MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives
Problem
The temperature T, measured in degrees Celsius, inside a chemical reactor during a heating phase is modeled by a differentiable function on the interval 0≤t≤10 minutes. Selected temperature readings are provided in the table below.
t(min)
4.00
4.20
4.35
4.45
4.55
4.65
4.80
5.00
T(t)(∘C)
62.1
63.7
64.7
65.3
65.8
66.1
66.5
66.8
Assume that T is twice differentiable on the given interval.
(1) Part 1: (a) Using the data in the table, construct two distinct sequences of average rates of temperature change that converge toward the instantaneous rate of change at t=4.5 minutes. Each sequence must contain at least three average rates and use progressively smaller symmetric intervals about t=4.5.
(2) Part 2: (b) From the sequences in part (a), estimate T′(4.5). Give your answer in degrees Celsius per minute and justify how your estimate is obtained from the data.
(3) Part 3: (c) Write, but do not evaluate, the limit expression that defines T′(4.5) using the same notation as in the table.
(4) Part 4: (d) Explain, in the context of the reactor, what the sign and magnitude of T′(4.5) indicate about the heating process at that moment.
(5) Part 5: (e) Suppose additional measurements reveal that T′′(4.5)<0. Discuss what this information, combined with your answer to part (d), implies about how the rate of temperature increase is changing at t=4.5 minutes.