91Ó°ÊÓ

In calculus, a derivative represents the rate at which a function is changing at any given point. It's like understanding the speed of a car at a specific moment. For the function given, which modeled the annual sales of bottled water, finding the first derivative helps us understand how quickly those sales were changing each year.
To find the derivative of the function \( R(t) = 12t^2 + 500t + 4700 \), we apply the power rule. This rule tells us to multiply the exponent by the coefficient of the term and then subtract one from the exponent. Therefore, the derivative \( R'(t) \) becomes:

• From \( 12t^2 \), we get \( 24t \).
• From \( 500t \), we get \( 500 \).
• Constant terms (like 4700) simply disappear since they don't change over time.

So, \( R'(t) = 24t + 500 \). This derivative tells us the rate of change of sales at any time \( t \). You can think of it as how sales were increasing (or decreasing) each year.

The second derivative provides insight into the acceleration, or the rate at which the rate of change is changing. It helps us answer questions not just about how fast sales are growing, but whether the speed of this growth is increasing or decreasing.
To find the second derivative from our first derivative \( R'(t) = 24t + 500 \), we apply the same differentiation technique:

• The derivative of \( 24t \) is \( 24 \).
• The derivative of constant \( 500 \) is \( 0 \).

Thus, the second derivative \( R''(t) = 24 \). This constant value tells us that the sales were accelerating at a constant rate, meaning that the speed of increase in sales counts was steady over the period. No matter what year we choose between 2000 and 2008, sales were constantly increasing their rate of growth.

Acceleration in calculus is the concept of speeding up or slowing down, a principle directly derived from the second derivative. In the context of the sales function, acceleration represents how quickly the change in sales is occurring.
In our problem, we found that \( R''(t) = 24 \), which is positive and remains constant. A positive second derivative indicates that sales are accelerating, not decelerating.
Specifically, during the year 2004, corresponding to \( t=4 \), the acceleration was still \( 24 \) million gallons per year². This implies that sales were picking up pace steadily and consistently, depicting a healthy growth pattern over the years.
In practical terms, people were buying more and more bottled water each year with increasing speed, which could be attributed to growing popularity or demand.