The second derivative of a twice-differentiable function was measured experimentally…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The second derivative of a twice-differentiable function p was measured experimentally at equally spaced points. Each recorded value has an absolute error not exceeding 0.3. The reported data are given in Table 4.
Table 4. Experimental measurements of p′′(x) (units: N/m)
x 0 1 2 3 4
p''(x) 2.4 0.6 −0.2 −1.1 −0.5
Complete the tasks below, taking the measurement uncertainty into account at every step.
(a) For each interval between consecutive x-values, decide whether the concavity of p can be guaranteed (concave up, concave down, or indeterminate) based on the error bound. Present your conclusions in an organized form and explain the reasoning that leads to each guarantee.
(b) Identify every open interval in which the sign of p′′ cannot be determined with certainty. For each such interval, describe the range of possible concavity behaviors consistent with the data and error bounds.
(c) Propose one additional x-value at which measuring p′′ would most efficiently resolve at least one indeterminate interval you found in part (b). Justify your choice.