# 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]

### Save your time - order a paper!

Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines

Order Paper Now

(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

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
University of Botswana MAT 244 Assignment 2 Department of Mathematics Issued: 23-10-2020 Due:03-11-2020 Question 1. (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]