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

 The values of $$\sin 31^\circ$$ and $$\sin 32^\circ$$ calculated in (G) are correct to only three decimal places. If greater accuracy than this is desired, we may use (I), which gives, for $$f\left( x \right) = \sin x$$,

These results are correct to four decimal places.

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


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Using formula (I) for interpolation by second differences, calculate the following functions:
 * (a) $$\cos 61^\circ$$, taking $$a = 60^\circ$$.
 * (c) $$\sin 85.1^\circ$$, taking $$a = 85^\circ$$.
 * (b) $$\tan 46^\circ$$, taking $$a = 45^\circ$$.
 * (d) $$\cot 70.3^\circ$$, taking $$a = 70^\circ$$.
 * }
 * }


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 * (a) $$\sin 11^\circ$$, taking $$a = 10^\circ$$.
 * (c) $$\cot 15.2^\circ$$, taking $$a = 15^\circ$$.
 * (b) $$\cos 86^\circ$$, taking $$a = 85^\circ$$.
 * (d) $$\tan 69^\circ$$, taking $$a = 70^\circ$$.
 * }
 * }

Draw the graphs of the functions $$x$$, $$x - \tfrac{x^3}{3!}$$, $$x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!}$$, respectively, and compare them with the graph of $$\sin x$$.

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

it is required to expand the function

in powers of $$h$$ and $$k$$. Consider the function