A function is continuous on the closed interval and differentiable on the open interval

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A function

h

is continuous on the closed interval

[0,6]

and differentiable on the open interval

(0,6)

. The values

h(0)=4

and

h(6)=9

are known, but no explicit formula for

h

is given. (a) Define the average value of

h

[0,6]

symbolically in terms of a definite integral. (b) Explain why there must exist at least one number

c

[0,6]

such that

h(c)

equals the average value you described in part (a). Your explanation should cite all conditions that guarantee the conclusion. (c) If

h

represents the temperature (in °C) of a chemical solution, interpret the meaning of the number

c

found in part (b) in practical terms.