MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A twice-differentiable function g satisfies g(2)=5, g′(2)=−1.4, and g′′(x)>0 for all x in the open interval (1.5,2.5).
A student uses the tangent line to the graph of g at x=2 to estimate g(1.9).
(a) Without performing any calculations, state whether the student’s estimate of g(1.9) is less than, equal to, or greater than the actual value of g(1.9).
(b) Explain your conclusion from part (a) by discussing the geometric relationship between concavity and the position of a tangent line relative to the graph of g in the given interval.
(c) The student’s numerical estimate is 5.14. Define the signed error E=(estimate)−(actual). Based on your work in part (a), indicate the sign of E and interpret its meaning in the context of this problem.
(d) Describe in words how knowledge of the second derivative could be used to bound ∣E∣ without explicitly computing g(1.9).