%i49) remvalue(a);
\[\tag{%o49} [a]\]
(%i50) a;
\[\tag{%o50} a\]
(%i51) a: 2;
\[\tag{a}2\]
(%i52) b: 3;
\[\tag{b}3\]
(%i53) values;
\[\tag{%o53} [a,b]\]
(%i54) remvalue(all);
\[\tag{%o54} [a,b]\]
(%i55) values;
\[\tag{%o55} []\]

slajd 13, kill

(%i56) a: 4;
\[\tag{a}4\]
(%i57) b: 5;
\[\tag{b}5\]
(%i58) f(x):=x^2;
\[\tag{%o58} \operatorname{f}(x):={{x}^{2}}\]
(%i59) values;
\[\tag{%o59} [a,b]\]
(%i60) kill(b);
\[\tag{%o60} \mathit{done}\]
(%i61) values;
\[\tag{%o61} [a]\]
(%i62) fundef(f);
\[\tag{%o62} \operatorname{f}(x):={{x}^{2}}\]
(%i63) kill(f);
\[\tag{%o63} \mathit{done}\]
(%i64) b: 7;
\[\tag{b}7\]
(%i65) values;
\[\tag{%o65} [a,b]\]
(%i66) kill(all);
\[\tag{%o0} \mathit{done}\]
(%i1) values;
\[\tag{%o1} []\]

slajd 14, jos o funkcijama i ev

(%i2) f(x):=x^2;
\[\tag{%o2} \operatorname{f}(x):={{x}^{2}}\]
(%i3) a: x^2;
\[\tag{a}{{x}^{2}}\]
(%i4) f(y);
\[\tag{%o4} {{y}^{2}}\]
(%i5) ev(a, x=y);
\[\tag{%o5} {{y}^{2}}\]
(%i6) f(4);
\[\tag{%o6} 16\]
(%i7) ev(a, x=4);
\[\tag{%o7} 16\]

slajd 15, fpprec i bfloat

(%i8) fpprec;
\[\tag{%o8} 16\]
(%i9) float(%pi);
\[\tag{%o9} 3.141592653589793\]
(%i10) bfloat(%pi);
\[\tag{%o10} 3.141592653589793b0\]
(%i11) fpprec: 50;
\[\tag{fpprec}50\]
(%i12) bfloat(%pi);
\[\tag{%o12} 3.1415926535897932384626433832795028841971693993751b0\]
(%i13) float(%pi);
\[\tag{%o13} 3.141592653589793\]
(%i14) fpprec: 3;
\[\tag{fpprec}3\]
(%i15) bfloat(%pi);
\[\tag{%o15} 3.14b0\]
(%i16) float(%pi);
\[\tag{%o16} 3.141592653589793\]

slajd 16, fpprintprec

(%i17) fpprintprec;
\[\tag{%o17} 0\]
(%i18) float(%pi);
\[\tag{%o18} 3.141592653589793\]
(%i19) float(%e);
\[\tag{%o19} 2.718281828459045\]
(%i20) fpprintprec: 3;
\[\tag{fpprintprec}3\]
(%i21) float(%pi);
\[\tag{%o21} 3.14\]
(%i22) float(%e);
\[\tag{%o22} 2.71\]
(%i23) fpprintprec: 5;
\[\tag{fpprintprec}5\]
(%i24) float(%pi);
\[\tag{%o24} 3.1415\]
(%i25) float(%e);
\[\tag{%o25} 2.7182\]
(%i26) fpprintprec: 0;
\[\tag{fpprintprec}0\]
(%i27) float(%pi);
\[\tag{%o27} 3.141592653589793\]
(%i28) float(%e);
\[\tag{%o28} 2.718281828459045\]

