A laboratory records the average rate of change of the temperature (in °C) with…
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A laboratory records the average rate of change of the temperature T (in °C) with respect to time t (in minutes) over shrinking intervals that all contain t=3. The computed values are shown below.
Interval length (min): 0.50\; \;0.25\; \;0.125
Average rate (°C/min):\; −4.2\; \;−4.5\; \;−4.6
Which statement best explains why these values are used to approximate the instantaneous rate of change of T at t=3?
(a) Because any sufficiently small interval around t=3 yields nearly the same slope, averaging the three computed rates gives a better approximation of the instantaneous rate than using just one of them.
(b) As intervals shorten, measurement error becomes proportionally larger, so the largest‐interval value (−4.2) is the most reliable estimate of the instantaneous rate.
(c) As the interval length approaches zero, the sequence of average rates appears to converge to a single value, so the last computed rate is a close estimate of the instantaneous rate.
(d) By the Mean Value Theorem, the slope of the secant line on the smallest interval equals the derivative at its midpoint, so that average rate equals the exact instantaneous rate at t=3.