AP Calculus BC · Limits and continuity
A continuous function T gives the outdoor temperature in degrees Fahrenheit t hours…
Problem
A continuous function \(T\) gives the outdoor temperature in degrees Fahrenheit \(t\) hours after 6:00 a.m., for \(0\le t\le 9\). Suppose \(T(0)=55\) and \(T(9)=88\). (a) Use the Intermediate Value Theorem to explain why there is at least one time in \((0,9)\) at which the temperature is \(70^\circ\mathrm F\). State the hypotheses you use. (b) Must there be a time in \([0,9]\) at which the temperature is \(40^\circ\mathrm F\)? If not, explain why the Intermediate Value Theorem does not guarantee this. (c) Suppose in addition that \(T(4)=38\). What can now be concluded about the existence of a time in \((0,4)\) at which the temperature is \(40^\circ\mathrm F\)? Justify your answer.
Hint
IVT applies only when the target temperature sits between the temperatures at the two endpoints of a closed interval of continuity.
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