The per capita health spending in the United States in year...

The per capita health spending in the United States in year $t$ for the years 2000 through 2011 is approximately $$C(t)=-1.1708 t^{3}+7.029 t^{2}+389.69 t+4780 \quad(0 \leq t \leq 11)$$ dollars in year $t,$ with $t=0$ corresponding to $2000 .$ a. What was the per capita health spending in $2010 ?$ b. How fast was the per capita health spending changing in $2010 ?$ Source: NHCM Foundation.

The per capita health spending in the United States in year $t$ for the years 2000 through 2011 is approximately
$$C(t)=-1.1708 t^{3}+7.029 t^{2}+389.69 t+4780 \quad(0 \leq t \leq 11)$$
dollars in year $t,$ with $t=0$ corresponding to $2000 .$
a. What was the per capita health spending in $2010 ?$
b. How fast was the per capita health spending changing in $2010 ?$
Source: NHCM Foundation.

Key Concepts

Polynomial Function Modeling

This concept involves representing real-life phenomena, such as health care spending, with algebraic expressions in the form of polynomials. In this context, the polynomial function encapsulates the trend of per capita health spending over time, enabling predictions and analyses across different years.

Function Evaluation

Evaluating a function means substituting a specific value into the function to calculate the corresponding output. Here, it involves plugging a particular year's value (translated into the variable t) into the health spending function to determine the spending amount for that year.

Derivative as an Instantaneous Rate of Change

The derivative of a function represents its instantaneous rate of change, which in real-world terms indicates how quickly the health spending is increasing or decreasing at a specific moment. This concept is crucial for understanding trends over time, such as the speed at which spending is changing in a particular year.

Differentiation of Polynomials

Differentiation is the mathematical process used to compute the derivative of a function. For polynomial functions, this involves applying the power rule to each term. This process allows for the straightforward computation of how a change in the input variable affects the output, which is key in determining the rate of change in the context of the problem.

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