AP Calculus AB · 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 \(0\le t\le 8\). Suppose \(T(0)=62\) and \(T(8)=89\). (a) Use the Intermediate Value Theorem to explain why there is at least one time in \((0,8)\) at which the temperature is \(70^\circ\mathrm F\). State the hypotheses you use. (b) Must there be a time in \([0,8]\) at which the temperature is \(55^\circ\mathrm F\)? If not, explain why the Intermediate Value Theorem does not guarantee this. (c) Suppose in addition that \(T(3)=48\). What can now be concluded about the existence of a time in \((0,3)\) at which the temperature is \(55^\circ\mathrm F\)? Justify your answer.
Hint
The Intermediate Value Theorem applies to a continuous function on a closed interval and to a target value lying between the endpoint values. A value outside that range is not guaranteed.
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