MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A twice-differentiable function p satisfies p(1)=3 and p′(1)=−0.8. For 1≤x≤1.1, the second derivative satisfies p′′(x)>0. Using the tangent line at x=1, a student finds the linear approximation L(1.1)=2.92. Which statement about the true value p(1.1) is most accurate?
(a) p(1.1) equals 2.92 because the tangent line is an exact predictor over such a small interval.
(b) p(1.1) is greater than 2.92 because the graph is concave up on [1,1.1].
(c) p(1.1) is less than 2.92 because a concave-up graph lies below its tangent line for x>1.
(d) p(1.1) is less than 2.92 because the negative slope at x=1 outweighs the effect of the curvature.