Let be a twice-differentiable function that gives the height
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let s be a twice-differentiable function that gives the height, in centimeters, of a plant t days after it is moved to a greenhouse. For 0≤t≤4, definite measurements are recorded and summarized.
t (days)
0.0
0.8
1.6
2.4
3.2
4.0
s(t) (cm)
5.0
5.6
6.3
7.1
8.0
9.0
A horticulturist wants to investigate the initial growth rate on the day the plant enters the greenhouse.
(a) Calculate an estimate of s′(0) using data from the table. Clearly identify each numerical value used in your computation.
(b) Provide the units of your answer in part (a) and interpret its meaning for the plant’s growth.
(c) Using at least three distinct values of h obtainable from the table, create a second estimate of s′(0) by extrapolating the pattern you observe. Explain your procedure.
(d) Discuss the reliability of your extrapolated estimate from part (c) by citing two specific aspects of the data set that support or weaken confidence in the result.