slajd 17, expand i factor

(%i29) expand((x+1)^2);
\[\tag{%o29} {{x}^{2}}+2 x+1\]
(%i30) expand((x+1)·(x−1));
\[\tag{%o30} {{x}^{2}}-1\]
(%i31) expand((x−3)^7);
\[\tag{%o31} {{x}^{7}}-21 {{x}^{6}}+189 {{x}^{5}}-945 {{x}^{4}}+2835 {{x}^{3}}-5103 {{x}^{2}}+5103 x-2187\]
(%i32) factor(%);
\[\tag{%o32} {{\left( x-3\right) }^{7}}\]
(%i33) eq: expand((x−4)·(x−5)·(x−6));
\[\tag{eq}{{x}^{3}}-15 {{x}^{2}}+74 x-120\]
(%i34) solve(eq, x);
\[\tag{%o34} [x=4,x=5,x=6]\]
(%i35) factor(eq);
\[\tag{%o35} \left( x-6\right) \, \left( x-5\right) \, \left( x-4\right) \]
(%i36) factor(4·x^5−4·x^4−13·x^3+x^2−17·x+5);
\[\tag{%o36} \left( 2 x-5\right) \, \left( {{x}^{2}}+1\right) \, \left( 2 {{x}^{2}}+3 x-1\right) \]
(%i37) factor(1001);
\[\tag{%o37} 7 11 13\]
(%i38) factor(123412341234);
\[\tag{%o38} 2 3 7 13 37 617 9901\]
(%i39) factor(2048);
\[\tag{%o39} {{2}^{11}}\]

slajd 18, parcijalni razlomci, partfrac(expression, variable)

(%i40) ex: (s^3+4·s^2+6·s+4)/(s^3+3·s^2+3·s+1);
\[\tag{ex}\frac{{{s}^{3}}+4 {{s}^{2}}+6 s+4}{{{s}^{3}}+3 {{s}^{2}}+3 s+1}\]
(%i41) factor(ex);
\[\tag{%o41} \frac{\left( s+2\right) \, \left( {{s}^{2}}+2 s+2\right) }{{{\left( s+1\right) }^{3}}}\]
(%i42) partfrac(ex, s);
\[\tag{%o42} \frac{1}{s+1}+\frac{1}{{{\left( s+1\right) }^{2}}}+\frac{1}{{{\left( s+1\right) }^{3}}}+1\]
(%i43) expand(%);
\[\tag{%o43} \frac{1}{{{s}^{3}}+3 {{s}^{2}}+3 s+1}+\frac{1}{{{s}^{2}}+2 s+1}+\frac{1}{s+1}+1\]
(%i44) ratsimp(%);
\[\tag{%o44} \frac{{{s}^{3}}+4 {{s}^{2}}+6 s+4}{{{s}^{3}}+3 {{s}^{2}}+3 s+1}\]

slajd 19, ratsimp i fullratsimp

(%i45) kill(all);
\[\tag{%o0} \mathit{done}\]
(%i1) eq: sin(x/(x^2+x))=exp((log(x)+1)^2−log(x)^2);
\[\tag{eq}\sin{\left( \frac{x}{{{x}^{2}}+x}\right) }={{\% e}^{{{\left( \log{(x)}+1\right) }^{2}}-{{\log{(x)}}^{2}}}}\]
(%i2) ratsimp(eq);
\[\tag{%o2} \sin{\left( \frac{1}{x+1}\right) }=\% e {{x}^{2}}\]
(%i3) ((x−1)^(3/2)−(x+1)·sqrt(x−1))/sqrt((x−1)·(x+1));
\[\tag{%o3} \frac{{{\left( x-1\right) }^{\frac{3}{2}}}-\sqrt{x-1}\, \left( x+1\right) }{\sqrt{\left( x-1\right) \, \left( x+1\right) }}\]
(%i4) ratsimp(%);
\[\tag{%o4} -\frac{2 \sqrt{x-1}}{\sqrt{{{x}^{2}}-1}}\]
(%i5) expr: (x^(a/2)+1)^2·(x^(a/2)−1)^2/(x^a−1);
\[\tag{expr}\frac{{{\left( {{x}^{\frac{a}{2}}}-1\right) }^{2}}\, {{\left( {{x}^{\frac{a}{2}}}+1\right) }^{2}}}{{{x}^{a}}-1}\]
(%i6) ratsimp(%);
\[\tag{%o6} \frac{{{x}^{2 a}}-2 {{x}^{a}}+1}{{{x}^{a}}-1}\]
(%i7) fullratsimp(%);
\[\tag{%o7} {{x}^{a}}-1\]

