A student seeks to approximate the derivative of the function g at x=2. Using several different values of the increment h in a symmetric difference quotient, the student obtains the following calculator results.
h
0.1
0.05
0.01
0.001
2hg(2+h)−g(2−h)
3.184
3.156
3.143
3.157
Assume that g is twice differentiable near x=2.
a. Part 1: (a) Using the data in the table, determine the best numerical estimate for g′(2) and state its value to three decimal places.
b. Part 2: (b) Provide a clear justification for your choice in part (a) by analyzing the competing effects of truncation error and round-off error as h varies. Discuss why the remaining three values are less reliable estimates.