Some nice plots:
sage: plot(sin(x), -2*pi, 2*pi, fill = 'axis') Graphics object consisting of 2 graphics primitives
Taylor approximation:
sage: f = sin(x) sage: g = f.taylor(x,0,3) sage: plot(g, -2*pi, 2*pi) Graphics object consisting of 1 graphics primitive
All the way to a full featured applet:
sage: x = var('x')
sage: @interact # not tested
....: def g(f=sin(x), c=0, n=(1..30),
....: xinterval=range_slider(-10, 10, 1, default=(-8,8), label="x-interval"),
....: yinterval=range_slider(-50, 50, 1, default=(-3,3), label="y-interval")):
....: x0 = c
....: degree = n
....: xmin,xmax = xinterval
....: ymin,ymax = yinterval
....: p = plot(f, xmin, xmax, thickness=4)
....: dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
....: ft = f.taylor(x,x0,degree)
....: pt = plot(ft, xmin, xmax, color='red', thickness=2, fill=f)
....: show(dot + p + pt, ymin=ymin, ymax=ymax, xmin=xmin, xmax=xmax)
....: html(r'$f(x)\;=\;%s$'%latex(f))
....: html(r'$P_{%s}(x)\;=\;%s+R_{%s}(x)$'%(degree,latex(ft),degree))