slajd 20, trigonometrija

(%i8) cos(%pi/3);
\[\tag{%o8} \frac{1}{2}\]
(%i9) sin(%pi/3);
\[\tag{%o9} \frac{\sqrt{3}}{2}\]
(%i10) ev(sin(%pi/3), numer);
\[\tag{%o10} 0.8660254037844386\]
(%i11) float(sin(%pi/3));
\[\tag{%o11} 0.8660254037844386\]
(%i12) csc(45·%pi/180);
\[\tag{%o12} \sqrt{2}\]
(%i13) tan(%pi/4);
\[\tag{%o13} 1\]
(%i14) tan(%pi/8);
\[\tag{%o14} \tan{\left( \frac{\ensuremath{\pi} }{8}\right) }\]
(%i15) acos(1/2);
\[\tag{%o15} \frac{\ensuremath{\pi} }{3}\]
(%i16) 180/%pi·asin(sqrt(3)/2);
\[\tag{%o16} 60\]
(%i17) acsc(1);
\[\tag{%o17} \frac{\ensuremath{\pi} }{2}\]

slajd 21, trigonometrija, izrazi

(%i18) ex: sin(x)^2+cos(x)^2;
\[\tag{ex}{{\sin{(x)}}^{2}}+{{\cos{(x)}}^{2}}\]
(%i19) trigsimp(ex);
\[\tag{%o19} 1\]
(%i20) kill(all);
\[\tag{%o0} \mathit{done}\]
(%i1) ex: sin(a+b);
\[\tag{ex}\sin{\left( b+a\right) }\]
(%i2) trigexpand(ex);
\[\tag{%o2} \cos{(a)} \sin{(b)}+\sin{(a)} \cos{(b)}\]
(%i3) trigrat(%);
\[\tag{%o3} \sin{\left( b+a\right) }\]
(%i4) ex: sin(x)^2;
\[\tag{ex}{{\sin{(x)}}^{2}}\]
(%i5) trigsimp(ex);
\[\tag{%o5} {{\sin{(x)}}^{2}}\]
(%i6) trigreduce(ex);
\[\tag{%o6} \frac{1-\cos{\left( 2 x\right) }}{2}\]
(%i7) trigrat(ex);
\[\tag{%o7} -\frac{\cos{\left( 2 x\right) }-1}{2}\]

slajd 22, linearni sistemi jednacina 1

(%i8) 3·x+2·y=7;
\[\tag{%o8} 2 y+3 x=7\]
(%i9) lhs(%);
\[\tag{%o9} 2 y+3 x\]
(%i10) rhs(%);
\[\tag{%o10} 0\]
(%i11) rhs(%o253);
\[\tag{%o11} 0\]
(%i12) e1: 3·x+2·y=7;
\[\tag{e1}2 y+3 x=7\]
(%i13) lhs(e1);
\[\tag{%o13} 2 y+3 x\]
(%i14) rhs(e1);
\[\tag{%o14} 7\]
(%i15) e2: 5·x−y=3;
\[\tag{e2}5 x-y=3\]
(%i16) linsolve([e1,e2], [x,y]);
\[\tag{%o16} [x=1,y=2]\]
(%i17) r: %;
\[\tag{r}[x=1,y=2]\]
(%i18) r[1];
\[\tag{%o18} x=1\]
(%i19) r[2];
\[\tag{%o19} y=2\]
(%i20) rhs(r[1]);
\[\tag{%o20} 1\]
(%i21) rhs(r[2]);
\[\tag{%o21} 2\]

