Page 96 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 4 NUMBER SERIES 73

where Fk(x) is the kth derivative of the function at the point x, and < is the
remainder or error term. As has been illustrated by examples we have seen
earlier, the magnitude of ( decreases as k (the number of terms) increases.
To obtain a result that closely approximates the true value of a function, we
need to sum a number of terms. Clearly, we will not have available to us
(without a lot of work) values of a large number of derivatives of the function F,
up to the kth derivative. Fortunately, we will usually need only the first
derivative, the first and second derivatives, or the first, second and third
derivatives to obtain results of sufficient accuracy. We will use the Taylor series
expansion of a function in several of the subsequent chapters.
The order of the approximation is determined by the highest-derivative term
that is included in the approximation; thus the first-order Taylor series
approximation is
F(x + h) = F(x) + hF'(x) (4-4)
the second-order approximation is
h2
F(x + h) = F(x) + hF'(x) + -F"(x) (4-5)
2
and the third-order approximation is
h2
h3
F(x + h) x F(x) + hF'(x) + - F"(x) + -F"'(x) (4-6)
2 6
Obviously, the accuracy of the approximation increases as the number of
terms is increased. It is also obvious that the accuracy of the approximation will
increase as h is made smaller. Higher-order terms will become more important
as h is increased, or if the function is nonlinear.

The Taylor Series: An Example
The following example will illustrate the use of the Taylor series to evaluate
a function. Consider the polynomial ax3 + bx2 + cx + d, with a = 1.25, b = 9, c =
-5 and d = 11. At x = 1, F(x) = 16.25. We wish to evaluate the function at x =
1.6. (Since we are dealing with a known function, we could just evaluate it at x =
1.6, but here we use a known function for purposes of illustration. In subsequent
chapters Taylor series will be used to evaluate functions whose value is known at
a certain point but whose form is unknown.)
From simple calculus, F'(x) = 3ux2 + 2bx + c = 3.75~~ + 18x - 5, F"(x) = 6ax
+
+ 2b = 7.5~ 18 and F"'(x) = 6a = 7.5. At x = 1, F'(x) = 16.75, Ff'(x) = 25.5 and
F"'(x) = 7.5. Substituting these values, along with h = 0.6, into equations 4-4, 4-
5 and 4-6 yields the results shown in Figure 4-2. As expected, the third-order
approximation provides the highest accuracy.