MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A differentiable function is defined by f(x)=sin(2x). The point of interest is a=4π. For several x–values near a, the corresponding function values are given.
x-value | f(x)
0.985398 | 0.9205
0.885398 | 0.9796
0.835398 | 0.9954
0.795398 | 0.9997
0.785398 | 1.0000
(All values are rounded to four decimal places.)
(a) For each interval [a,x] listed in the table, calculate the average rate of change x−af(x)−f(a). Record your results in a new column next to the table.
(b) Describe the numerical trend exhibited by your average rates as the interval length decreases. Based on this trend, estimate the instantaneous rate of change of f at x=a. State your estimate correct to three decimal places.
(c) Explain, in complete sentences, why the sequence of average rates you computed in part (a) is expected to approach the instantaneous rate of change obtained in part (b). In your explanation, refer to the behavior of f on shrinking intervals and the concept of a limiting value.