slajd 23, linearni sistemi jednacina 2

(%i22) e1;
\[\tag{%o22} 2 y+3 x=7\]
(%i23) e1: 10·x−2·y=6;
\[\tag{e1}10 x-2 y=6\]
(%i24) e2;
\[\tag{%o24} 5 x-y=3\]
(%i25) linsolve([e1,e2], [x,y]);
\[\mbox{}\\\mbox{solve: dependent equations eliminated: (2)}\] \[\tag{%o25} [x=\frac{\mathit{\% r1}+3}{5},y=\mathit{\% r1}]\]
(%i26) e1: 10·x−2·y=5;
\[\tag{e1}10 x-2 y=5\]
(%i27) linsolve([e1,e2], [x,y]);
\[\tag{%o27} []\]

slajd 24, eliminacija

(%i28) remvalue(all);
\[\tag{%o28} [\mathit{ex},\mathit{bestlength},\mathit{trylength},\mathit{e1},\mathit{e2},r]\]
(%i29) a1: x+y+2·t=7;
\[\tag{a1}y+x+2 t=7\]
(%i30) a2: x−y−t=2;
\[\tag{a2}-y+x-t=2\]
(%i31) eliminate([a1,a2], [t]);
\[\tag{%o31} [-y+3 x-11]\]

slajd 25, matrice

(%i32) A: matrix([1,2],[2,1]);
\[\tag{A}\begin{pmatrix}1 & 2\\ 2 & 1\end{pmatrix}\]
(%i33) B: invert(A);
\[\tag{B}\begin{pmatrix}-\frac{1}{3} & \frac{2}{3}\\ \frac{2}{3} & -\frac{1}{3}\end{pmatrix}\]
(%i34) A·B;
\[\tag{%o34} \begin{pmatrix}-\frac{1}{3} & \frac{4}{3}\\ \frac{4}{3} & -\frac{1}{3}\end{pmatrix}\]
(%i35) a·B;
\[\tag{%o35} \begin{pmatrix}-\frac{a}{3} & \frac{2 a}{3}\\ \frac{2 a}{3} & -\frac{a}{3}\end{pmatrix}\]
(%i36) A.B;
\[\tag{%o36} \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\]
(%i37) determinant(A);
\[\tag{%o37} -3\]
(%i38) determinant(B);
\[\tag{%o38} -\frac{1}{3}\]
(%i39) b: matrix([3],[3]);
\[\tag{b}\begin{pmatrix}3\\ 3\end{pmatrix}\]
(%i40) x: B.b;
\[\tag{x}\begin{pmatrix}1\\ 1\end{pmatrix}\]
(%i41) A.x;
\[\tag{%o41} \begin{pmatrix}3\\ 3\end{pmatrix}\]
(%i42) A.x−b;
\[\tag{%o42} \begin{pmatrix}0\\ 0\end{pmatrix}\]
(%i43) kill(all);
\[\tag{%o0} \mathit{done}\]

slajd 26, Kronecker-Capelli

(%i1) A: matrix([5,−1, 3],[10,−2,6]);
\[\tag{A}\begin{pmatrix}5 & -1 & 3\\ 10 & -2 & 6\end{pmatrix}\]
(%i2) echelon(A);
\[\tag{%o2} \begin{pmatrix}1 & -\frac{1}{5} & \frac{3}{5}\\ 0 & 0 & 0\end{pmatrix}\]
(%i3) A[2][3]: 5;
\[\tag{%o3} 5\]
(%i4) A;
\[\tag{%o4} \begin{pmatrix}5 & -1 & 3\\ 10 & -2 & 5\end{pmatrix}\]
(%i5) echelon(A);
\[\tag{%o5} \begin{pmatrix}1 & -\frac{1}{5} & \frac{3}{5}\\ 0 & 0 & 1\end{pmatrix}\]

