A1 Let the inner product for polynomial functions on the interval 0 ≤ x ≤ 1 be given by 1 (f, 9) = f* f(x)g(x)x dx. 0 (a) Applying the Gram-Schmidt method to the sequence of powers 1, x, x², x³,. find the first three members of the orthonormal basis of polynomials. (1) (b) Find the best in the mean approximation ao + a1x + a2x² of the function √√x in the norm defined by the inner product (1).
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- Consider the function f(x) = cos x, xe [-, ]. Construct the Lagrange interpolation polynomial which interpolates the function f(x) at x = -, *₁ = 0 and ₂ = . 02²+1 Or 0-² + 0 - ²+1 O 0 - ² + 1find the least value of a such that the function f given f(x)=x²+ax+1 is strictly increasing on (1,2)Orthogonalize {1, t, t²} in P₂ (R) with 2 1 (f,g) = [*, f(x)g(x) dx. -1 Then normalize each one so that its graph passes through (1,1). These are the first three Legendre polynomials. " " 2 }
- Let f(x) = e^x. Find the polynomial P2(x) of degree at most 2 such thatP2(1) = f(1), P2(2) = f(2), P2(3) = f(3) by using a divided-difference table. (Donot use decimal approximations for the number e.)Calculate the 1st order Newton polynomial that interpolates the two data points (1,1),(2,3) . Now extend this to a 2nd order Newton polynomial that in addition interpolates the data point (3,2). (In both cases leave the polynomials in Newton form, for example c, +c,(x-x,)+c,(x-x,)(x-x,)Find all residues of the function 2z+1 z(2-2)' not including the residue at infinity.
- What are the Taylor polynomials P0,P1, P2, P3, and P4 f(x)= e^2x, c=0, [-2,4]Find the factorization of the polynomial (P(z) = z^4 + z^3 + z + 1) into quadratic factors over (R) and linear factors over (C).Verify the Cauchy-Schwartz inequality for the polynomials p(x) = 1 + x^2 and q(x) = 1 − x − 2x^2 using the inner product of p(x), q(x) is <P(x),Q(x)> = -3.
- Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.Let's assume that f(x) = c.x² c=10 Pi (1)= sin(z), p2(x) = sin (2x) (p, q) = f p(x)q(x) dx Find approximation of the function by using given orthogonal polynomial in the following form f(a)~ a₁p₁(x) + a₂p₂(x) where Q1 = Find 92 = 10 8 2 (1.P.) (P1, P₁) (F.P₂) (p2.p2) A₂ = 05 (J.P₂) (P2, P2) 1.0 1.5 2.0 2.5 3.0 JuryQ8) Assume that the roots of a characteristic polynomial of a homogeneous ODE with constant coefficients are 0,0,0,2+5i,2-5i, then the general solutiom of this ODE, is: a) C₁ + ₂x + C3x² + e²x [Acos(x) + Bsin(x)] b) C₁+C₂x + C3x² + esx[Acos (2x) + Bsin(2x)] c) C₁x + ₂x² + c3x³ + e*[Acos (5x) + Bsin(5x)] d) C₁ + C₂x + C3x² + e2x [Acos(5x) + Bsin(5x)]