Problem 35 - Applied Calculus

Question

Suppose \(C(r)\) is the total cost of paying off a car loan borrowed at an annual interest rate of \(r \% .\) What are the units of \(C^{\prime}(r) ?\) What is the practical meaning of \(C^{\prime}(r) ?\) What is its sign?

Step-by-Step Solution

Answer

The units of \( C^{\prime}(r) \) are dollars per percent. It shows the sensitivity of cost to changes in interest rate, typically positive.

Step 1: Understanding the Function and Its Derivative

The function given is \( C(r) \), where \( r \) is the annual interest rate in percentage. The function \( C(r) \) represents the total cost of paying off a car loan at this interest rate. The derivative of this function, \( C^{\prime}(r) \), represents the rate of change of the total cost with respect to changes in the interest rate \( r \).

Step 2: Determine the Units of the Derivative

The units of the function \( C(r) \) are in dollars, as it represents a cost. The units of the variable \( r \) are in percentage, since it represents an interest rate. When taking the derivative \( C^{\prime}(r) \), the result will have units of dollars per percentage point. Thus, the units of \( C^{\prime}(r) \) are dollars per percent.

Step 3: Practical Meaning of the Derivative

The practical meaning of \( C^{\prime}(r) \) is that it describes how much the total cost of the loan, \( C(r) \), changes when the interest rate \( r \) changes by one percentage point. In other words, \( C^{\prime}(r) \) provides the sensitivity of the loan cost to changes in the interest rate.

Step 4: Determine the Sign of the Derivative

Since increasing the interest rate \( r \) generally increases the total cost \( C(r) \) of a loan, we expect \( C^{\prime}(r) \) to be positive. This positive sign indicates that as the interest rate goes up, the total cost of the loan also increases.

Key Concepts

Cost Function

A cost function, like our given example of \( C(r) \), captures how a cost varies with a particular factor—in this case, the interest rate \( r \). For loans, the cost function expresses the total amount you will pay for the loan when factors like time and interest rate are considered.
Understanding how this function behaves can help borrowers make informed financial decisions. By knowing \( C(r) \), you grasp how much you will be repaying as the interest rate changes. This function is expressed in terms of dollars since it represents the total financial outlay.
In practical terms, the cost function can help you plan your budget and evaluate different loan options. It's essentially the blueprint of what your loan will cost over its life and allows you to compare different interest rates effectively.

Interest Rate

Interest rate \( r \) is a crucial variable in finance, representing the cost of borrowing money. When looking at a car loan, the interest rate tells you how much extra you will pay each year on the borrowed amount.
The interest rate is often given in percentages, linking directly to the cost function \( C(r) \). Changes in \( r \) affect the total cost \( C \) due to how interest accumulates over time. It's like a lever that can significantly increase or decrease your total repayment based on its value.
It becomes important to monitor the interest rate when assessing loan offers. Small changes in \( r \) can lead to significant differences in overall cost, as revealed by the derivative \( C^{\prime}(r) \). Understanding this concept can save you money and stress.

Economic Interpretation

The derivative \( C^{\prime}(r) \) provides valuable insights into the economic impact of interest rate changes on loan costs. This derivative measures the sensitivity of the cost function \( C(r) \) to changes in the interest rate \( r \).
In simple terms, \( C^{\prime}(r) \) tells us how much the total cost changes when the interest rate increases by one percent. The unit of measurement is dollars per percent, which clarifies the cost fluctuation for each percentage change in \( r \).
This economic interpretation underlines the significance of negotiating for better rates. A positive \( C^{\prime}(r) \) signifies that costs rise with increasing rates, highlighting the need to keep interest rates as low as possible for financial efficiency. By understanding this, borrowers can aim for strategies that minimize costs related to their loans.