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Derivatives are a fundamental concept in calculus, serving as tools to compute the rate at which a function is changing. To calculate the derivative of a polynomial function such as \( f(x) = \frac{1}{4} x^2 + 5x + 6 \), we apply basic differentiation rules. This process involves:

• Deriving the power rule, which states the derivative of \( x^n \) is \( nx^{n-1} \).
• Applying it to each term separately.

For instance, to find the derivative \( f'(x) \):

• The derivative of \( \frac{1}{4} x^2 \) is \( \frac{1}{2} x \) because we multiply the power 2 by the coefficient \( \frac{1}{4} \) and decrease the power by 1.
• The derivative of \( 5x \) is 5 since the power of \( x \) is 1.
• Constant terms like 6 have a derivative of 0.

By adding these results, we obtain the derivative \( f'(x) = \frac{1}{2} x + 5 \). This tells us how the percentage of U.S. households with broadband access changes with time.

The concept of rate of change is essential when analyzing how a quantity evolves over time. In this context, we focus on the rate at which the percentage of U.S. households with broadband access changes annually. Mathematically, this is described by the function's derivative.

The derivative \( f'(x) \) provides a direct measure of this rate. For example, by substituting a specific year after 2000 into \( f'(x) = \frac{1}{2} x + 5 \), we can determine how quickly the percentage is increasing within that year. When for the year 2010 (where \( x = 10 \)) the calculation \( f'(10) = 10 \) indicates:

• The percentage is increasing by 10% per year.

Understanding rates of change helps in predicting trends and making informed decisions, especially in technological advancements and market analysis.

Polynomial functions are expressions composed of variables and coefficients, involving terms like \( x^n \). The structure of polynomial functions can vary from simple linear equations to complex higher-degree polynomials.In the given exercise, the function \( f(x) = \frac{1}{4} x^2 + 5x + 6 \) is a quadratic polynomial. Quadratics are second-degree polynomials characterized by their parabolic graphs.

Key features of polynomial functions include:

• Degree: The highest power of the variable (here, 2).
• Coefficients: Numbers multiplying the variables (\( \frac{1}{4} \) for \( x^2 \), 5 for \( x \)).
• Constant term: A standalone number (6), affecting the vertical shift of the graph.

Polynomial functions like these offer a smooth and continuous representation of real-world phenomena—e.g., modeling internet access trends as shown. Leveraging this understanding is vital for forecasting and strategic planning.