Answer to Question #155736 in Quantitative Methods for usman

Question #155736

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


1
Expert's answer
2021-01-17T17:42:54-0500

The difference table is


xyΔyΔ2yΔ3yΔ4y00.00000.1736100.17360.00520.16840.0052200.34200.01040.00040.15800.0048300.50000.01520.1428400.6428\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=10180π=π18h=\dfrac{10}{180}\pi=\dfrac{\pi}{18}

y(10)=1h(Δy012Δ2y0+13Δ3y014Δ4y0)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)

=18π(0.173612(0.0052)+13(0.0052)=\dfrac{18}{\pi}\bigg(0.1736-\dfrac{1}{2}(-0.0052)+\dfrac{1}{3}(-0.0052)

14(0.0004))0.9990-\dfrac{1}{4}(0.0004)\bigg)\approx0.9990

y(10)=0.9990y'(10)=0.9990



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