Answer to Question #155736 in Quantitative Methods for usman

Question

find the numerical value of y'(10) for y = sinx given that:

sin 0 = 0.000 , sin 10 = 0.1736 , sin 20 = 0.3420 , sin 30 = 0.5000 , sin 40 =0.6428

Answer 1

The difference table is

$$\begin{matrix} x & y & \Delta y & \Delta^2y & \Delta^3y & \Delta^4y \\ 0 & 0.0000 & & & & \\ & &0.1736 & & & \\ 10 & 0.1736 & & -0.0052 & & \\ & &0.1684 & & -0.0052 & \\ 20 & 0.3420 & & -0.0104 & & 0.0004 \\ & & 0.1580 & & -0.0048 & \\ 30 & 0.5000 & & -0.0152 & & \\ & & 0.1428 & & & \\ 40 & 0.6428 & & & & \\ \end{matrix}$$

$$h=\dfrac{10}{180}\pi=\dfrac{\pi}{18}$$

$$y'(10)=\dfrac{1}{h}(\Delta y_0-\dfrac{1}{2}\Delta^2y_0+\dfrac{1}{3}\Delta^3y_0-\dfrac{1}{4}\Delta^4y_0)$$

$$=\dfrac{18}{\pi}\bigg(0.1736-\dfrac{1}{2}(-0.0052)+\dfrac{1}{3}(-0.0052)$$

$$-\dfrac{1}{4}(0.0004)\bigg)\approx0.9990$$

$y'(10)=0.9990$