slajd 27, nelinearni sistemi

(%i6) remvalue(all);
\[\tag{%o6} [A]\]
(%i7) a1: x^2+y^2=41;
\[\tag{a1}{{y}^{2}}+{{x}^{2}}=41\]
(%i8) a2: y=x+1;
\[\tag{a2}y=x+1\]
(%i9) algsys([a1,a2],[x,y]);
\[\tag{%o9} [[x=4,y=5],[x=-5,y=-4]]\]
(%i10) t: solve(a1, y);
\[\tag{t}[y=-\sqrt{41-{{x}^{2}}},y=\sqrt{41-{{x}^{2}}}]\]
(%i11) y1: rhs(t[1]);
\[\tag{y1}-\sqrt{41-{{x}^{2}}}\]
(%i12) y2: rhs(t[2]);
\[\tag{y2}\sqrt{41-{{x}^{2}}}\]
(%i13) solve(a2, y);
\[\tag{%o13} [y=x+1]\]
(%i14) y3: rhs(solve(a2, y)[1]);
\[\tag{y3}x+1\]

slajd 28, plotovanje

(%i15) plot2d([y1,y2,y3],[x,−10,10],[y,−15,15]);
\[\mbox{}\\\mbox{plot2d: expression evaluates to non-numeric value somewhere in plotting range.}\mbox{}\\\mbox{plot2d: expression evaluates to non-numeric value somewhere in plotting range.}\] \[\tag{%o15} [/tmp/maxout11052.gnuplot\_ pipes]\]
(%i16) wxplot2d([y1,y2,y3],[x,−10,10],[y,−15,15]);
\[\mbox{}\\\mbox{plot2d: expression evaluates to non-numeric value somewhere in plotting range.}\mbox{}\\\mbox{plot2d: expression evaluates to non-numeric value somewhere in plotting range.}\] \[\tag{%t16} \] (Graphics)
\[\tag{%o16} \]

slajd 29, limesi

(%i17) limit((1+1/x)^(2·x), x, inf);
\[\tag{%o17} {{\% e}^{2}}\]
(%i18) float(%);
\[\tag{%o18} 7.38905609893065\]
(%i19) (x−2)/(x^2−4);
\[\tag{%o19} \frac{x-2}{{{x}^{2}}-4}\]
(%i20) limit(%, x, 2);
\[\tag{%o20} \frac{1}{4}\]
(%i21) f(x):=atan(x);
\[\tag{%o21} \operatorname{f}(x):=\operatorname{atan}(x)\]
(%i22) limit(f(x), x, inf);
\[\tag{%o22} \frac{\ensuremath{\pi} }{2}\]
(%i23) limit(f(x), x, minf);
\[\tag{%o23} -\frac{\ensuremath{\pi} }{2}\]
(%i24) limit(x^3, x, inf);
\[\tag{%o24} \infty \]
(%i25) limit(x^3, x, minf);
\[\tag{%o25} -\infty \]
(%i26) limit(sin(3·x)/x, x, 0);
\[\tag{%o26} 3\]
(%i27) limit(sin(3·x)/x, x, inf);
\[\tag{%o27} 0\]
(%i28) limit(sin(3·x)/x, x, minf);
\[\tag{%o28} 0\]

slajd 30, kombinacije . . .

(%i29) f(x):=x^3·tan(x);
\[\tag{%o29} \operatorname{f}(x):={{x}^{3}} \tan{(x)}\]
(%i30) rd: (f(x+h)−f(x))/h;
\[\tag{rd}\frac{{{\left( x+h\right) }^{3}} \tan{\left( x+h\right) }-{{x}^{3}} \tan{(x)}}{h}\]
(%i31) limit(rd, h, 0);
\[\tag{%o31} 3 {{x}^{2}} \tan{(x)}+\frac{{{x}^{3}}}{{{\cos{(x)}}^{2}}}\]
(%i32) trigsimp(%);
\[\tag{%o32} \frac{3 {{x}^{2}} \cos{(x)} \sin{(x)}+{{x}^{3}}}{{{\cos{(x)}}^{2}}}\]
(%i33) trigrat(%);
\[\tag{%o33} \frac{3 {{x}^{2}} \sin{\left( 2 x\right) }+2 {{x}^{3}}}{\cos{\left( 2 x\right) }+1}\]

