SES Assignment 2

$x$	f[]	f[,]	f[,,]	f[,,,]
$0$	$0.75$
$1.5$	$0.65$	$F[x_1,x_2]$ = $\frac{F[x_2]-F[x_1]}{x_2-x_1}=\frac{0.65-0.75}{1.5-0}=-\frac{1}{15}$
$4.5$	$1.0$	$F[x_2,x_3]$ = $\frac{F[x_3]-F[x_2]}{x_3-x_2}=\frac{1.0-0.65}{4.5-1.5}=\frac{7}{60}$	$F[x_1,x_2,x_3]$ = $\frac{F[x_2,x_3]-F[x_1,x_2]}{x_3-x_1}=\frac{7/60-(-1/15)}{4.5-0}=\frac{11}{270}$

| $7.5$ | $0.8$ | $F[x_3,x_4]$ = $\frac{F[x_4]-F[x_3]}{x_4-x_3} = \frac{0.8 - 1.0}{7.5 - 4.5} = -\frac{1}{15}$ | $F[x_2,x_3,x_4]$ = $\frac{F[x_3,x_4]-F[x_2,x_3]}{x_4-x_2} =\frac{-1/15 - 7/60}{7.5 -1.5} = -\frac{11}{360}$ | $F[x_1,x_2,x_3,x_4]$ = $\frac{F[x_2,x_3,x_4]-F[x_1,x_2,x_3]}{x_4-x_1} = \frac{-11/360-11/270}{7.5 - 0} = -\frac{77}{8100}$

| 8.9 | $0.5$ |$F[x_4,x_5]$ = $\frac{F[x_5]-F[x_4]}{x_5-x_4}=\frac{0.5-0.8}{8.9-7.5}=-\frac{3}{14}$ | $F[x_3,x_4,x_5]$ = $\frac{F[x_3,x_4]-F[x_4,x_5]}{x_5-x_3}=\frac{(-3/14)-(-1/15)}{8.9-4.5}=-\frac{31}{924}$ | $F[x_2,x_3,x_4,x_5]$ = $\frac{F[x_3,x_4,x_5]-F[x_4,x_5,x_6]}{x_5-x_2}=\frac{-31/924-(-11/360)}{8.9-1.5}=-\frac{83}{205128}$ | | 10 | $0.3$ | $F[x_5,x_6]$ = $\frac{F[x_6]-F[x_5]}{x_6-x_5}=\frac{0.3-0.5}{10-8.9}=-\frac{2}{11}$ | $F[x_4,x_5,x_6]$ = $\frac{F[x_5,x_6]-F[x_4,x_5]}{x_6-x_4}=\frac{(-2/11)-(-3/14)}{10-7.5}=\frac{1}{77}$ | $F[x_3,x_4,x_5,x_6]$ = $\frac{F[x_4,x_5,x_6]-F[x_3,x_4,x_5]}{x_6-x_3}=\frac{1/77-(-31/924)}{10-4.5}=\frac{43}{5082}$ |

f[,,,,]	f[,,,,,]
$F[x_1,x_2,x_3,x_4,x_5]$ = $\frac{F[x_2,x_3,x_4,x_5]-F[x_1,x_2,x_3,x_4]}{x_5-x_1}=\frac{-83/205128-(-77/8100)}{8.9-0}=\frac{420071}{410768820}$
$F[x_2,x_3,x_4,x_5,x_6]$ = $\frac{F[x_3,x_4,x_5,x_6]-F[x_2,x_3,x_4,x_5]}{x_6-x_2}=\frac{43/5082-(-83/205128)}{10-1.5}=\frac{20005}{19179468}$	$F[x_1,x_2,x_3,x_4,x_5,x_6]$ = $\frac{F[x_2,x_3,x_4,x_5,x_6]-F[x_1,x_2,x_3,x_4,x_5]}{x_6-x_1}$ = $\frac{20005/19179468-(420071/410768820)}{10-0}=\frac{391687}{192034423350}$

$a_n$	$F[]$	Value	Approx Value	$x_n$
$a_0$	$F[x_1]$	$0.75$	$0.75$	0
$a_1$	$F[x_1,x_2]$	$-\frac{1}{15}$	$-0.066667$	1.5
$a_2$	$F[x_1,x_2,x_3]$	$\frac{11}{270}$	$0.040741$	4.5
$a_3$	$F[x_1,x_2,x_3,x_4]$	$-\frac{77}{8100}$	$-0.0095062$	7.5
$a_4$	$F[x_1,x_2,x_3,x_4,x_5]$	$\frac{420071}{410768820}$	$0.001023$	8.9
$a_5$	$F[x_1,x_2,x_3,x_4,x_5,x_6]$	$-\frac{27170723}{192034423350}$	$-0.000141$	10

$P_6(x)=0.75+x(-0.066667+(x-1.5)(0.040741+(x-4.5)(-0.0095062+(x-7.5)(0.001023+(x-8.9)(-0.000141+(x-10))))))$ $G_c=1.61$

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