The number of NCAA women's college basketball teams in the...

The number of NCAA women's college basketball teams in the United States can be modeled by: $$ n(t)=-0.98 t^{2}+32 t+1,850 \quad(0 \leq t \leq 8) $$ where $t$ is time in years since $2000 .^{4}$ a. Find $n^{\prime}(t)$. HINT [See Example 2.] b. How fast (to the nearest whole number) was the number of women's college basketball teams increasing in 2005 ? c. According to the model, did the rate of increase of the number of teams increase or decrease with time? Explain. HINT [See Example 4.]

The number of NCAA women's college basketball teams in the United States can be modeled by:
$$
n(t)=-0.98 t^{2}+32 t+1,850 \quad(0 \leq t \leq 8)
$$
where $t$ is time in years since $2000 .^{4}$
a. Find $n^{\prime}(t)$. HINT [See Example 2.]
b. How fast (to the nearest whole number) was the number of women's college basketball teams increasing in 2005 ?
c. According to the model, did the rate of increase of the number of teams increase or decrease with time? Explain. HINT [See Example 4.]

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Key Concepts

Differentiation of Polynomial Functions

This concept involves using calculus rules such as the power rule and the constant multiple rule to find the derivative of a polynomial. Differentiation transforms a function into another function that describes its instant rate of change with respect to its variable. The process simplifies the evaluation of how a function’s output changes as its input varies.

Instantaneous Rate of Change

The instantaneous rate of change refers to how a quantity is changing at a specific moment in time. In mathematical modeling, this is represented by the derivative of a function evaluated at a particular point. It provides a precise measure of the rate of increase or decrease, which in real-world scenarios helps in understanding trends or dynamics at an exact time.

Analyzing the Behavior of a Rate Function

This concept involves interpreting the derivative function to determine how the rate of change itself behaves over time. By examining the sign and slope of the derivative, one can infer whether the rate of change is increasing, decreasing, or remaining constant. This analysis is crucial for understanding the overall behavior and long-term trends in the modeled phenomenon.

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