slajd 31, 0, 0- i 0+

(%i34) limit(1/x, x, 0);
\[\tag{%o34} \mathit{infinity}\]
(%i35) limit(1/x, x, 0, plus);
\[\tag{%o35} \infty \]
(%i36) limit(1/x, x, 0, minus);
\[\tag{%o36} -\infty \]

slajd 32, izvodi

(%i37) remvalue(all);
\[\tag{%o37} [\mathit{a1},\mathit{a2},t,\mathit{y1},\mathit{y2},\mathit{y3},\mathit{rd},\mathit{bestlength},\mathit{trylength}]\]
(%i38) diff(x^2, x);
\[\tag{%o38} 2 x\]
(%i39) diff(sin(x), x);
\[\tag{%o39} \cos{(x)}\]
(%i40) diff(sin(x), x, 2);
\[\tag{%o40} -\sin{(x)}\]
(%i41) diff(sin(x), x, 3);
\[\tag{%o41} -\cos{(x)}\]
(%i42) diff(sin(x), x, 4);
\[\tag{%o42} \sin{(x)}\]
(%i43) diff(sin(x·y), x);
\[\tag{%o43} y \cos{\left( x y\right) }\]
(%i44) diff(sin(w·t), t);
\[\tag{%o44} w \cos{\left( t w\right) }\]

slajd 33, razvoj u red

(%i45) taylor(sin(x), x, 0, 5);
\[\tag{%o45)/T} x-\frac{{{x}^{3}}}{6}+\frac{{{x}^{5}}}{120}+\mbox{...}\]
(%i46) taylor(cos(x), x, 0, 7);
\[\tag{%o46)/T} 1-\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{24}-\frac{{{x}^{6}}}{720}+\mbox{...}\]
(%i47) f(x):=%e^x−cos(x);
\[\tag{%o47} \operatorname{f}(x):={{\% e}^{x}}-\cos{(x)}\]
(%i48) taylor(f(x), x, 0, 7);
\[\tag{%o48)/T} x+{{x}^{2}}+\frac{{{x}^{3}}}{6}+\frac{{{x}^{5}}}{120}+\frac{{{x}^{6}}}{360}+\frac{{{x}^{7}}}{5040}+\mbox{...}\]

slajd 34, integrali

(%i49) integrate(x^2, x);
\[\tag{%o49} \frac{{{x}^{3}}}{3}\]
(%i50) integrate(sin(x), x);
\[\tag{%o50} -\cos{(x)}\]
(%i51) integrate(x^2, x, 1, 2);
\[\tag{%o51} \frac{7}{3}\]
(%i52) integrate(sin(x), x, 0, %pi);
\[\tag{%o52} 2\]
(%i53) integrate(1/(1+x^2), x, 0, 1);
\[\tag{%o53} \frac{\ensuremath{\pi} }{4}\]

slajd 35, operator '

(%i54) kill(all);
\[\tag{%o0} \mathit{done}\]
(%i1) a: 4;
\[\tag{a}4\]
(%i2) a;
\[\tag{%o2} 4\]
(%i3) 'a;
\[\tag{%o3} a\]
(%i4) 'diff(x^2, x);
\[\tag{%o4} \frac{d}{d x} {{x}^{2}}\]
(%i5) 'integrate(x^2, x);
\[\tag{%o5} \int {\left. {{x}^{2}}dx\right.}\]
(%i6) ev(%, integrate);
\[\tag{%o6} \frac{{{x}^{3}}}{3}\]
(%i7) 'integrate(x^2, x, 0, 1);
\[\tag{%o7} \int_{0}^{1}{\left. {{x}^{2}}dx\right.}\]
(%i8) ev(%, integrate);
\[\tag{%o8} \frac{1}{3}\]

