Suppose f(x, y) 13-6x² +10xy — 12y² + .. is the multivariate power series expansion of ƒ centered at (0,0). Is (0,0) a critical point for f? If so, classify (0, 0) as either a local minimum, maximum or saddle point. =
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- Let Y, be a stationary time series with autocovariance function Yr (s) a. Show that the new series X, = a + bt +Y, where a and b are fixed non-zero constants, is not stationary. b. Express the autocovariance function of AX, = X, - X-1 in terms of 7r (S) and show that this new series is stationary. c. Show that if Y, is a moving average process of order 1, then the series AXt is not invertible and has variance larger than that of Y.Q 3. Find the Taylor series of f at the point c for : (c) f(x) = sin x, c= T (d) f(x) = cos x, c = 5 (a) f(x) = e", c= 2 (b) f(x) = }, c = 1s) =\s- 27, 7f(t) {3t, 0<t<2} f(t){0, -2<t<0} f(t)= f(t+4) i) Sketch the graph of f (t) over the interval[−6, 4].ii) Find the Fourier series expansion off (t ) .Expand the boundary conditions u(x,2)in Fourier series.Find the trigonometric Fourier series for the square 2n-periodic wave defined on the interval [-11,1]: f(x) = {0, if-SX<0 f(x) = {1, if 0(a) Sum the series y = 1+ 2x + 3x² + 4x³ + · .. , l¤| < 1. (b) If f(x) = re-z/^ over the interval 0 < x < ∞, find the mean and most probable values of r. f(r) is the probability density of r. (c) Evaluate I = S a (d) Find the eigenvalues and normalized eigenvectors of the matrix 1 2 4 2 3 0 5 0 3 Are the eigenvectors orthogonal? Comment on this.Express f(x) = |x| Fourier series. Hence obtain -TSubject: calculusRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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