4426.sagews

var('x')

x

x0=0

f = sin(x)*e^(-x)

f(x=0.5)

0.290786288212692

p=point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))

pl=plot(f,-1,5,thickness=2)

show(pl)

@interact
def _(order=(1..12)):
  ft = f.taylor(x,x0,order)
  pt = plot(ft,-1, 5, color='green', thickness=2)
  html('$f(x)\;=\;%s$'%latex(f))
  html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
  show(p + pl + pt, ymin = -.5, ymax = 1)

order