# Application: Using Taylor's Theorem to Approximate Functions.

The material for the remainder of this book was taken from Sean Mauch's Applied mathematics text13.9.

Theorem 13.15.1   Taylor's Theorem of the Mean. If is times continuously differentiable in then there exists a point such that

 (13.20)

For the case , the formula is

which is just a rearrangement of the terms in the theorem of the mean,

One can use Taylor's theorem to approximate functions with polynomials. Consider an infinitely differentiable function and a point . Substituting for into Equation 13.20 we obtain,

If the last term in the sum is small then we can approximate our function with an order polynomial.

The last term in Equation 13.15 is called the remainder or the error term,

Since the function is infinitely differentiable, exists and is bounded. Therefore we note that the error must vanish as because of the factor. We therefore suspect that our approximation would be a good one if is close to . Also note that eventually grows faster than ,

So if the derivative term, , does not grow to quickly, the error for a certain value of will get smaller with increasing and the polynomial will become a better approximation of the function. (It is also possible that the derivative factor grows very quickly and the approximation gets worse with increasing .)

Example 13.15.1   Consider the function . We want a polynomial approximation of this function near the point . Since the derivative of is , the value of all the derivatives at is . Taylor's theorem thus states that

for some . The first few polynomial approximations of the exponent about the point are

The four approximations are graphed in Figure 13.7.

Figure 13.7: Finite Taylor Series Approximations of , , to .

Note that for the range of we are looking at, the approximations become more accurate as the number of terms increases.

Here is one way to compute these approximations using Sage:

[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: x = var("x")
sage: y = exp(x)
sage: a = lambda n: diff(y,x,n)(0)/factorial(n)
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1/2
sage: a(3)
1/6
sage: taylor = lambda n: sum([a(i)*x^i for i in range(n)])
sage: taylor(2)
x + 1
sage: taylor(3)
x^2/2 + x + 1
sage: taylor(4)
x^3/6 + x^2/2 + x + 1


Example 13.15.2   Consider the function . We want a polynomial approximation of this function near the point . The first few derivatives of are

It's easy to pick out the pattern here,

Since and the -term approximation of the cosine is,

Here are graphs of the one, two, three and four term approximations.

Figure 13.8: Taylor Series Approximations of , , to .

Note that for the range of we are looking at, the approximations become more accurate as the number of terms increases. Consider the ten term approximation of the cosine about ,

Note that for any value of , . Therefore the absolute value of the error term satisfies,

Note that the error is very small for , fairly small but non-negligible for and large for . The ten term approximation of the cosine, plotted below, behaves just we would predict.

Figure 13.9: Taylor Series Approximation of to .

The error is very small until it becomes non-negligible at and large at .

Example 13.15.3   Consider the function . We want a polynomial approximation of this function near the point . The first few derivatives of are

The derivatives evaluated at are

for

By Taylor's theorem of the mean we have,

Below are plots of the 1, 2, and 3 term approximations.

Figure 13.10: Taylor series (about ) approximations of , , to .

Note that the approximation gets better on the interval and worse outside this interval as the number of terms increases. The Taylor series converges to only on this interval.

david joyner 2008-11-22