PLEASE HELP QUESTION 3 Derive the Maclaurin series expansion

PLEASE HELP QUESTION 3

Derive the Maclaurin series expansion for sin x. Then estimate the value of sin(pi/3) and include enough number of Maclaurin expansion terms so that the approximate error estimate falls below an error criterion conforming to three significant figures. Use zero- through third-order Taylor series expansions to predict f(2.5) for: f(x) = ln x using a base point at x = 1. Compute the true percent relative error epsilon_t for each approximation. Determine the maximum of following function by employing bisection method for [0, 1]. Calculate the approximate percent relative error and continue the iterations until this error is less than 5%: f(x) = -2x^6 - 1.5x^4 + 10x + 2 Use simple fixed-point iteration to find the roots of the following equation: f(x) = sin (Squareroot x) - x Use an initial guess of x_0 = 0.5 and iterate until epsilon_a

Solution

F(x)=-2x^6-1.5x^4+10x+2 a b F(a) F(b) c=a+b/2 F(c) Update, If F(a)*F(c)<0 then b=c or else a=c New(b-a) %Error, (c1-c0/c1)*100 0 1 2 8.5 0.5 6.875 a=c=0.5 0.5 0.5 1 6.875 8.5 0.75 8.669434 a=c=0.75 0.25 33.33333333 0.75 1 8.669434 8.5 0.875 8.973137 a=c=0.875 0.125 14.28571429 0.875 1 8.973137 8.5 0.9375 8.858417 a=c=0.9375 0.0625 6.666666667 0.9375 1 8.858417 8.5 0.96875 8.713288 a=c=0.96875 0.03125 3.225806452