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Explorations of Lagrange's Remainder

The command LagrangeRem(f,n,a,x) returns the difference between f(x) and the (n-1) degree Taylor polynomial of f expanded about a.

**> ** |
**LagrangeRem := (f,n,a,x) -> evalf(eval(f(t) - convert(taylor(f(t),t=a,n),polynom),t=x),20);** |

**> ** |
**LagrangeRem(sin,10,0,x);** |

**> ** |
**LagrangeRem(sin,10,0,0.5);** |

**> ** |
**plot(****LagrangeRem(sin,10,0,x),x=-2..2);** |

The problem with investigating Lagrange's remainder theorem is that the Lagrange remainder is so extremely small near x=a. Rather than comparing this remainder with the nth derivative times (x-a)^n divided by n!, it makes more sense to divide the remainder by (x-a)^n, multiply it by n!, and then compare the resulting function to the nth derivative of f.

**> ** |
**ModifiedLagrangeRem := (f,n,a,x) -> LagrangeRem(f,n,a,x)*n!/(x-a)^n;** |

We now compare the plot of the constant function y = ModifiedLagrangeRem with the nth derivative of f over the interval [a,x]. Lagrange's Remainder theorem is simply the statement that the graphs of these two functions interesect.

**> ** |
**ModifiedLagrangeRem(cosh,10,0,0.5);** |

**> ** |
**plot([ModifiedLagrangeRem(cosh,10,0,0.5),eval(diff(cosh(x),x$10),x=t)],t=0..0.5);** |