13 pt) 12) Suppose k and I are real numbers and let u(x, t) = sin(kx)e-'t....
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i + 2 j +
t k
2. Let u(t) - ti+In(t)j+ et k and Compute the derivative of the dot product f u(t)v() in two ways and confirm they agree: Compute the dot product u(t) v(t) first and then differentiate the result. . Alternatively, use the following "Dot Product Rule" u(t) v(t)] u'(t) v(t)+ u(t) v'(t) Aside: It's worth noting that there are other forms...
Let U ={a, b, c, d, e, f, g, h, i, j, k}. Let A={d, f, g, h, i, k}. Let B={a, d, f, g, h}. Let C={a, c, f. i, k} Determine (AUC) U ( AB). Choose the correct answer below and, if necessary, fill in the answer box in your choice. OA. (AUC) U(ANB)= } (Use a comma to separate answers as needed.) OB. (A'UC) U (ANB) is the empty set. LE This Question: 1 pt Let U={x|XEN...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I denote the identity transformation I(u) = u for each u EU and let 0 denote the zero transformation. Show that there is a natural number N, and constants C1, ..., CN+1 such that C1I + c2T + ... + CN+1TN = 0 (Hint: Given dim(U) = n, what is the dimension of L(U,U)? consider ciI + c2T + ... + Cn+11'" = 0, where...
let two vectors be a(t) = e^t i + (sin 2t) j + t^3 k and b(t) = (e^-t , cos 3t, - 2 t^3) in euclidean three space R^3. Find d/dt [a(t) * b(t)].
2.11. Let x(t) 11(1-3)-u(t-5) and h(t) = e-3t11(1). (a) Compute y(i) - x(t) * h(t)
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B Find functions g and h such that X, has the same covariance as a Brownian bridge.
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B...