2518.sagews

n=var('n')
a_n=(1+1/n)**n
a_n.limit(n=oo)

e

limit(sin(n*pi/3),n=oo)

ind

n=var('n')
limit((n**2-1)/(n+5),n=oo)

+Infinity

x=var('x')
diff(sin(x),x)

cos(x)

diff(sin(x),x,2)

-sin(x)

x,y=var('x,y')
f=x^2*y+x*exp(y)
f.diff(x,y)

2*x + e^y

var('x,a')
f=exp(sin(a-x^2))/x
f.derivative(x)

-2*e^(sin(-x^2 + a))*cos(-x^2 + a) - e^(sin(-x^2 + a))/x^2

x=var('x')
f=sin(x)
taylor(f,x,0,6)

1/120*x^5 - 1/6*x^3 + x

x,y=var('x,y')
f=sin(x*y)
taylor(f,(x,0),(y,-1),4)

-1/2*(y + 1)*x^3 + 1/6*x^3 + (y + 1)*x - x

x=var('x')
integral(x*sin(x^2),x)

-1/2*cos(x^2)

integral(x^2,x,0,1)

1/3

x=var('x')
integral(x**(-2),1,infinity)

1

var('x,k,w')

(x, k, w)

f=x^3*exp(k*x)*sin(w*x)
f.integral(x)

-(((k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 - 24*k^3*w + 24*k*w^3 - 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 + 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*e^(k*x)*cos(w*x) - ((k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 - 6*k^4 + 36*k^2*w^2 - 6*w^4 - 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 + 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)

m=var('m')
sum(1/(m*(m+1)),m,1,oo)

1

sum(1/(m*(m+1)),m,1,10)

10/11

sum(1/(m*(m+1)),m,1,100)

100/101

t=var('t')
x=function('x',t)
a,b=var('a,b')
DE=a*diff(x,t)+b*x-1
desolve(DE,[x,t])

(c + e^(b*t/a)/b)*e^(-b*t/a)

plot(sin(x),(x,0,2*pi),color='red')

p=plot(sin(x),color=hue(1.0))
for a in range(2,8+1):
   p+=plot(sin(a*x),color=hue(1.0/a))

p.show(xmin=-1,xmax=1,ymin=-1,ymax=1)

var('t')
parametric_plot((1-2*sin(t),t^2),(t,-4,4))

var('x,y')
implicit_plot(x^2/4-y^2/9-1,(x,-5,5),(y,-6,6)).show(aspect_ratio=1)

var('a')
polar_plot(a/5,(a,0,6*pi),color='green').show(aspect_ratio=1)

var('x,y')
plot3d(x^2/6-y^2/9,(x,-10,10),(y,-10,10))