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- Two trigonometric identities are given by: (sin x + cos x)2 = 1+ 2 sin x cos x 1- 2 cos x - 3cos2 x = 1- 3cos x sin2 x 1- cos x For each part, verify that the identity is correct by calculating the values of the left and right sides of the equation using MATLAT, substituting x=200 Find the solution to the systems + = ... .. . y + 3 y + 2 y y u Where y (0) =0; y’ (0) =0. u (t) is a square wave.(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.3- The variation of specific volume (ν) internal energy (u) enthalpy (h) and entropy (s) of superheated water with temperature at P=0.01 MPa pressure are presented in the table. By using this data, rebuild the given table for ν, u, h and s between T=50 °C and T=500 °C with ΔT=10 °C steps by using Newton’s divided difference interpolating polynomials. Present your results in a new table.
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