Let . For verification purposes, the exact value of is known to be . In this problem…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let p(x)=ln(2x+1). For verification purposes, the exact value of p′(2) is known to be 52. In this problem you will approximate p′(2) numerically and analyze the associated error.
(1) Part 1: (a) Using a step size of 0.1, compute an estimate of p′(2). Show all intermediate values used.
(2) Part 2: (b) Repeat the estimation in part (a) with a step size of 0.05.
(3) Part 3: (c) Using the two approximations from parts (a) and (b), produce a refined estimate of p′(2) and clearly state your reasoning.
(4) Part 4: (d) For each of the three estimates from parts (a), (b), and (c), calculate the absolute error relative to the exact derivative value. Which estimate is most accurate, and why does this outcome make sense in terms of the numerical method employed?