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

The number of NCAA men's college basketball teams in the United States can be modeled by: $$ n(t)=-0.56 t^{2}+14 t+930 \quad(0 \leq t \leq 8) $$ where $t$ is time in years since $2000 .^{3}$ a. Find $n^{\prime}(t) .$ HINT [See Example 2.] b. How fast (to the nearest whole number) was the number of men's college basketball teams increasing in 2006 ? 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 men's college basketball teams in the United States can be modeled by:
$$
n(t)=-0.56 t^{2}+14 t+930 \quad(0 \leq t \leq 8)
$$
where $t$ is time in years since $2000 .^{3}$
a. Find $n^{\prime}(t) .$ HINT [See Example 2.]
b. How fast (to the nearest whole number) was the number of men's college basketball teams increasing in 2006 ?
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.]

Key Concepts

Polynomial Function Modeling

This concept involves representing real-world phenomena using polynomial equations. In the context of the problem, the quadratic function is used to model the number of NCAA men's college basketball teams over time, capturing the trend and variation in the number of teams with respect to years since 2000.

Differentiation

Differentiation is a fundamental tool in calculus that finds the derivative of a function. The derivative represents the instantaneous rate of change of the function with respect to its independent variable. In this problem, differentiating the polynomial provides a formula for how rapidly the number of teams is changing at any given time.

Power Rule

The power rule is a basic technique in differentiation that is used for finding the derivative of a term of the form t^n. It states that the derivative of t^n is n*t^(n-1). This rule is applied to each term in the quadratic model to compute the overall derivative.

Instantaneous Rate of Change

The instantaneous rate of change is the value of the derivative at a specific point in time. In context, evaluating the derivative at a particular year, such as 2006, provides the rate at which the number of teams was increasing at that moment.

Interpretation of the Derivative in Context

The derivative not only gives a numerical rate but also provides insight into the behavior of the model over time. By analyzing whether the derivative increases or decreases as time progresses, one can determine if the rate of change of the modeled quantity (here, the number of teams) is itself accelerating or decelerating.

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