The average credit card debt per U.S. household between...

The average credit card debt per U.S. household between $1990(t=0)$ and $2003(t=13)$ was growing at the rate of approximately $D(t)=-4.479 t^{2}+69.8 t+279.5 \quad(0 \leq t \leq 13)$ dollars/year. The average credit card debt per U.S. household stood at $$\$ 2917$$ in 1990 . a. Find an expression giving the approximate average credit card debt per U.S. household in year $t$ $(0 \leq t \leq 13) .$ b. Use the result of part (a) to estimate the average credit card debt per U.S. household in 2003 .

The average credit card debt per U.S. household between $1990(t=0)$ and $2003(t=13)$ was growing at the rate of approximately
$D(t)=-4.479 t^{2}+69.8 t+279.5 \quad(0 \leq t \leq 13)$
dollars/year. The average credit card debt per U.S. household stood at $$\$ 2917$$ in 1990 .
a. Find an expression giving the approximate average credit card debt per U.S. household in year $t$ $(0 \leq t \leq 13) .$
b. Use the result of part (a) to estimate the average credit card debt per U.S. household in 2003 .

Key Concepts

Estimation Using Function Evaluation

Once a function is derived through integration and adjusted with an initial condition, it can be evaluated at a specific point to estimate the value of the quantity of interest at that moment. This technique is widely used in forecasting and prediction in various fields.

Differential Equations in Modeling

Differential equations are fundamental tools in modeling physical, economic, or other real-world processes where a quantity changes over time. They relate rates of change to the quantities themselves, and solving these equations through integration can predict future behavior based on observed trends.

Antiderivative

An antiderivative is a function obtained by integrating a rate of change function. It represents the total accumulated change and is essentially the reverse process of differentiation. In many applications, finding the antiderivative allows one to determine the original function from which the rate of change was derived.

Initial Value Problem

An initial value problem involves finding a specific solution to a differential equation by using a known value of the function at a starting point. This process determines the constant of integration arising from the integration process, ensuring that the model accurately fits the given initial condition.

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