A function p that models the air pressure (in kilopascals) inside a laboratory chamber is defined for t≥0 minutes. The chamber operator records the pressure every half-minute for the first two minutes, producing the numerical data set shown.
t (min)
0.0
0.5
1.0
1.5
2.0
p(t) (kPa)
101.3
100.7
100.1
99.4
98.8
Assume p is differentiable for t>0.
(a) Using values from the table, determine an approximation for p′(0) in units of kPa per minute. Show analytical set-up and computation.
(b) Interpret your result from part (a) in the context of the laboratory chamber.
(c) Explain why the method you used in part (a) would not be appropriate for approximating p′(1) and propose a modification to obtain an estimate of p′(1) instead.