Problem 75
Question
Consumer credit. The total outstanding consumer credit of the United States (in billions of dollars) can be modeled by the function $$C(x)=9.26 x^{4}-85.27 x^{3}+287.24 x^{2}-309.12 x+2651.4$$ where \(x\) is the number of years since 2008 . 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 2014 .
Step-by-Step Solution
Verified Answer
In 2014, consumer credit was decreasing at approximately -74.76 billion dollars per year.
1Step 1: Differentiate the Function
To find \( \frac{dC}{dx} \), we need to take the derivative of the function \( C(x) = 9.26x^4 - 85.27x^3 + 287.24x^2 - 309.12x + 2651.4 \). Use the power rule for differentiation, which states that the derivative of \( ax^n \) is \( nax^{n-1} \).Differentiating each term:- The derivative of \( 9.26x^4 \) is \( 4 \cdot 9.26x^{3} = 37.04x^3 \)- The derivative of \( -85.27x^3 \) is \( 3 \cdot -85.27x^{2} = -255.81x^2 \)- The derivative of \( 287.24x^2 \) is \( 2 \cdot 287.24x^{1} = 574.48x \)- The derivative of \( -309.12x \) is \( -309.12 \)- The derivative of the constant \( 2651.4 \) is \( 0 \)So, \( \frac{dC}{dx} = 37.04x^3 - 255.81x^2 + 574.48x - 309.12 \).
2Step 2: Interpret the Derivative
The derivative \( \frac{dC}{dx} \) represents the rate of change of the consumer credit \( C \), measured in billions of dollars, with respect to time \( x \), measured in years since 2008. This tells us how quickly the consumer credit is changing at any particular year.
3Step 3: Calculate for 2014
To find out how quickly outstanding consumer credit was rising in 2014, substitute \( x = 6 \) into the equation \( \frac{dC}{dx} \). This is because 2014 is 6 years after 2008.Using the derivative found in Step 1:\[\frac{dC}{dx} = 37.04(6)^3 - 255.81(6)^2 + 574.48(6) - 309.12\]- Calculate \( 37.04 \times 216 = 7996.64 \)- Calculate \( -255.81 \times 36 = -9209.16 \)- Calculate \( 574.48 \times 6 = 3446.88 \)Plug these into the equation:\[\frac{dC}{dx} = 7996.64 - 9209.16 + 3446.88 - 309.12 = -74.76\]Thus, in 2014, the outstanding consumer credit was decreasing at approximately \(-74.76\) billion dollars per year.
Key Concepts
DifferentiationRate of ChangePolynomial Derivatives
Differentiation
Differentiation is a fundamental concept in calculus, mainly concerned with understanding how functions change. It involves finding the derivative of a function, which is the function that describes the rate of change.
To differentiate a given function like the one in our exercise, we apply rules such as the power rule. The power rule states that if you have a term in the form of \( ax^n \), the derivative is \( nax^{n-1} \). For example, if you have \( 9.26x^4 \), differentiating it yields \( 37.04x^3 \), because you multiply the exponent with the coefficient and subtract one from the exponent.
Differentiation helps us understand not just where a function's slope is, but also how steep the slope is at any given point, which is essential for interpreting real-world data, as seen with consumer credit.
To differentiate a given function like the one in our exercise, we apply rules such as the power rule. The power rule states that if you have a term in the form of \( ax^n \), the derivative is \( nax^{n-1} \). For example, if you have \( 9.26x^4 \), differentiating it yields \( 37.04x^3 \), because you multiply the exponent with the coefficient and subtract one from the exponent.
Differentiation helps us understand not just where a function's slope is, but also how steep the slope is at any given point, which is essential for interpreting real-world data, as seen with consumer credit.
Rate of Change
The rate of change is one of the primary reasons to differentiate a function. For our consumer credit function, the derivative \( \frac{dC}{dx} \) gives us how the consumer credit changes year over year.
This derivative tells us if the total outstanding consumer credit is increasing or decreasing and by how much, which is measured in billions of dollars in our problem statement. If \( \frac{dC}{dx} \) is positive, credit is rising – negative means it's falling.
Understanding the rate of change is crucial, as it helps us make predictions and informed decisions based on trends indicated by the model. For instance, in 2014, as calculated, the credit was surprisingly decreasing, showcasing the model's importance in forecasting economic trends.
This derivative tells us if the total outstanding consumer credit is increasing or decreasing and by how much, which is measured in billions of dollars in our problem statement. If \( \frac{dC}{dx} \) is positive, credit is rising – negative means it's falling.
Understanding the rate of change is crucial, as it helps us make predictions and informed decisions based on trends indicated by the model. For instance, in 2014, as calculated, the credit was surprisingly decreasing, showcasing the model's importance in forecasting economic trends.
Polynomial Derivatives
Polynomial derivatives are a specific application of differentiation on polynomial functions. These functions, like our consumer credit function, consist of several terms with varying powers of \( x \). Each term in a polynomial can be individually differentiated using the power rule.
The resulting derivative, \( 37.04x^3 - 255.81x^2 + 574.48x - 309.12 \), is a new polynomial that provides the slope at any given point along the curve described by the original function.
Polynomial derivatives are straightforward compared to other functions because the rules are consistent and systematic. Each term follows the same differentiation rules, allowing us to deal with complex real-world models, such as national consumer credit, efficiently. Understanding these helps us analyze trends and make sense of changes over time in various applications.
The resulting derivative, \( 37.04x^3 - 255.81x^2 + 574.48x - 309.12 \), is a new polynomial that provides the slope at any given point along the curve described by the original function.
Polynomial derivatives are straightforward compared to other functions because the rules are consistent and systematic. Each term follows the same differentiation rules, allowing us to deal with complex real-world models, such as national consumer credit, efficiently. Understanding these helps us analyze trends and make sense of changes over time in various applications.
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