a. Approximate f (0.05) using the following data and the Newton forward-difference

Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (8.4) if f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f (8.7) = 18.82091 b. f (0.9) if f (0.6) = 0.17694460, f (0.7) = 0.01375227, f (0.8) = 0.22363362, f (1.0) = 0.65809197

Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (0.43) if f (0) = 1, f (0.25) = 1.64872, f (0.5) = 2.71828, f (0.75) = 4.48169 b. f (0) if f (0.5) = 1.93750, f (0.25) = 1.33203, f (0.25) = 0.800781, f (0.5) = 0.687500

Use Newton the forward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f 1 3  if f (0.75) = 0.07181250, f (0.5) = 0.02475000, f (0.25) = 0.33493750, f (0) = 1.10100000 b. f (0.25) if f (0.1) = 0.62049958, f (0.2) = 0.28398668, f (0.3) = 0.00660095, f (0.4) = 0.24842440

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

Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (1/3) if f (0.75) = 0.07181250, f (0.5) = 0.02475000, f (0.25) = 0.33493750, f (0) = 1.10100000 b. f (0.25) if f (0.1) = 0.62049958, f (0.2) = 0.28398668, f (0.3) = 0.00660095, f (0.4) = 0.24842440

Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (0.43) if f (0) = 1, f (0.25) = 1.64872, f (0.5) = 2.71828, f (0.75) = 4.48169 b. f (0.25) if f (1) = 0.86199480, f (0.5) = 0.95802009, f (0) = 1.0986123, f (0.5) = 1.2943767

a. Use Algorithm 3.2 to construct the interpolating polynomial of degree three for the unequally spaced points given in the following table: x f (x) 0.1 5.30000 0.0 2.00000 0.2 3.19000 0.3 1.00000 b. Add f (0.35) = 0.97260 to the table, and construct the interpolating polynomial of degree four.

a. Use Algorithm 3.2 to construct the interpolating polynomial of degree four for the unequally spaced points given in the following table: x f (x) 0.0 6.00000 0.1 5.89483 0.3 5.65014 0.6 5.17788 1.0 4.28172 b. Add f (1.1) = 3.99583 to the table, and construct the interpolating polynomial of degree five.

a. Approximate f (0.05) using the following data and the Newton forward-difference formula: x 0.0 0.2 0.4 0.6 0.8 f (x) 1.00000 1.22140 1.49182 1.82212 2.22554 b. Use the Newton backward-difference formula to approximate f (0.65). c. Use Stirlings formula to approximate f (0.43).

Show that the polynomial interpolating the following data has degree 3. x 2 1 0 1 2 3 f (x) 1 4 11 16 13 4

a. 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 x 2 1 0 1 2 f (x) 1 3 1 1 3 b. Why does part (a) not violate the uniqueness property of interpolating polynomials?

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 given for a polynomial P(x) of unknown degree. x 0 1 2 P(x) 2 1 4 Determine the coefficient of x2 in P(x) if all third-order forward differences are 1.

The following data are given for a polynomial P(x) of unknown degree. x 0 1 2 3 P(x) 4 9 15 18 Determine the coefficient of x3 in P(x) if all fourth-order forward differences are 1.

The Newton forward-difference formula is used to approximate f (0.3) given the following data. x 0.0 0.2 0.4 0.6 f (x) 15.0 21.0 30.0 51.0 Suppose it is discovered that f (0.4) was understated by 10 and f (0.6) was overstated by 5. By what amount should the approximation to f (0.3) be changed?

For a function f , the Newton divided-difference formula gives the interpolating polynomial P3(x) = 1 + 4x + 4x(x 0.25) + 16 3 x(x 0.25)(x 0.5), on the nodes x0 = 0, x1 = 0.25, x2 = 0.5 and x3 = 0.75. Find f (0.75).

For a function f , the forward-divided differences are given by x0 = 0.0 f [x0] f [x0, x1] x1 = 0.4 f [x1] f [x0, x1, x2] = 50 7 f [x1, x2] = 10 x2 = 0.7 f [x2] = 6 Determine the missing entries in the table.

a. The introduction to this chapter included a table listing the population of the United States from 1950 to 2000. Use appropriate divided differences to approximate the population in the years 1940, 1975, and 2020. b. The population in 1940 was approximately 132,165,000. How accurate do you think your 1975 and 2020 figures are?

Given Pn(x) = f [x0] + f [x0, x1](x x0) + a2(x x0)(x x1) + a3(x x0)(x x1)(x x2) + + an(x x0)(x x1)(x xn1), use Pn(x2) to show that a2 = f [x0, x1, x2].

Show that f [x0, x1, ... , xn, x] = f (n+1) ((x)) (n + 1)! , for some (x). [Hint: From Eq. (3.3), f (x) = Pn(x) + f (n+1) ((x)) (n + 1)! (x x0)(x xn). Considering the interpolation polynomial of degree n + 1 on x0, x1, ... , xn, x, we have f (x) = Pn+1(x) = Pn(x) + f [x0, x1, ... , xn, x](x x0)(x xn).]

Let i0, i1, ... , in be a rearrangement of the integers 0, 1, ... , n. Show that f [xi0 , xi1 , ..., xin ] = f [x0, x1, ..., xn]. [Hint: Consider the leading coefficient of the nth Lagrange polynomial on the data {x0, x1, ... , xn}={xi0 , xi1 , ... , xin }.]