slajd 36, uvod u diferencijalne jednacine

(%i9) eq1: 'diff(y, t, 2) + 4 · y = 0;
\[\tag{eq1}\frac{{{d}^{2}}}{d {{t}^{2}}} y+4 y=0\]
(%i10) ode2(eq1, y, t);
\[\tag{%o10} y=\mathit{\% k1} \sin{\left( 2 t\right) }+\mathit{\% k2} \cos{\left( 2 t\right) }\]
(%i11) eq2: 'diff(y, t, 2) − 4 · y = 0;
\[\tag{eq2}\frac{{{d}^{2}}}{d {{t}^{2}}} y-4 y=0\]
(%i12) ode2(eq2, y, t);
\[\tag{%o12} y=\mathit{\% k1}\, {{\% e}^{2 t}}+\mathit{\% k2}\, {{\% e}^{-2 t}}\]
(%i13) eq3: 'diff(y, t, 2) − 2 · 'diff(y, t) + y = 0;
\[\tag{eq3}\frac{{{d}^{2}}}{d {{t}^{2}}} y-2 \left( \frac{d}{d t} y\right) +y=0\]
(%i14) ode2(eq3, y, t);
\[\tag{%o14} y=\left( \mathit{\% k2} t+\mathit{\% k1}\right) \, {{\% e}^{t}}\]
(%i15) eq4: 'diff(y,t,2)+2·'diff(y,t)+4·y=8·sin(4·t);
\[\tag{eq4}\frac{{{d}^{2}}}{d {{t}^{2}}} y+2 \left( \frac{d}{d t} y\right) +4 y=8 \sin{\left( 4 t\right) }\]
(%i16) ode2(eq4, y, t);
\[\tag{%o16} y={{\% e}^{-t}}\, \left( \mathit{\% k1} \sin{\left( \sqrt{3} t\right) }+\mathit{\% k2} \cos{\left( \sqrt{3} t\right) }\right) -\frac{6 \sin{\left( 4 t\right) }+4 \cos{\left( 4 t\right) }}{13}\]

slajd 37, provera resenja, substitute

(%i17) eq: 'diff(y, t, 2) + y = 0;
\[\tag{eq}\frac{{{d}^{2}}}{d {{t}^{2}}} y+y=0\]
(%i18) s: ode2(eq, y, t);
\[\tag{s}y=\mathit{\% k1} \sin{(t)}+\mathit{\% k2} \cos{(t)}\]
(%i19) s: rhs(s);
\[\tag{s}\mathit{\% k1} \sin{(t)}+\mathit{\% k2} \cos{(t)}\]
(%i20) p: subst(s, y, eq);
\[\tag{p}\frac{{{d}^{2}}}{d {{t}^{2}}} \left( \mathit{\% k1} \sin{(t)}+\mathit{\% k2} \cos{(t)}\right) +\mathit{\% k1} \sin{(t)}+\mathit{\% k2} \cos{(t)}=0\]
(%i21) ev(p, diff);
\[\tag{%o21} 0=0\]
(%i22) is(%);
\[\tag{%o22} \mbox{true}\]

