Question 1: (a) Determine the root of the following equation f(x) = 0.011- sin(x) (3+0.1x2)3 =0 using bisection method with two initial guesses of a=1 and b=4. Perform the computation until the percentage relative error is less than 1%. x is in unit of radians. (10 marks) (b) The minimum of a two-variable function f(x,y) can be determined by using an iterative approach described as follows: (i) Given the initial coordinates (xo, yo), calculate the values of first derivatives u and v: u = of ( x, y ) V = of ( x, y) Ix=x0 .V=Yo dy x=xo.=yo (ii) Construct the single-variable function g(h) from f(x,y) by substituting x =X, tu . h, y = yo tv.h ; (iii) Determine the minimum of g(h) at h*, which is g(h*) and occurs with the coordinates (x*, y"): x = X, tu . h , y = yo + v. h"; and the minimum of f(x,y) is then evaluated to be f(x , y"). (iv) Let (x*, y") be the new initial coordinates in (i) and repeat (ii)-(iv) until the minimum of f(x,y) is obtained. In step (iii), the minimum of g(h) is determined by solving the equation g'(h)=0 with the Newton- Raphson's method; The initial guess is ho=0 and the prescribed error is 0.01%. Given (xo, yo)=(0, 2), find the minimum of f(x,y)=2+2x+2x2+2xyty by using the iterative method described above. The prescribed error is 3%. (20 marks)