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Study Resources (Calculus)

Consider Maxwell%u2019s equations of electrodynamics where units have been chosen s.t. the speed of light c = 1. These equations, resulting from many years of experimental and theoretical investigation by Ampere, Faraday, Gauss, Maxwell and others, relate the interconnectivity of electricity.
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Compute the following integrtals with regards to real analysis, local integrability, and riemann integrability. A.)       \(h(x)= x^{-\alpha} \text{ on the interval } [1,\infty)\) also are there any restrictions on alpha B.)       \(g(x)= \frac{1}{x}\ln(x)\text{ on the interval }[1,\infty)\)  .
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Consider the equation x=g_c(x)=cx(1-x), with c a nonzero constant. This equation has two solutions, and we let alpha_c denote the nonzero solution. What is alpha_c? For what values of c will the iteration x_(n+1)=g_c(x_n) converge to alpha_c (provided that x_0 is chosen sufficiently close to alpha)?.
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Compute the renewal type equation for E[A(t)B(t}], where A(t) (B(t)) is the age (excess/residual life) at time t in a renewal process. Show that the key renewal theorem is applicable and compute the limiting value of E[A(t)B(t}] as t goes to infinity, assuming that the renewal process is aperiodic. Using.
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