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).  (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.  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.  (b) Find the error bound of the interpolating polynomial.  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.  (b) Using the scatter plot determine the data trend and law of the curve of best fit for the data.  (c) Use the least squares method find the curve of best fit for the data.  (d) Hence estimate the population of Botswana in the year 2036.  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  Question 5. (a) Evaluate the integral I = Z 3 1 1 x d x using the trapezoidal rule method with accuracy ε
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