Question 1: The Bisection method: Consider the function f(x) = 3x + sin(x) ex. (1.1) Use the bisection method to determine a root of f(x) in the interval (0, 2), using up to ten iterations. (10) (1.2) Repeat the above question by using Mathematica commands. Give a command to generate each iteration. Present all commands and results generated. (10) (1.3) Give a single Mathematica command that determines a root of f(x) in the interval (0, 2), and run it. Present all the results. (3) (1.4) Give a single Mathematica command that plots f(x) in the interval 1 < x < 1.5 and 0 < f(x) < 2, label the axes accordingly as x and f(x). Present both the command and the plot. (2) [25]

Question 2: Newton's method: Consider the same function f(x) = 3x + sin(x) ex. (2.1) Use the Newton's method method to determine a root of f(x) in the interval (0, 2), using up to ten iterations. (10) (2.2) Repeat the above question by using Mathematica commands. Give a command to generate each iteration, using Newton's method. Present all commands and results generated. (10) (2.3) Give a single Mathematica command that plots f(x) = x3 + 2x2 x + 5 in the interval 5 < x < 5 and 5 < f(x) < 10, label the axes accordingly as x and f(x). Determine the root graphically, by estimating it off the plot. Present both the command and the plot. (5) [25] 16

APM4813/101 Question 3: The x = g(x) method: Use the x = g(x) method to determine the root of f(x) mentioned in Question 1. Provide the same details asked for. (25)

Question 4: Lagrangian interpolation: Suppose that we have the following data pairs-x-values and f(x)-values- where f(x) is some unknown function: Table 1: Data for curve fitting x f(x) 3.2 22 2.7 17.8 1.0 14.2 4.8 38.3 5.6 51.7 (4.1) Use the Lagrangian polynomial to interpolate the data in the table. (5) (4.2) Repeat the above question by using Mathematica commands. Give a command to generate the whole interpolation. Present all commands and results generated. (5) (4.3) Give a single Mathematica command that uses Lagrangian polynomial to interpolate the data in the table. (5) (4.4) Plot the points in the table using mathematica. (5) (4.5) Plot the interpolation polynomial you obtained using mathematica. (5) [25] TOTAL: [100]