MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A differentiable function f is sampled near x=1, producing the numerical values shown below.
Numerical data (no units):
x f(x)
0.96 1.9817
0.98 1.9900
1.00 2.0000
1.02 2.0097
1.04 2.0190
Throughout this question, round your numerical work to at least five decimal places before presenting a final answer rounded to four decimal places.
(a) Using the data, write one difference quotient that uses only values of f at x>1 and another that uses values at x<1 to approximate the slope of f at x=1. Do not evaluate yet.
(b) Evaluate each quotient you wrote in part (a).
(c) Construct a third quotient that uses points on both sides of x=1. Evaluate it and explain, in words, why this symmetric calculation is expected to give a better estimate of the slope at x=1 than either one–sided calculation.
(d) Suppose additional measurements were taken at x=0.99 and x=1.01. Describe qualitatively—without further computation—how incorporating these values could affect the accuracy of your estimate from part (c). Identify at least two distinct sources of numerical error that might still remain even after using smaller increments.