1. Use the Newton forward-difference formula to construct interpolating polynomials ofdegree one, two, and three for the following data. Approximate the specified value using each of the polynomials.
   1. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169
   2. f(0.18) if f(0.1) = 0.29004986, f(0.2) = 0.56079734, f(0.3) = 0.81401972, f(0.4) = 1.0526302
2. Use the Newton backward-difference formula to construct interpolating polynomials ofdegree one, two, and three for the following data. Approximate the specified value using each of the polynomials.
   1. f(1/3) if f(0.75) = 0.07181250, f(0.5) = 0.02475000, f(0.25) = 0.33493750,

f(0) = 1.10100000

1. f(0.25) if f(0.1) = 0.62049958 , f(0.2) = 0.28398668 , f(0.3) = 0.00660095, f(0.4) = 0.24842440

3. A fourth-degree polynomial P(x) satisfies ⁴P(0) = 24, ³P(0) = 6, and ²P(0) = 0, where P(x) = P(x + 1) P(x). Compute ²P(10). The following data are part of a table for.

Calculate g(0.25) as accurately as possible

1. by forward difference interpolating directly in this table,
2. by first tabulating xg(x) and then forward difference interpolating in that table,

• explain the difference between the results in (i) and (ii) respectively.

5. Show that the cubic polynomials

P(x) = 3 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x 1)

and

Q(x) = 1 + 4(x + 2) 3(x + 2)(x + 1) + (x + 2)(x + 1)(x)

both interpolate the data

f(2) = 1,f(1) = 3,f(0) = 1,f(1) = 1,f(2) = 3

1. Why does part (i) not violate the uniqueness property of interpolating polynomials?
2. The following data are given for a polynomial P(x) of unknown degree.

P(0) = 4,P(1) = 9,P(2) = 15,P(3) = 18

Determine the coefficient of x³ in P(x), if all fourth-order forward differences are 1.

• For a function f , the Newton divided-difference formula gives the interpolating polynomial

,

on the nodes x₀ = 0,x₁ = 0.25,x₂ = 0.5, and x₃ = 0.75. Find f(0.75).

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