slajd 38, scripting

(%i23) batch("dj.mac");
\[\mbox{}\\\mbox{read and interpret file: \neq p/home/peja/Desktop/skloni/PSAE-sve/PSAE-sources-2018/11-2018/primeri-11-2017/wxMaxima/dj.mac}\mbox{}\\\mbox{(\% i24) eq1:'diff(y,t,2)+4\cdot y = 0}\] \[\tag{%o24} \frac{{{d}^{2}}}{d {{t}^{2}}} y+4 y=0\mbox{}\\\mbox{(\% i25) ode2(eq1,y,t)}\] \[\tag{%o25} y=\mathit{\% k1} \sin{\left( 2 t\right) }+\mathit{\% k2} \cos{\left( 2 t\right) }\mbox{}\\\mbox{(\% i26) eq2:'diff(y,t,2)-4\cdot y = 0}\] \[\tag{%o26} \frac{{{d}^{2}}}{d {{t}^{2}}} y-4 y=0\mbox{}\\\mbox{(\% i27) ode2(eq2,y,t)}\] \[\tag{%o27} y=\mathit{\% k1}\, {{\% e}^{2 t}}+\mathit{\% k2}\, {{\% e}^{-2 t}}\mbox{}\\\mbox{(\% i28) eq3:'diff(y,t,2)-2\cdot 'diff(y,t)+y = 0}\] \[\tag{%o28} \frac{{{d}^{2}}}{d {{t}^{2}}} y-2 \left( \frac{d}{d t} y\right) +y=0\mbox{}\\\mbox{(\% i29) ode2(eq3,y,t)}\] \[\tag{%o29} y=\left( \mathit{\% k2} t+\mathit{\% k1}\right) \, {{\% e}^{t}}\mbox{}\\\mbox{(\% i30) eq4:'diff(y,t,2)+2\cdot 'diff(y,t)+4\cdot y = 8\cdot sin(4\cdot t)}\] \[\tag{%o30} \frac{{{d}^{2}}}{d {{t}^{2}}} y+2 \left( \frac{d}{d t} y\right) +4 y=8 \sin{\left( 4 t\right) }\mbox{}\\\mbox{(\% i31) ode2(eq4,y,t)}\] \[\tag{%o31} y={{\% e}^{-t}}\, \left( \mathit{\% k1} \sin{\left( \sqrt{3} t\right) }+\mathit{\% k2} \cos{\left( \sqrt{3} t\right) }\right) -\frac{6 \sin{\left( 4 t\right) }+4 \cos{\left( 4 t\right) }}{13}\] \[\tag{%o31} dj.mac\]

slajd 39, Laplasova transformacija

(%i32) kill(all);
\[\tag{%o0} \mathit{done}\]
(%i1) laplace(1, t, s);
\[\tag{%o1} \frac{1}{s}\]
(%i2) laplace(sin(w·t), t, s);
\[\tag{%o2} \frac{w}{{{w}^{2}}+{{s}^{2}}}\]
(%i3) laplace(cos(w·t), t, s);
\[\tag{%o3} \frac{s}{{{w}^{2}}+{{s}^{2}}}\]
(%i4) laplace(exp(a·t), t, s);
\[\tag{%o4} \frac{1}{s-a}\]
(%i5) laplace(exp(a·t)·sin(w·t), t, s);
\[\tag{%o5} \frac{w}{{{w}^{2}}+{{s}^{2}}-2 a s+{{a}^{2}}}\]
(%i6) laplace(exp(a·t)·cos(w·t), t, s);
\[\tag{%o6} \frac{s-a}{{{w}^{2}}+{{s}^{2}}-2 a s+{{a}^{2}}}\]
(%i7) laplace(exp(t−T), t, s);
\[\tag{%o7} \frac{{{\% e}^{-T}}}{s-1}\]

slajd 40, inverzna Laplasova transformacija

(%i8) ilt(1/(s+2), s, t);
\[\tag{%o8} {{\% e}^{-2 t}}\]
(%i9) ilt(2/(s^2+4), s, t);
\[\tag{%o9} \sin{\left( 2 t\right) }\]
(%i10) ilt((s^2+3·s+3)/(s^3+3·s^2+3·s+1), s, t);
\[\tag{%o10} \frac{{{t}^{2}}\, {{\% e}^{-t}}}{2}+t\, {{\% e}^{-t}}+{{\% e}^{-t}}\]

slajd 40, wxMaxima

(%i11) plot3d(x^2−y^2, [x,−1,1], [y,−1,1]);
\[\tag{%o11} [/tmp/maxout11052.gnuplot\_ pipes]\]
(%i12) wxplot3d(x^2−y^2, [x,−1,1], [y,−1,1]);
\[\tag{%t12} \] (Graphics)
\[\tag{%o12} \]
Created with wxMaxima.