The average annual medical costs for a family of four in the...

The average annual medical costs for a family of four in the United States from 2001 through 2011 is approximated by the function $$ C(t)=22.9883 t^{2}+830.358 t+7513 \quad(1 \leq t \leq 11) $$ where $t$ is measured in years, with $t=1$ corresponding to 2001 , and $C(t)$ is measured in dollars. a. What was the approximate average annual medical costs for a family of four in $2010 ?$ b. How fast was the approximate average medical costs for a family of four increasing in $2010 ?$

The average annual medical costs for a family of four in the United States from 2001 through 2011 is approximated by the function
$$
C(t)=22.9883 t^{2}+830.358 t+7513 \quad(1 \leq t \leq 11)
$$
where $t$ is measured in years, with $t=1$ corresponding to 2001 , and $C(t)$ is measured in dollars.
a. What was the approximate average annual medical costs for a family of four in $2010 ?$
b. How fast was the approximate average medical costs for a family of four increasing in $2010 ?$

Key Concepts

Function Evaluation

This concept involves substituting a specific value into a function to determine its output at that point. In real-world models, such as cost functions or population models, evaluating the function at a given input (e.g., a particular year) yields the approximate value of the quantity being modeled at that specific time.

Polynomial Functions

Polynomial functions, particularly quadratic functions in this context, are used to model behaviors that are not linear. Their characteristics, such as curvature and turning points, make them useful for approximating trends over time in various applications, such as economic or medical cost forecasting.

Differentiation

Differentiation is the process of finding the derivative of a function, which provides a mathematical description of the rate at which the function's output changes with respect to changes in its input. This is fundamental in understanding trends, such as how quickly medical costs are increasing at a given time.

Interpreting Derivatives in Real-World Contexts

This concept involves using the derivative to interpret the rate of change of a modeled quantity in practical terms. For example, in a cost function the derivative tells us how fast costs are rising or falling per unit time, thereby providing valuable insight into trends and enabling prediction and analysis of future behavior.

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