3. Consider f(x) = x lnx and interpolating points with abscissae xo = 1 and 2₁ = 2. (Keep your calculations accurate to at least 4 decimal places.) (a) Compute the Hermite polynomial H(x) that approximates f(x) using the basis functions Hnj and Hnj in the lecture notes. (b) Compute the divided-difference table and construct the Hermite polynomial G(x) that approximates f(x). (c) The Hermite polynomial can also be obtained using the monomial basis. List the linear system for the coefficients (under the monomial basis) of the Hermite polynomial F(x) that approximates f(x). Use the backslash in matlab to solve this system. Verify that the polynomials obtained in (a), (b), and (c) are actually the same. (d) Suppose we use G(1.5) to approximate f(1.5). Find the error f(1.5) - G(1.5). (e) Use the error formula to find the error bound at x = 1.5, and compare the error bound with the actual error in (d).

Q: 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T)…

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Q: If q(x) is an arbitrary polynomial in Pn it follows that                                      q(x) =…

A: Given, if q(x) is an arbitrary polynomial in Pn it follows that                   q(x) = c0p0(x) + .…

Q: Consider the function h(x) = e³ (x-7) with x E [-7,7]. %3D Consider n + 1 points xo < x1 < ... < x,…

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Q: 6 Use the Linear Approximation to estimate Af for f(x) = when a = 2 and Ax = 3+x2 (Use decimal…

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Q: (a) Determine the missing values in the divided difference table provided below. xo = ? ƒ[x] = 1 x₁…

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Q: Determine the NATURAL cubic spline S(x) that interpolates the data f (1) = 1, f (2) = 1, and f (3) =…

A: The given data  is x0=1 , x1=2 , x2=3 and fx0=1 , fx1=1 , fx2=0. The spline polynomial will consist…

Q: 1. Find the Newton-Hermite interpolation polynomial for f(x) with f(1) = 8, f'(1) = 16, ƒ(3) = 16,…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Q4) Consider the Lagrange coefficient polynomial L2x(x) that are used for quadratic interpolation at…

A: The Lagrange coefficient polynomials are given as,…

Q: Determine the main base function as one of f(x) = x, f(x) =x², f(x) = Vx , or f(x) = -. 1 а) h(x) —…

A: A)h(x)=x-7The function h(x) is obtained byvertical transformations of 7 units downof f(x)=xThe base…

Q: Let f(x) = x sin (2x) – x², xo = 0, x1 = 0.3, X2 = 0.7. - %3D (a) Find Lagrange interpolating…

A: Use the following formulae, to obtain the solution. Lagrange interpolating polynomial:…

Q: 2. (а) Find the function P(x) = a +b cos(n x) +c sin(nx), which interpolates the data 0.5 1 y 4 (b)…

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Q: The Maclaurin polynomial of degree 4 for the function f(x) = cos(x) is %3D

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Q: 2. Consider the Newton polynomial Pn(x) = ao+a1(xr-xo) + a₂(x-xo)(x − x₁) +... +an(x-xo)(x -…

A: Given that   that interpolate then we have Now  ,       putting