{{{id=1| x /// x }}} {{{id=2| parent(x) /// Symbolic Ring }}} {{{id=3| x^2+4 /// x^2 + 4 }}} {{{id=4| var('y') /// y }}} {{{id=5| parent(y) /// Symbolic Ring }}} {{{id=6| var('u v') /// (u, v) }}} {{{id=7| parent(u) /// Symbolic Ring }}} {{{id=8| f=(x+y)^3 f /// (x + y)^3 }}} {{{id=9| g=expand(f) g /// x^3 + 3*x^2*y + 3*x*y^2 + y^3 }}} {{{id=10| f.expand() /// x^3 + 3*x^2*y + 3*x*y^2 + y^3 }}} {{{id=11| g /// x^3 + 3*x^2*y + 3*x*y^2 + y^3 }}} {{{id=12| factor(g) /// (x + y)^3 }}} {{{id=13| factor(442) /// 2 * 13 * 17 }}} {{{id=14| show(diff(exp(-2*x),x)) /// $\newcommand{\Bold}[1]{\mathbf{#1}}-2 \, e^{\left(-2 \, x\right)}$ }}}

Integral: $\int_0^{\pi}\sin(x) dx$

{{{id=15| integrate(sin(x),x,0,pi) /// 2 }}} {{{id=16| integral(sin(x),x,0,pi) /// 2 }}} {{{id=18| integral(sin(x),(x,0,pi)) /// 2 }}} {{{id=19| parent(v) /// Symbolic Ring }}} {{{id=20| reset('v') /// }}} {{{id=21| parent(v) /// Traceback (most recent call last): File "", line 1, in File "_sage_input_29.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGFyZW50KHYp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/var/folders/6d/rpqgvbc16j94tm0q81g0j_n40000gn/T/tmpjANSnR/___code___.py", line 2, in exec compile(u'parent(v)' + '\n', '', 'single') File "", line 1, in NameError: name 'v' is not defined }}} {{{id=22| f(x)=x^2+x /// }}} {{{id=23| print f show(f) /// x |--> x^2 + x $\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + x$ }}} {{{id=24| parent(f) /// Callable function ring with argument x }}} {{{id=25| F(n)=n*(n-1) show(F) /// $\newcommand{\Bold}[1]{\mathbf{#1}}n \ {\mapsto}\ {\left(n - 1\right)} n$ }}} {{{id=26| parent(n) /// Symbolic Ring }}} {{{id=27| show(f) /// $\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + x$ }}} {{{id=28| g(x)=x^4 /// }}} {{{id=29| show(f+g) /// $\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{4} + x^{2} + x$ }}} {{{id=30| show(expand(g(f))) /// $\newcommand{\Bold}[1]{\mathbf{#1}}x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}$ }}} {{{id=31| print f(7) print parent(f(7)) print factor(f(7)) print parent(56) print factor(Integer(f(7))) /// 56 Symbolic Ring 56 Integer Ring 2^3 * 7 }}} {{{id=32| diff(f) /// x |--> 2*x + 1 }}} {{{id=33| h(x,y,z)=sin(x*y+z) diff(h) /// (x, y, z) |--> (y*cos(x*y + z), x*cos(x*y + z), cos(x*y + z)) }}} {{{id=34| diff(h,y) /// (x, y, z) |--> x*cos(x*y + z) }}}

Integral $\int_1^\infty ae^{-x}dx$

{{{id=35| var('a') integral(a*exp(-x),x,1,oo) /// a*e^(-1) }}} {{{id=36| integral(exp(-a*x),x,1,oo) /// Traceback (most recent call last): File "", line 1, in File "_sage_input_54.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("aW50ZWdyYWwoZXhwKC1hKngpLHgsMSxvbyk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/var/folders/6d/rpqgvbc16j94tm0q81g0j_n40000gn/T/tmp6oFD32/___code___.py", line 3, in exec compile(u'integral(exp(-a*x),x,_sage_const_1 ,oo)' + '\n', '', 'single') File "", line 1, in File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/misc/functional.py", line 721, in integral return x.integral(*args, **kwds) File "sage/symbolic/expression.pyx", line 12275, in sage.symbolic.expression.Expression.integral (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/symbolic/expression.cpp:69945) File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 799, in integrate return definite_integral(expression, v, a, b, hold=hold) File "sage/symbolic/function.pyx", line 996, in sage.symbolic.function.BuiltinFunction.__call__ (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/symbolic/function.cpp:11397) File "sage/symbolic/function.pyx", line 486, in sage.symbolic.function.Function.__call__ (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/symbolic/function.cpp:6440) File "sage/symbolic/function.pyx", line 1085, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/symbolic/function.cpp:12757) File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 181, in _eval_ return integrator(*args) File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py", line 34, in maxima_integrator result = maxima.sr_integral(expression, v, a, b) File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py", line 813, in sr_integral self._missing_assumption(s) File "/Applications/SageMath-8.0.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py", line 1049, in _missing_assumption raise ValueError(outstr) ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a>0)', see `assume?` for more details) Is a positive, negative or zero? }}} {{{id=38| assume(a>0) integral(exp(-a*x),x,1,oo) /// e^(-a)/a }}}

