MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function g is defined on the closed interval [2,8]. Values of g at several points near the left–hand endpoint are shown in the table.
x (seconds)
2.0
2.3
2.6
2.9
g(x) (meters)
12.4
13.1
13.8
14.6
The derivative g′(2) is of interest to a researcher studying the initial motion of an object whose position is modeled by g.
Throughout this problem, use the data provided to analyze the behavior of g at x=2.
(a) Explain why evaluating g′(2) requires consideration of a limit involving only x>2 rather than values of x on both sides of 2.
(b) Write an algebraic expression, in terms of h, that represents an approximation of g′(2) using the tabulated data. Indicate any domain restrictions on h implied by the table.
(c) Using the two smallest positive values of h that can be formed from the table, compute numerical approximations of g′(2). Show your work.
(d) Based on your work in part (c), describe the apparent instantaneous rate of change of g at x=2, including appropriate units, and discuss one factor that limits the accuracy of this estimate.