MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A twice–differentiable function f is defined for 1.5<x<2.5. A table of selected values is given below.
x
1.8
2.0
2.2
f(x)
0.5878
0.6931
0.7885
It is known that f is concave down on the interval shown.
Three students estimate f′(2) by computing the slopes of the following secant lines:
• Student I uses the points (2,f(2)) and (2.2,f(2.2)).
• Student II uses the points (1.8,f(1.8)) and (2,f(2)).
• Student III uses the points (1.8,f(1.8)) and (2.2,f(2.2)).
Answer the questions that follow, giving clear justifications in each case.
(a) For each student’s estimate, indicate whether the value obtained is an overestimate, an underestimate, or could be either for f′(2). Support each conclusion by referencing appropriate properties of f.
(b) Without performing numerical calculations, determine which student’s estimate is expected to be closest to the actual value of f′(2). Explain your reasoning in terms of competing sources of error.
(c) Describe qualitatively how decreasing or increasing the interval width around x=2 would affect both truncation error and the influence of round-off error when estimating f′(2) from tabulated data.