State Cigarette Taxes The average state cigarette tax per...

State Cigarette Taxes 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?

State Cigarette Taxes 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

Power Functions

Power functions are a class of functions that involve an independent variable raised to a constant exponent. The properties of power functions, including how to differentiate them using the power rule, are essential to analyzing their behavior over specific domains.

Differentiation

Differentiation is a mathematical technique used to determine the derivative of a function, which represents the instantaneous rate of change. It is crucial for understanding how functions behave, particularly in determining if and how rapidly a function is increasing or decreasing.

Increasing Functions

A function is considered increasing on an interval if its derivative is positive throughout that interval. This concept is important because it provides a formal way to verify that the function’s values continue to grow as the input variable increases.

Rate of Change

The derivative of a function gives the rate of change at any given point. By analyzing the derivative, one can gain insight into not only whether a function is increasing or decreasing but also how the speed of this change varies over different intervals.

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