2541.sagews

var('n')
limit((exp(n))/(n^2),n=oo)

+Infinity

var('n')
limit((n**2+1)/(2*n+1),n=oo)

+Infinity

var('n,t')
a=(sum(t,t,1,n+1))/n^3
a.limit(n=oo)

0

var('n')
a=ln(1/n)
a.limit(n=oo)

-Infinity

var('x')
f=e^x/x^2
f.limit(x=oo)

+Infinity

var('x')
limit((1+1/x)^x,x=oo)

e

var('x')
limit((x^2+1)/(2*x+1),x=oo)

+Infinity

var('x')
limit(ln(1/x),x=oo)

-Infinity

var('x,y,n')
y=(sin(n*x))^n
y.derivative(x)

n^2*sin(n*x)^(n - 1)*cos(n*x)

var('x,y')
y=x^x
y.derivative(x)

(log(x) + 1)*x^x

var('x,y')
diff(ln(x^x),x,2)

1/x

var('x,y')
diff(x*sin(x*y),x)

x*y*cos(x*y) + sin(x*y)

var('x,y')
diff(x*sin(x*y),x,y)

-x^2*y*sin(x*y) + 2*x*cos(x*y)

var('x,y')
diff(x*sin(x*y),x,1,y,2)

-x^3*y*cos(x*y) - 3*x^2*sin(x*y)

var('x')
integral(1/(4+5*cos(x)),x)

-1/3*log(sin(x)/(cos(x) + 1) - 3) + 1/3*log(sin(x)/(cos(x) + 1) + 3)

var('x')
integral(sin(x)/(1+sin(x)^2),x)

1/4*sqrt(2)*log(-(sqrt(2) - cos(x))/(sqrt(2) + cos(x)))

var('x')
integral(x*exp(x)/(1+x)^2,x)

e^x/(x + 1)

var('x')
integral(sin(ln(x)),x)

1/2*(sin(log(x)) - cos(log(x)))*x

var('x')
integral(x/cos(x)^2,x,(-pi)/4,pi/4)

0

var('x')
integral(x^2*ln(x-1),x,1,e+1)

1/2*e^2 + 2/9*e^3

var('n')
sum((2*n-1)/2^n,n,1,oo)

3

var('n')
sum(n^2/2^n,n,1,oo)

6

var('n')
sum((n^2+1)/(n^2-1),n,2,+infinity)

+Infinity

p=plot(m*x/(1+(m*x)^2),color=hue(1.0))
for a in range(1,8+1):
   p+=plot(m*x/(1+(m*x)^2),color=hue(1.0/m))
p.show(xmin=0,xmax=4)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_61.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cD1wbG90KG0qeC8oMSsobSp4KV4yKSxjb2xvcj1odWUoMS4wKSkKZm9yIGEgaW4gcmFuZ2UoMSw4KzEpOgogICBwKz1wbG90KG0qeC8oMSsobSp4KV4yKSxjb2xvcj1odWUoMS4wL20pKQpwLnNob3coeG1pbj0wLHhtYXg9NCk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpBPdZBp/___code___.py", line 3, in <module> p=plot(m*x/(_sage_const_1 +(m*x)**_sage_const_2 ),color=hue(_sage_const_1p0 )) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/plot/misc.py", line 283, in wrapper return func(*args, **kwds) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/plot/misc.py", line 138, in wrapper return func(*args, **options) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/plot/plot.py", line 2488, in plot G = funcs.plot(*args, **original_opts) File "expression.pyx", line 7189, in sage.symbolic.expression.Expression.plot (sage/symbolic/expression.cpp:28371) File "expression.pyx", line 7219, in sage.symbolic.expression.Expression._plot_fast_callable (sage/symbolic/expression.cpp:28688) File "expression.pyx", line 7069, in sage.symbolic.expression.Expression._fast_float_ (sage/symbolic/expression.cpp:27699) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 1200, in fast_float return FastFloatConverter(ex, *vars)() File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 213, in __call__ return self.arithmetic(div, div.operator()) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 1151, in arithmetic fops = map(self, operands) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 213, in __call__ return self.arithmetic(div, div.operator()) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 1151, in arithmetic fops = map(self, operands) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 208, in __call__ return self.symbol(ex) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/symbolic/expression_conversions.py", line 1118, in symbol raise ValueError, "free variable: %s" % repr(ex) ValueError: free variable: x

var('t')
parametric_plot((sin(3*t)*cos(t),sin(3*t)*sin(t)),(t,0,pi))

var('x,y')
implicit_plot(x^2/16+y^2/4-1,(x,-4,4),(y,-2,2)).show(aspect_ratio=1)

var('a')
polar_plot(a^2/sin(a^2),color='purple').show(aspect_ratio=1)