The Lagrange Highway

Recently, there was an interesting discussion on the AP Calculus Community discussion boards about the Lagrange error bound. You may link to it by clicking here. The replies by James L. Hartman and Daniel J. Teague were particularly enlightening and included files that you may download with the proof of Taylor’s Theorem (Hartman) and its geometric interpretation (Teague).

There are also two good Kahn Academy videos on Taylor’s theorem and the error bound on YouTube. The first part is here (11:26 minutes) and the second part is here (15:08 minutes).

I wrotean earlier blog post on the topic of error bounds on February 22, 2013 that you can find here.

Taylor’s Theorem says that

If f is a function with derivatives through order n + 1 on an interval I containing a, then, for each x in I , there exists a number c between x and a such that

\displaystyle f\left( x \right)=\sum\limits_{k=1}^{n}{\frac{{{f}^{\left( k \right)}}\left( a \right)}{k!}{{\left( x-a \right)}^{k}}}+\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}}

The number \displaystyle R=\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}} is called the remainder.

The equation above says that if you can find the correct c the function is exactly equal to Tn(x) + R.

Tn(x) is called the n th  Taylor Approximating Polynomial. (TAP). Notice the form of the remainder is the same as the other terms, except it is evaluated at the mysterious c that we don’t know and usually are not able to find without knowing the value we are trying to approximate.

Lagrange Error Bound. (LEB)

\displaystyle \left| \frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n-1}} \right|\le \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-a \right|}^{n+1}}}{\left( n+1 \right)!}

The number \displaystyle \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-c \right|}^{n+1}}}{\left( n+1 \right)!}\ge \left| R \right| is called the Lagrange Error Bound. The expression \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right) means the maximum absolute value of the (n + 1) derivative on the interval between the value of x and c.

The LEB is then a positive number greater than the error in using the TAP to approximate the function f(x). In symbols \left| {{T}_{n}}\left( x \right)-f\left( x \right) \right|<LEB.

Here is a little story that I hope will help your students understand what all this means.

Building A Road

Suppose you were tasked with building a road through the interval of convergence of a Taylor Series that the function could safely travel on. Here is how you could go about it.

Build the road so that the graph of the TAP is its center line. The edges of the road are built LEB units above and below the center line. (The width of the road is about twice the LEB.) Now when the function comes through the interval of convergence it will travel safely on the road. I will not necessarily go down the center, but will not go over the edges. It may wander back and forth over the center line, but will always stay on the road. Thus you know where the function is; it is less than LEB units (vertically) from the center line, the TAP.

Building a Wider Road

As shown in the example at the end of my previous post, it is often necessary to use a number larger than the minimum we could get away with for the LEB. This is because the maximum value of the derivative may be difficult to find. This amounts to building a road that is wider than necessary. The function will still remain within LEB units of the center line, but will not come as close to the edges of our wider road as it may on the original road.  As long as the width of the wider road is less than the accuracy we need, this will not be a problem: the TAP will give an accurate enough approximation of the function.

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