Gradients and radial fields Prove that for a real number p. with
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- 3 : Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this i=1 purpose, compute the Hessian matrix of the function f(x).arrow_forward- 4. Let = (1, 2, 1), 7 (-1,-1, 2) and w= (0,3,2). Show that ü, 7 and w are linearly dependent or independent. 5. Let = (0, 2, 1), 7 (-1,0, 2) and = (2,3,0). Find the coordinates of (2,2,2) with respect to the basis u, u and w.arrow_forwardWe have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b] . Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1, X2, ..., Xn,..., XN. Each interval is Ax = (b – a)/N. The Simpon numerical integration rule is derived as: N-2 Li f(x)dx = * f(x0) + 4 (2n odd f(xn)) + 2 ( En=2,n even N-1 f(x,) + f(xn)] . Now complete the Python function InterageSimpson(N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is f (x) = 2x³ (Already defined, don't change it). In [ ]: # Complete the function given the variables N,a,b and return the value as "TotalArea". # Don't change the predefined content, only fill your code in the region "YOUR CODE" from math import * def InterageSimpson (N, a, b): # n is…arrow_forward
- We have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b] . Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1, x2,.. Xn,..., XN. Each interval is Ax = (b − a)/N. The equation for the Simpson numerical integration rule is derived as: f f(x) dx N-1 Ax [ƒ(x0) + 4 (Σ1,n odd f(xn)) ƒ(x₂)) + f(xx)]. N-2 + 2 (n=2,n even Now complete the Python function InterageSimpson (N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is ƒ(x) = 2x³ (Already defined in the function, no need to change).arrow_forwardWe have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b]. Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1,x2,..., X., XN. Each interval is Ax = (b − a)/N. The equation for the Simpson numerical integration rule is derived as: f f(x)dx ≈ [ƒ(x0) + 4 (EN-1,n odd S(x)) + 2 (Σ2²n even f(x)) + f(XN)]. Now complete the Python function InterageSimpson (N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is f(x) = 2x³ (Already defined in the function, no need to change). *Complete the function given the variables N, a,b and return the value as "TotalArea"." "Don't change the predefined content' only fill your code in the region *YOUR CODE"" from math import * def InterageSimpson (N, a,…arrow_forwardGenerate H (Hilbert Matrix) for n = 15.Compute the right-hand size vector b = Hx so that the exact solution is xˆ = 1 (thevector of ones). Solve the linear system in MATLAB as a least squares problem usingSVD and report the relative error in the approximate solution in the Euclidean norm.Compare your result to the solution obtained by H\b.arrow_forward
- Theorem 14.8. If P is a polynomial over a field K, then we can find a field L2 K in which P factorises completely into linear factorsarrow_forwardConsider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?arrow_forward1. Assume that a function f is in polynomial time and can be computed in time 0(n) and that g is in polynomial time and can be computed in time 0(n°). Prove that f composed with g, that is f(g(x)), can be computed in time 0(n5).arrow_forward
- Consider the function f(x) = arctan(2(x − 1)) – In |x|. Plot (using PYTHON ) the graph of the function f(x) and describe the intervals of monotonicity of the function f. Speak also about the roots of the equation f(x) = 0. (a) (b) Prove by analytical means (using Calculus), that the equation f(x) = 0 has exactly four real roots P1 < P2 < P3 < P4.arrow_forwardFor each of the following pairs of functions ff and gg, circle one of the answers f \in o(g), f \in \Theta(g),fEo(g),fE0(g), or g \in o(f)g€o(f) f(n) = 2", g(n) = (;) Oƒ € o(g9) Oƒ€ 0(g) O gE o(f) %3D %3D i=log2 n f(n) = E" 2', g(n) =n Oƒ € o(g) Oƒ€ 0(g) O gE o(f) %3D %3D f(n) = (210 + 3n)², g(n) = n² Oƒe o(g) Oƒ€ 0(g) Oge o(f) %3D %3Darrow_forwardProve or Disprove: For all real valued functions f, if f(n) is O(2¹), then f(2m) is 0(2m).arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole