f(x) = 2x² + x + 3. a) Show that if you use the newton method on the function f(x) it not converges, instead is being cyclical if you start it with *o = 0 . b) Given function: f(x) = ax² +x+ e with a ± 0 and c ± 0 Which requirements need the coefficients a and c need to fulfill, in order for the newton method applied to the function f(x) to not converge, but instead being cyclical if you start it with xo = 0

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A particle of unit mass moves on a straight line under the action of a force which is a function f(v) of the velocity v of the particle, but the form of this function is not known. A motion is observed, and the distance x covered in time t is found to be connected with t by the formula                                       x = at + bt2 + ct3 where a, b, and c have numerical values determined by observation of the motion. Find the function f(v) for the range of v covered by the experiment.

Find the root of the function f(x) = e-x- x using Newton-Raphson Method. (Please include the table with iteration, xi, xi+1, f(xi+1), f(xi+1), xr, f(xr) and f(xi)f(xr)

Ex 6. Find the points at which the given function f has a local extremum: (i) f(x)=x²-11 on I = [-4,4]. (ii) f(x) = xr² - 12 on I = [-2,3]. (iii) f(x) = x(x-8)1/3 on I = [0,9].

Use Newton's method to find the zeros of f(x) = 4x² - 4x² using the given starting values in (a) through (c). (a) xo = -1.5 and Xo = - 0.9, lying in = O A. For Xo = B. For Xo -1.5 and Xo 1.5 and Xo √2 (-∞, -1/2) == - 0.9, Newton's method does not converge. The values of x; alternate between 1.5 and 0.9 as i increases. == -0.9, X₁→ as i gets large.

B8

Consider the function f(x) = x3 − 3x2 + 3. (a) Use a graphing utility to graph f. (b) Use Newton’s Method to approximate a zero with x1 = 1 as the initial guess. (c) Repeat part (b) using x1 = 1/4 as the initial guess and observe that the result is different. (d) To understand why the results in parts (b) and (c) are different, sketch the tangent lines to the graph of f at the points (1, f(1)) and ( 1/4, f( 1/4)). Describe why it is important to select the initial guess carefully.

Newton’s method for finding a root of f is to calculate x1, x2, ... via the iterative formula xk+1 = xk - f(xk) / f'(xk) starting from x0. For each of the following functions, calculate x1 and x2: (a) f(x) = 10 - x, with x0 =7 (b) f(x) = -x2 + 1, with x0 = 0

1. Consider the following function: f(x) = 3+ 6x + 5x² + 3x3 + 4x* Locate the minimum by finding the root of the derivative of this function. a. Newton's method xo = -1 (at least 5 iterations) b. Using Quadratic interpolation with initial conditions: x, = -2, x1 = 0, and X2 = 1, iterations = 5