You are given the expression for Taylor Series Expansion And the following set o

Question

You are given the expression for Taylor Series Expansion And the following set of points are given at an interval of Delta x. You have been given the liberty of expressing the Taylor Series at any point about any points (i.e. an interval of 3 Delta x is acceptable). Ignore the 0( Delta x5) in your expressions. Use the Taylor Series Expansions and find the mathematical expressions for First, Second, Third & Fourth order derivatives of / (i.e. f', f", f"', f"" ) at xi. Write a small script in MATLAB to calculate the derivatives using the expressions that you have fond. Next assume, f{x) = 2x4 + 3x3 + Delta x2 + 5x - 24 and calculate the First, Second, Third & Fourth Derivative of f at x=0.25, Delta x=0.5, by plugging the numbers in your MATLAB code. Calculate the same using MATLAB 'diff function and compare your results. Present your results in a tabular form.

Explanation / Answer

syms x taylor(exp(x)) taylor(sin(x)) taylor(cos(x)) ans = x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1 ans = x^5/120 - x^3/6 + x ans = x^4/24 - x^2/2 + 1 plotT6 = ezplot(t6, [-4, 4]); hold on set(plotT6,'Color','red') plotT8 = ezplot(t8, [-4, 4]); set(plotT8,'Color','magenta') plotT10 = ezplot(t10, [-4, 4]); set(plotT10,'Color','cyan') plotF = ezplot(f, [-4, 4]); set(plotF,'Color','blue','LineWidth', 2) legend('approximation of sin(x)/x up to O(x^6)',... 'approximation of sin(x)/x up to O(x^8)',... 'approximation of sin(x)/x up to O(x^1^0)',... 'sin(x)/x',... 'Location', 'South') title('Taylor Series Expansion') hold off

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