Consumer credit. The total outstanding consumer credit of...

Consumer credit. The total outstanding consumer credit of the United States (in billions of dollars) can be modeled by the function $C(x)=0.21 x^{4}-5.92 x^{3}+50.53 x^{2}-18.92 x+1114.93$ where $x$ is the number of years since 1995 (figure can't copy) a) Find $d C / d x$ b) Interpret the meaning of $d C / d x$. c) Using this model, estimate how quickly outstanding consumer credit was rising in 2010

Consumer credit. The total outstanding consumer credit of the United States (in billions of dollars) can be modeled by the function $C(x)=0.21 x^{4}-5.92 x^{3}+50.53 x^{2}-18.92 x+1114.93$ where $x$ is the number of years since 1995 (figure can't copy)
a) Find $d C / d x$
b) Interpret the meaning of $d C / d x$.
c) Using this model, estimate how quickly outstanding consumer credit was rising in 2010

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which mathematically represents how the function's value changes with respect to changes in its input. In polynomial functions, differentiation is performed term by term using power rules, providing a new function that describes the slope of the original function at any given point.

Instantaneous Rate of Change

The derivative of a function at a specific point is interpreted as the instantaneous rate of change of the quantity being modeled. In the context of consumer credit, it measures how rapidly consumer credit is increasing or decreasing at a particular time. This is a critical concept because it translates mathematical output into real-world behavior, indicating trends such as growth or decline.

Polynomial Function Modeling

This concept involves representing real-world quantities with polynomial functions. In contexts like consumer credit trends, a polynomial can approximate the behavior over time by fitting curves that capture the underlying complexity of the data. It helps in predicting future values based on historical data and understanding the dynamics of change.

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