Chapter 1: Limits
Section 1.7: Intermediate Value Theorem
Suppose the temperature at midnight is 55°F, increases to a high of 85°F, then returns to 55°F at midnight the next day. Assuming that the temperature throughout the day is a continuous function of the time of day, show that there is at least one time in the morning when the temperature is the same as the temperature exactly 12 hours later.
The curve in Figure 1.7.2(a) represents a temperature profile Tt for one 24-hour period, where t is the number of hours past midnight. It starts at T=55 when t=0, reaches t=85, then returns to T=55 at t=24.
The black line represents a span of 12 hours. The endpoints sit at temperatures that are the same for two moments in time during the 24-hour period.
If Tt represents the temperature at time t, then the endpoints of the black line are solutions of the equation Ta+12=Ta, 0≤a≤12, or the alternative equation
Figure 1.7.2(a) Representative temperature profile for 1 day
Since the actual temperature function Tt isn't known, this equation can't be solved explicitly, so an argument built around it's behavior has to be used. As part of this argument, the Intermediate Value theorem will be invoked.
If f0=0, then T12=T0, so the two times when the temperature is equal are t=0,12.
If f0≠0, then f12=T24−T12=T0−T12=−T12−T0=−f0
Now Tt is continuous (because temperatures are continuous), so fa is continuous, and f0⋅f12<0 because f12 and f0 have opposite signs. Hence, the Intermediate Value theorem can be applied to f and there is an a^≠0 for which fa^=0=Ta^+12−Ta^. Thus, the two times at which the temperatures are the same are a^ and a^+12.
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