Question 1. (a) Using the formula Tn(x) = cos(n cos−1 x), n ≥ 0, find the Chebyshev polynomials T0(x), T1(x), T2(x), T3(x), and T4(x). [8] (b) Find the Chebyshev interpolating polynomial that attains

(a) Using the formula T n( x ) = cos( ncos 1 x); n 0, nd the Chebyshev polynomials T 0( x ), T 1( x ), T 2( x ), T 3( x ), and T 4( x ). [8] (b) Find the Chebyshev interpolating polynomial that attains the values 6, 1, 3, and 66 at the points 1, 0, 2 and 6. Reduce the polynomial to its natural form. [12] Question 2.

(a) Find the Hermite interpolating polynomial for the function f(x ) = p x satisfying the conditions H 5( x i) = p x i; i = 0 ;1; 2 and H 0 5 ( x i) = 1 2 p x i; i = 0 ;1; 2 for the points x 0 = 1, x 1 = 4 and x 2 = 9.

Reduce the polynomial to its natural form. [15] (b) Find the error bound of the interpolating polynomial. [10] Question 3. The population of Botswana (in millions) for the years 1970 to 2020 is given in the table. Year 1970 1980 1990 2000 2010 2020 Population, P 0.628 0.898 1.287 1.643 1.987 2.254 (a) Make a scatter plot (population versus years) for the data. [3] (b) Using the scatter plot determine the data trend and law of the curve of best t for the data. [3] (c) Use the least squares method nd the curve of best t for the data. [11] (d) Hence estimate the population of Botswana in the year 2036. [3] Question 4. Fit the curvey= a (1 bx)2 to the data x 4 6 8 10 11 12 y 4.89 5.49 6.62 9 11.4 16.1 [11] Question 5.

(a) Evaluate the integral I= Z 3 1 1 x d x using the trapezoidal rule method with accuracy " <0:05. [15] (b) Find the error of the results given the exact value of the integral is ln(3). [1] (c) Improve the results in part (a) using the Romberg method for the same number of subintervals. Find the error of the results by the Romberg method. [8]