Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/264

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The values of and calculated in (G) are correct to only three decimal places. If greater accuracy than this is desired, we may use (I), which gives, for ,


(J)
Let radian.
Then

These results are correct to four decimal places.

EXAMPLES

1. Using formula (H) for interpolation by first differences, calculate the following functions:

(a) , taking . (c) , taking .
(b) , taking . (d) , taking .

2. Using formula (I) for interpolation by second differences, calculate the following functions:

(a) , taking . (c) , taking .
(b) , taking . (d) , taking .

3. Draw the graphs of the functions , , , respectively, and compare them with the graph of .


148. Taylor's Theorem for functions of two or more variables. The scope of this book will allow only an elementary treatment of the expansion of functions involving more than one variable by Taylor's Theorem. The expressions for the remainder are complicated and will not be written down.

Having given the function

(A)

it is required to expand the function

(B)

in powers of and .

Consider the function

(C)