A function q is defined for all real numbers with ,
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A function q is defined for all real numbers with q(0)=1, q′(0)=−0.3, and q′′(x)=sinx−2x.
Two different teams wish to approximate q(0.15).
• Team A bases its work at x=0.
• Team B bases its work at x=0.1.
(1) Part 1: (a) Construct the approximation each team would obtain for q(0.15).
(2) Part 2: (b) For Team A, determine a bound on the magnitude of the error in its approximation by finding the maximum value of ∣q′′(x)∣ on the interval [0,0.15]. Show all supporting work, including justification that your maximum is valid.
(3) Part 3: (c) Repeat the error-bound analysis in part (b) for Team B on the interval [0.1,0.15].
(4) Part 4: (d) Compare the two error bounds and decide which team’s approximation is provably more accurate. Provide a clear argument that references the results from parts (b) and (c).
(5) Part 5: (e) Suppose both teams could instead choose any base point in [0,0.15]. Describe qualitatively how the choice of base point influences the bound on error when approximating q(0.15).