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

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 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) = √ x satisfying the conditions H5(xi) = √ xi , i = 0, 1, 2 and H 0 5 (xi) = 1/ 2 √ xi , i = 0, 1, 2 for the points x0 = 1, x1 = 4 and x2 = 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 fit for the data. [3] 

(c) Use the least squares method find the curve of best fit for the data. [11] 

(d) Hence estimate the population of Botswana in the year 2036. [3] 

Question 4. Fit the curve y = 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 ε