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

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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 $f\left(x\right)=\sin x$ ,

(J)	$\sin x$	$=\sin a+cosa\left(x-a\right)-{\frac {\sin a}{2!}}\left(x-a\right)^{2}.$
Let	$a$	$=30^{\circ }=.5236$ radian.
Then	$\sin 31^{\circ }$	$=\sin 30^{\circ }+\cos 30^{\circ }\left(.01745\right)-{\frac {\sin 30^{\circ }}{2}}\left(0.1745\right)^{2}$
		$=.50000+.01511-.00008$
		$=.51503.$
	$\sin 32^{\circ }$	$=\sin 30^{\circ }+\cos 30^{\circ }\left(.03490\right)-{\frac {\sin 30^{\circ }}{2}}\left(.03490\right)^{2}$
		$=.50000+.03022-.00030$
		$=.52992.$

These results are correct to four decimal places.

EXAMPLES

1. Using formula (H) for interpolation by first 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 $a=70^{\circ }$ .

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

(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 $a=70^{\circ }$ .

3. 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 $\sin x$ .

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)	$f\left(x,y\right),$

it is required to expand the function

(B)	$f\left(x+h,y+k\right)$

in powers of $h$ and $k$ .

Consider the function

(C)	$f\left(x+ht,y+kt\right).$