Chapter 3.1, Problem 8ES

The data for Exercise 6 were generated using the following functions. Use the error formula to find a bound for the error and compare the bound to the actual error for the cases n = 1 and n = 2.

1. a. f (x) = e^2x
2. b. f (x) = x⁴ − x³ + x² − x + 1
3. c. f (x) = x² cos x − 3x
4. d. f (x) = ln(e^x + 2)

6. Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following:

1. a. f(0.43) if f(0) = 1, f (0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169
2. b. f(0) if f(−0.5) = 1.93750, f(−0.25) = 1.33203, f(0.25) = 0.800781, f(0.5) = 0.687500
3. c. f(0.18) if f (0.1) = −0.29004986, f (0.2) = −0.56079734, f (0.3) = −0.81401972, f (0.4) = −1.0526302
4. d. f (0.25) if f (−1) = 0.86199480, f (−0.5) = 0.95802009, f (0) = 1.0986123, f (0.5) = 1.2943767