5: The Hessian matrix for f(X) = x}xy + x3x3 at the point (1 2 3], is given by: [6 0 31 [3 0 61 [18 0 31 A: 02 2 B: 2 20 C: 0 6 4 D: Non of these 13 2 0 lo 3 4 0 О А О в О с OD 2 31
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- Show that the volume of a triclinic unit cell of sides a, b, and c and angles α, β, and γ is V = abc(1 − cos2α − cos2β − cos2γ + 2 cos α cos β cos γ)1/2 . There is this answered question already over here, but I have a dout regarding the process it took to get to the result, specifically, once we have our 3x3 matrix, how do we determine that this matrix equals to the linear eq? this is the image :DLet A(t) be a n × n matrix whose coefficients are continuous functions of t on the interval (α, β). Let t0 ∈ (α, β) and asssume that y(t) is a solution of the initial value problem x' = A(t)x, x(t0) = 0. Show that y(t) = 0 for all t ∈ (α, β).You are given 6 data points (xi; yi) below, where i = 1,..., 6 that are observed from a model: y = a3x3 + a2x2 + a1x + a0 + n, where n is a zero-mean Gaussian noise. Estimate the coefficients: a0,...,a3 by setting up a linear system of equations, and solving it using least squares. (x1,y1) = (-3, -171.17478734011377) (x2, y2) = (-1.5, -26.283603386675573) (x3, y3) = (0, 10.342025892375206) (x4, y4) = (2, 34.71284310507833 (x5, y5) = (5.5, 733.5702844691194) (x6, y6) = (7, 1546.4530015969324)
- For a linear algebra matrix with initial value, how would I set up a general formula for xk to determine the limit as k approaches infinity for xk?Find the general solution of the linear system (1) when A is then x n diagonal matrix A = diag[Aj, A21... ,An]- What condition onthe eigenvalues Al, ... , An will guarantee that limt_,,. x(t) = 0 for allsolutions x(t) of (1)?The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint pmf of X and Y is modeled as fxy (x,y)=c(x+y), x=1,2,3 and y=1,2,3. Check if the number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are independent.
- 1. Find the natural cubic spline sN (x) passing through the 3 points (xj, yj) given by (0, 2), (2, 3), and (3, 1).Then evaluate sN (1).Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.Given the following R2→R function:b) f(x,y)= x2+5y2-2x-20y+24Find and analyze the nature of the critical points using the algebraic method and Hessian matrix and show that both methods lead to the same results.
- Maximize f(x) = 4x 1.8x² + 1.2x³- 0.3x4 a) Using Golden-Section Search (x₁ = −2,xu = 4, εs = 1%) b) Using Newton's Method (xo = 3, &s=1%)Let X(t) be a fundamental matrix for x′ = A(t)x on the interval I. Q. If t0 ∈ I, show that the solution to the initial-value problem x′= Ax, x(t0)=x0, can be written as x= X(t)X−1(t0)x0The function f (x, y) = 2xy certainly has a saddle point and not a minimum at (0, 0). What symmetric matrix S produces this f? What are its eigenvalues?