(a)(i) Given that at a = 20 + ph, Question 4 f(a) = f(xo+ph) = f(ro) +pAf(ro) + 2! f(x)+(+2) 2)(1). (+3) (2) prove that for n = 2, pronh (a)dx = f(x)da f(x)dx=(fo+4/1+2)- flix)(1). 115 90 (12 marks) 20 30 (3) (ii) Name the formula for numerical integration given in equation (3) and give its corre- sponding composite formula. (2 marks) (b) Use the Composite formula, for numerical integration given in (ii) above at n = 4. to evaluate 4.z C tan 3r dr, The calculator should be set in degrees, and the functional values should be rounded off to the nearest whole number. (6 marks) Question 5 a)(i) When required to obtain the root to f(x) = 0, on the interval [a, b], there exists a point CE [a, b], that solves f(x) = 0. If f(a)f(b) < 0, the Regular Falsi method locates the root c by the equation C= af (b) - bf (a) f(b) - f(a) (4) with a suitable graph, derive the given regular Falsi formula, and use it to state the regula Falsi recurrence formula for determining the root c, n = 0,1,2, (ii) With the given formula in (4), state the regular Falsi algorithm. (5 marks (1 mark C1. (iii) Determine the root at c₁ to 3- decimal places, to r³+x-1=0 on the interval [0.5, 1] by the regular Falsi method, i.e obtain the values at co and (6 marks (i)) Outline the bisection algorithm for determining the root to f(x) = 0, on the interv [a, b]. (2 mark 42² + 8 = 0, on the i ii) Determine the root to 3- decimal places to the equation r³ terval [3,4], by the Bisection method, with 3- iterations, i.e stop at (the root pa Computational values during the iterations should be rounded off to 3- decim places. (6 mark​