A twice-differentiable function satisfies the following conditions for in the…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A twice-differentiable function h satisfies the following conditions for x in the interval [2,2.1]:
• h(2)=3,
• h′(2)=−0.6,
• −4≤h′′(x)≤−2.
A table of selected values near x=2 is provided.
x 2.00 2.05 2.10
h′′(x)−3.8−3.0−2.5
(Note: All values of h′′(x) on [2,2.1] lie between −4 and −2.)
(a) Use the information given to construct an approximation of h(2.1).
(b) Using the bound on h′′ given above, determine an upper bound for the magnitude of the error in your approximation from part (a). Express your answer as an inequality of the form ∣E∣<k.
(c) Using your bound from part (b), determine the maximum possible percent error of the approximation. Express your answer as a percentage rounded to two decimal places.
(d) Discuss whether the actual error is likely to be closer to the upper or lower end of the bound found in part (b). Justify your response by referring to the variation of h′′(x) on the interval.