The temperature (in ) along a 100 ,cm metal rod is modeled by a function that is…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

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}

. (1) 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. (2) Using your result from part (a), state the interval of all possible values of

T(100)

. (3) 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.