The average state cigarette tax per pack (in dollars) from...

The average state cigarette tax per pack (in dollars) from 2001 through 2007 is approximated by the function $$T(t)=0.43 t^{0.43} \quad(1 \leq t \leq 7)$$ where $t$ is measured in years, with $t=1$ corresponding to 2001 . a. Show that the average state cigarette tax per pack was increasing throughout the period in question. b. What can you say about the rate at which the average state cigarette tax per pack was increasing over the period in question?

The average state cigarette tax per pack (in dollars) from 2001 through 2007 is approximated by the function
$$T(t)=0.43 t^{0.43} \quad(1 \leq t \leq 7)$$
where $t$ is measured in years, with $t=1$ corresponding to 2001 .
a. Show that the average state cigarette tax per pack was increasing throughout the period in question.
b. What can you say about the rate at which the average state cigarette tax per pack was increasing over the period in question?

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Key Concepts

Rate of Change and Concavity

The analysis of the rate of change involves not only calculating the first derivative but also understanding how this rate itself changes over time. Examining the second derivative, or the concavity of the function, shows whether the increasing rate (slope) is becoming steeper or shallower. A decreasing first derivative or a negative second derivative would indicate that even though the tax is increasing, it is doing so at a slowing rate.

Monotonicity

Monotonicity refers to a function's consistent behavior in either increasing or decreasing across its domain. By analyzing the sign of the first derivative of the tax function, one can determine that if the derivative is positive throughout the interval, the function is increasing, meaning that the average state cigarette tax per pack is continuously rising over the given period.

Differentiation

Differentiation is the process of finding the derivative of a function, which gives the rate at which the function’s output changes with respect to its input. In the context of this problem, differentiating the tax function helps us determine whether the tax increases or decreases over time and at what rate.

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