f = 1/x
a = 1
fp = f.derivative()
fpp = fp.derivative()
fppp = fpp.derivative()
G = plot(f,(x,0.2,5),color='red',thickness='5') + point((a,f(x=a)),size=50,color='red')
p1= f(x=a) + fp(x=a)*(x-a)
G += plot(p1,(x,0.2,5),color='blue')
p2 = f(x=a) + fp(x=a)*(x-a) + fpp(x=a)*(x-a)^2/2
G += plot(p2,(x,0.2,4),color='green')
p3 = p2 + fppp(x=a)*(x-a)^3/6
G += plot(p3,(x,0.2,3),color='orange')
show(G)
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# what about piecewise?
f = 1/x
G = plot(f,(x,0.2,5),color='red')
x1 = 1
x2 = 2
x3 = 3
x4 = 5
p1 = (x-x2)/(x1-x2)*f(x=x1) + (x-x1)/(x2-x1)*f(x=x2)
G += plot(p1,(x,x1,x2),color='blue')
p2 = (x-x3)/(x2-x3)*f(x=x2) + (x-x2)/(x3-x2)*f(x=x3)
G += plot(p2,(x,x2,x3),color='blue')
p3 = (x-x4)/(x3-x4)*f(x=x3) + (x-x3)/(x4-x3)*f(x=x4)
G += plot(p3,(x,x3,x4),color='blue')
show(G)
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