Let be twice–differentiable on with on that interval
MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Let f be twice–differentiable on (1,3) with f′′(x)>0 on that interval. A student wants to approximate f(2.2) using the tangent line at either x=2 or x=2.5. Which of the following statements gives the best reason for selecting x=2 as the point of tangency?
(a) Because f′(x) varies less between 2 and 2.2 than between 2.5 and 2.2, the slope used at x=2 is closer to the true slope at x=2.2, giving a more accurate linearization.
(b) Because ∣2.2−2∣<∣2.2−2.5∣ and, for a fixed concavity, the error in a tangent–line approximation increases as ∣x−x0∣ increases.
(c) Because ∣2.2−2∣<∣2.2−2.5∣ and, since f′′(x)>0, the tangent line at x=2 is guaranteed to lie below f at x=2.2, avoiding overestimation.
(d) Because the error term is proportional to (x−a) when f′′(x)>0, halving the distance from 0.3 to 0.2 roughly halves the error, so x=2 is significantly more accurate.