18.2 Fit a second-order Newton's interpolating polynomial to estimate log 10 using the data from Prob. 18.1 at x=8, 9, and 11. Compute the true percent relative error. 18.3 Fit a third-order Newton's interpolating polynomial to esti- mate log 10 using the data from Prob. 18.1. 18.4 Repeat Probs. 18.1 through 18.3 using the Lagrange polynomial. 18.5 Given the data 18.11 polyno spond: X y (18.12 corre X 1.6 2 2.5 3.2 4 4.5 Fix 2 8 14 15 8 2 your (a) Calculate f(2.8) using Newton's interpolating polynomials of order 1 through 3. Choose the sequence of the points for estimates to attain the best possible accuracy. (b) Utilize Eq. (18.18) to estimate the error for each prediction. 18.6 Given the data Fix Not fix (a) X fix (b) (c) 2 3 8 19 99 291 444 (d) Calculate f(4) using Newton's interpolating polynomials of order 1 through 4. Choose your base points to attain good accuracy. What do your results indicate regarding the order of the polynomial used to generate the data in the table? Pr 18.7 Repeat Prob. 18.6 using Lagrange polynomials of order 1 E through 3. 18.8 The following data come from a table that was measured with high precision. Use the best numerical method (for this type of prob- lem) to determine y at x=3.5. Note that a polynomial will yield an exact value. Your solution should prove that your result is exact. 12 0 1.8 5 92 8.2 26 16.415 5.375 3.5 2.015 254 8