(a) Find the linearization of Eq. (i) at x = 0. (b) In t
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(a) Find the linearization of Eq. (i) at x = 0. (b) In the case that h > 0, what is the natural frequency of the lineari
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(a) Find the linearization of Eq. (i) at x = 0. (b) In the case that h > 0, what is the natural frequency of the linearized system? Explain how the natural frequency of the linearized system depends on h and L. (c) On the same set of coordinate axes, plot the graphs of y = f(x) = 2kx [ 1 ? L ?(L + h)2 + x2 ] , its first degree Taylor polynomial y1 = f ? (0)x, and its third degree Taylor polynomial y3 = f ? (0)x + f ??(0)x2? 2 + f ???(0)x3?6. Construct these plots for each of the following sets of parameter values: i. L = 1, k = 1, h = 1 ii. L = 1, k = 1, h = 0.5 iii. L = 1, k = 1, h = 0.1 iv. L = 1, k = 1, h = 0 (d) Find the dependence on h of the approximate range of values of x such that |y ? y1| ? |y3 ? y1| < 0.1. (e) In the case that h = 0, explain why the solution of the linearized equation obtained in Problem 2 is a poor approximation to the solution of the nonlinear equation (i) for any initial conditions other than x(0) = 0, x? (0) = 0
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