Considering fx tan1x determine the value of fx and f x at x

Considering f(x) = tan^-1(x), determine the value of f\'(x), and f\'\" (x) at x = 1.0, using central difference scheme, with a step size of h = 0.1 and h = 0.01 using a calculator. Now. determine the relative error epsilon_t, (i.e. expressed as a percentage) from the exact value of the first and second derivative of f(x). You should specify the number of significant figure in your calculator. Now develop a MATLAB code (or any programming language) to determine the value of the numerical difference scheme used in Problem 1 above, with step sums given by equation 1, i = 1.......14. Compare the results obtained with Problem 1 and give your comments on the trend of relative emirs against step size h along with the plot. h = 10^-1/2 For each value of h, display the results in the following tabular form. i, h, y, y\', epsilon_t(y\'), y\" epsilon_t(y\") Evaluate e^-8.3 using two approaches given by equations 2 and 3 and compare with the true value, using MATLAB code. Discuss the results using 25 teruns to evaluate each series, and compute true and approximate relative errors as the terms are added. e^-x = 1 - x + x^2/2! - x^3/3! + ........ and e^-x = 1/e^x = 1/1 + x + x^2/2! + x^3/3! + .......

Solution

e^-8.3 = 1-8.3 + 8.3²/2 -8.3³/6 + 8.3⁴/24.... = 129.588 from equation (2) upto 4 units

1/e^-8.3 = 1/ ( 1+8.3+8.3²/2+8.3³/6+8.3⁴/24.....)

= 1/ (1+8.3+34.445+95.297+197.74+...)= 0.0029761 from equation (3) upto 4 units

from matcode e^-8.3 = -5.58