Use Newton\'s Method to determine where the function f(x) = 15x - x^4/3 - 17x^3/

Question

Use Newton's Method to determine where the function f(x) = 15x - x^4/3 - 17x^3/4 = 51. Recall that Newton's Method is an iterative method defined by: X_n + 1 = X_n - f(x_n)/f' (X_n) And that the expression must be reformulated as f(x) - 51 = 0 And that the first value for x is one that you simply have to guess at. Find the expression for f'(x) analytically by hand. State your initial guessed value for x: x_0 = Use "format long" and state your successive values to six decimal places. Show the results of each of your iterations. After your sequence of values converge? If not, choose a different initial guess for X_0.

Explanation / Answer

f(x) = 15x - x4/3 - 17x3/4 = 51

Reformulated f(x) = 15x - x4/3 - 17x3/4 - 51

f'(x) = 15 - (4/3)x1/3 - (51/4) x-1/4

Initial guess x0 = 10

xn+1 = xn - f(xn)/f'(xn)

f1 = 13.572269

f2 = 13.572237

f3 = 13.572237

Yes the values converge and they converge after two iterations.

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