A twice–differentiable function satisfies for all in the open interval
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A twice–differentiable function g satisfies g′′(x)>0 for all x in the open interval (c,c+0.4). Two students wish to approximate g(c+0.3) using a tangent line.
• Student A uses the tangent line to g at x=c.
• Student B uses the tangent line to g at x=c+0.2.
Answer the following, providing clear reasoning grounded in the given concavity information.
(a) For each student, indicate whether the tangent–line approximation is an overestimate or an underestimate of g(c+0.3). Justify your answers without calculating specific numerical values.
(b) Compare the likely accuracies of the two approximations. Explain which student you expect to obtain the smaller absolute error and cite two distinct features of the problem statement that support your conclusion.