12 Approximate x - 11| dx, by computing Lf (P) and Uf (P) for the partition P = {10, 10.5, 11, 11.5, 12}. 10 Lf (P) = 1 Uf (P) = 1.75 Submit Answer
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- Consider a random process X(t) defined by 4 X(t) = Ucost + (V + 1)sint, where U and V are independent random variables for which E(U) = E(V) = 0; E (U²) = E (V²) = 1 a) Find the auto covariance function of X(t) b) Is X(t) wide sense stationary? Explain your answer Jf the Wwss process (Y(+ is given by Y(+) - 100 cos(10t where uniformlsWhat is the form of the partial fraction expansion of arational function P(x)/Q(x) if the degree of P is less thanthe degree of Q and Q(x) has only distinct linear factors?What if a linear factor is repeated? What if Q(x) has an irreduciblequadratic factor (not repeated)? What if the quadraticfactor is repeated?Let pk(n) denote the number of partitions of the integer n into atmost k parts. Find the generating function summation pk(n)x^n
- In this problem we show how a general partial fraction expansion can be used to calculate many inverse Laplace transforms. Suppose that F(s)=P(s)Q(s),F(s)=P(s)Q(s), where Q(s) is a polynomial of degree n with n distinct zeros r1, …, rn, and P(s) is a polynomial of degree less than n. In this case it is possible to show that P(s)/Q(s) has a partial fraction expansion of the form (36) P(s)Q(s)=A1s−r1+⋯+Ans−rn,P(s)Q(s)=A1s−r1+⋯+Ans−rn, where the coefficients A1, …, An must be determined. Show that L−1{F(s)}=n∑k=1P(rk)Q′(rk)erkt.Use partial fraction decomposition to evaluate the integral. ∫6x2-13x+3/x2(x-3) dx. Let Dn be as in Example 10. Find a Hamiltonian circuit inCay({(r, 0), (f, 0), (e, 1)}:D4 ⨁ Z5).Does your circuit generalize to the case Dn ⨁ Zn+1 for all n ≥ 4?