Consider the function f(x)=x-5x+7x-3. (a) (i) Find f'(x) and (x). [1 mark] (ii) Using Newton's method,...

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Consider the function f(x)=x³-5x²+7x-3. (a) (i) Find f'(x) and "(x). [1 mark] (ii) Using Newton's method, find a root of f(x) starting with po =0.9 by completing the following table. Use FOUR decimal places for all workings. [3 marks] Iteration, Pn f(p) f'(p.) n 0 1 2 (iii) Given the root of f(x) is r=1, is the convergence linear or quadratic as Newton's method is used to find the root? [1.5 marks] (iv) Based on the answer obtained in (iii), find the convergence rate. Hence express the error e, in terms of the previous error e.. [1.5 marks] (b) Consider the definite integral dx. Use FOUR decimal places for all workings. (i) Find the actual value for the definite integral. [1.5 marks] (ii) (iii) Find the number of subintervals if Composite Trapezoidal Rule is used to approximate the definite integral with an error of at most 2x10³. [Hint: error formula: -b-h²f "(x)] 12 [4.5 marks] With the number of subintervals obtained in (ii), find the approximated value for the definite integral using the Composite Trapezoidal Rule. [2 marks] Consider the function f(x)=x³-5x²+7x-3. (a) (i) Find f'(x) and "(x). [1 mark] (ii) Using Newton's method, find a root of f(x) starting with po =0.9 by completing the following table. Use FOUR decimal places for all workings. [3 marks] Iteration, Pn f(p) f'(p.) n 0 1 2 (iii) Given the root of f(x) is r=1, is the convergence linear or quadratic as Newton's method is used to find the root? [1.5 marks] (iv) Based on the answer obtained in (iii), find the convergence rate. Hence express the error e, in terms of the previous error e.. [1.5 marks] (b) Consider the definite integral dx. Use FOUR decimal places for all workings. (i) Find the actual value for the definite integral. [1.5 marks] (ii) (iii) Find the number of subintervals if Composite Trapezoidal Rule is used to approximate the definite integral with an error of at most 2x10³. [Hint: error formula: -b-h²f "(x)] 12 [4.5 marks] With the number of subintervals obtained in (ii), find the approximated value for the definite integral using the Composite Trapezoidal Rule. [2 marks]

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a i To find fx and fx differentiate fx x3 5x2 7x 3 with respect to x fx 3x2 10x 7 fx 6x 10 ii Using ... View the full answer