The equation f (x) = x2 − 2ex = 0 has a solution in the interval [-1,1].With p0 = −1 and p1 = 1 calculate p2 using the BisectionWith p2 from part (a) calculate p3 using Newton’s

Mathematical methods used to solve equations or evaluate integrals or solve differential equations can be classified broadly into two types. Analytical Methods Numerical Methods Analytical methods are those which by an analysis of the equation obtain a solution directly as a readymade formula in terms of say, the coefficients present in the equations. In numerical solution, instead of directly writing the answer in terms of some formula, we perform stepwise calculations using some algorithms or numerical procedures and arrive at the same results. The advantage of numerical methods is that usually, these procedures work on a much wider range of problems as compared to analytical solutions which work in a limited class of problems.The bisection method says that for a function f(x) and two numbers a and b if fa·fb<0, then, there exists at least one root of f(x) between x=a and x=b. The iteration formula for the bisection method is: xn = a+b2 The first two guesses to calculate the root of the given function f(x) using the bisection method are p0=-1 and p1=1. Therefore, the next value, p2, according to the bisection method will be: p2 = p0+p12= -1+12= 02= 0The formula to find the next iteration according to Newton's method is: xn+1 = xn-fxnf'xn We need to find, p3, for p2=0 and the given function f(x). Therefore, p3, according to Newton's method will be given by: p3 = p2-fp2f'p2= 0-f0f'0= -f0f'0 ···1 First, we need to find f(0). Substitute 0 for x in the given function f(x) and simplify to find f(0). fx = x2-2exf0 = 02-2e0= 0-2= -2 Now we need to find f'(0). So, first of all, differentiate the function f(x) with respect to x, substitute 0 for x, and simplify to find f'(0). fx = x2-2exf'x = 2x-2exf'0 = 20-2e0= 0-2= -2Substitute the obtained values of f(0) and f'(0) in equation 1 and simplify to obtain p3. p3 = -f0f'0= --2-2= -1 Answer: p2 = 0p3 = -1