Mathematics

Problem

The temperature (in

^{\circ}\!\text{C}

) along a 100\,cm metal rod is modeled by a function

T

that is continuous on

[0,100]

and differentiable on

(0,100)

. Laboratory tests have established the bound

|T'(x)|\le 0.8\quad\text{(degrees Celsius per centimeter)}\qquad\text{for }0\le x\le 100.

A technician records a temperature of

22^{\circ}\!\text{C}

at the midpoint of the rod,

x=50\text{ cm}

. a. Determine the greatest possible difference between the temperature at the end of the rod,

T(100)

, and the midpoint temperature,

T(50)

. Show all reasoning used. b. Using your result from part (a), state the interval of all possible values of

T(100)

. c. The engineering specification for the rod requires that the absolute temperature difference between the midpoint and either end must not exceed

35^{\circ}\!\text{C}

. Decide whether the derivative bound alone guarantees compliance with this specification at

x=100

. Justify your conclusion.

Intuitive explanation

The key insight is that a bound on the derivative $|T'(x)| \le 0.8$ limits how fast temperature can change along the rod—this is the Mean Value Theorem in action. Since you know the temperature at one point (the midpoint) and how steeply the temperature can vary everywhere, you can build a "cone" of possible temperatures spreading outward from $x=50$ . The maximum and minimum temperatures anywhere on the rod are constrained by how far the slope could have carried you from that central measurement.

Intuitive explanation

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