$\int_1^{\pi/3} \frac{\tan(x)}{x}dx$

{{{id=39| h=integral(tan(x)/x,(x,1,pi/3)) show(h) /// $\newcommand{\Bold}[1]{\mathbf{#1}}\int_{1}^{\frac{1}{3} \, \pi} \frac{\tan\left(x\right)}{x}\,{d x}$ }}} {{{id=41| h.n() /// 0.07571599101702896 }}} {{{id=42| f /// x |--> x^2 + x }}} {{{id=43| p=plot(f,x,-3,5,color="orange") p /// }}} {{{id=44| parent(p) /// }}} {{{id=45| p.save("Funktionsplot.pdf") /// }}} {{{id=46| p_ableitung= plot(diff(f),x,-3,5,color="red") p_ableitung /// }}} {{{id=47| show(p+p_ableitung+point((3,5),color="green",size=50)) /// }}} {{{id=48| h(x,y)=exp(-x^2-y^2) plot3d(h,(x,-2,2),(y,-3,3)) /// }}} {{{id=49| h1(x,y)=sin(x+y) plot3d(h1,(x,-10,10),(y,-10,10)) /// }}} {{{id=50| h1(x,y)=x^2*y-x*y^3+2*x plot3d(h1,(x,-10,10),(y,-10,10)) /// }}} {{{id=51| f(x)=1/x limit(f,x=+oo) limit(f,x=0,dir="minus") /// x |--> -Infinity }}} {{{id=52| f1(x)=(x+3)/(x^2+5) limit(f1,x=-oo) /// x |--> 0 }}} {{{id=53| f2(x)=sin(1/x) limit(f2,x=0) /// x |--> ind }}} {{{id=54| plot(f2,x,0,1) /// }}} {{{id=55| f3(x)=sin(x)/x limit(f3,x=0,dir="plus") /// x |--> 1 }}} {{{id=56| plot(f3,x,0,10) /// }}}

Reihen: $\sum_{n\geq 1} a_n$

{{{id=57| a(n)=1/n^2 sum(a(n),n,1,oo) /// 1/6*pi^2 }}} {{{id=59| f(x)=x^2-2*x-3 solutions = solve(f(x)==0,x) print solutions /// [ x == 3, x == -1 ] }}} {{{id=60| solutions[0] /// x == 3 }}} {{{id=61| f.subs(solutions[0]) /// x |--> 0 }}} {{{id=62| f(solutions[0]) /// (x == 3)^2 - 2*(x == 3) - 3 }}} {{{id=63| solutionlist = solve(f(x)==0,x,solution_dict=True) print solutionlist /// [{x: 3}, {x: -1}] }}} {{{id=64| print solutionlist[0] print solutionlist[0][x] /// {x: 3} 3 }}} {{{id=65| N1=point((solutionlist[0][x],0),color="red",size=50) N2=point((solutionlist[1][x],0),color="red",size=50) plot(f,x,-2,4)+N1+N2 /// }}} {{{id=66| find_root(f,-2,0) /// -0.9999999999999986 }}} {{{id=67| f5(x)=x^3-5*x+5 plot(f5,x,-5,3) /// }}} {{{id=68| solve(f5(x)==0,x) /// [x == -1/2*(5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-5*I*sqrt(3) + 5)/(5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3), x == -1/2*(5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(5*I*sqrt(3) + 5)/(5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3), x == (5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3) + 5/3/(5/18*sqrt(7)*sqrt(3) - 5/2)^(1/3)] }}} {{{id=69| find_root(f5,-3,-2) /// -2.627365084711833 }}} {{{id=71| Px=QQ[x] /// }}} {{{id=72| parent(Px) /// }}} {{{id=73| var('z') QQ[z] /// Univariate Polynomial Ring in z over Rational Field }}} {{{id=74| z^3+z /// z^3 + z }}} {{{id=75| parent(z) /// Symbolic Ring }}} {{{id=76| x1=x /// }}} {{{id=77| x=Px(x) /// }}} {{{id=78| parent(x) /// Univariate Polynomial Ring in x over Rational Field }}} {{{id=79| parent(x1) /// Symbolic Ring }}} {{{id=80| P2.=QQ[] /// }}} {{{id=81| P2 /// Multivariate Polynomial Ring in u, v over Rational Field }}} {{{id=82| parent(u^2) /// Multivariate Polynomial Ring in u, v over Rational Field }}} {{{id=83| parent(x^2) /// Univariate Polynomial Ring in x over Rational Field }}} {{{id=84| u^2+x^2 /// Traceback (most recent call last): File "", line 1, in File "_sage_input_141.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("dV4yK3heMg=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/var/folders/6d/rpqgvbc16j94tm0q81g0j_n40000gn/T/tmp9Suhs_/___code___.py", line 3, in exec compile(u'u**_sage_const_2 +x**_sage_const_2 ' + '\n', '', 'single') File "", line 1, in File "sage/structure/element.pyx", line 1195, in sage.structure.element.Element.__add__ (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/structure/element.c:10675) File "sage/structure/coerce.pyx", line 1104, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (/Applications/SageMath-8.0.app/Contents/Resources/sage/src/build/cythonized/sage/structure/coerce.c:10448) TypeError: unsupported operand parent(s) for +: 'Multivariate Polynomial Ring in u, v over Rational Field' and 'Univariate Polynomial Ring in x over Rational Field' }}} {{{id=85| p=x^2-9 /// }}} {{{id=86| p.roots() /// [(3, 1), (-3, 1)] }}} {{{id=87| p1=x^2-3 /// }}} {{{id=88| p1.roots(ring=RR) /// [(-1.73205080756888, 1), (1.73205080756888, 1)] }}} {{{id=89| pe=SR(p1) /// }}} {{{id=90| parent(pe) /// Symbolic Ring }}} {{{id=91| var('x') solve(pe==0,x) /// [x == -sqrt(3), x == sqrt(3)] }}} {{{id=92| g(x,y)=x*y+3 h(x,y)=x-4*y solve((g(x,y)==0,h(x,y)==0),(x,y)) /// [[x == -2*I*sqrt(3), y == -1/2*I*sqrt(3)], [x == 2*I*sqrt(3), y == 1/2*I*sqrt(3)]] }}} {{{id=93| /// }}}