A differentiable function is defined on an open interval containing
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A differentiable function g is defined on an open interval containing x=2.
A set of secant‐line slopes for g, each computed over an interval of the form [2,2+h], is recorded in the table below.
h
−0.200
−0.100
−0.050
−0.020
−0.010
0.010
0.020
0.050
0.100
0.200
hg(2+h)−g(2)
4.53
4.37
4.30
4.26
4.25
4.24
4.23
4.20
4.16
4.01
(1) Part 1: (a) Estimate g′(2). State the value you are using and justify your choice with reference to the data.
(2) Part 2: (b) Describe, in complete sentences, the relationship between the slopes of the secant lines shown in the table and the slope of the tangent line to the graph of g at x=2.
(3) Part 3: (c) Explain why the process illustrated in the table embodies the formal definition of the derivative at a point.