Problem 34 - 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

\(C'(r)\) has units of dollars per percentage point and is usually positive.

Step 1: Understanding the function

The function given, \(C(r)\), represents the total cost of paying off a car loan based on the annual interest rate \(r\). The variable \(r\) is expressed in percentage, and it influences the total cost \(C(r)\).

Step 2: Introducing the derivative

The derivative \(C'(r)\) tells us how the total cost \(C\) changes with respect to a change in interest rate \(r\). Essentially, it measures the rate of change of cost with respect to the rate of interest.

Step 3: Determining the units of the derivative

The units of \(C'(r)\) are determined by taking the units of the numerator from \(C(r)\) which is in dollars (cost) and dividing by the units of the denominator \(r\) which is in percentage. Thus, the units of \(C'(r)\) are dollars per percentage point.

Step 4: Analyzing the practical meaning of the derivative

The practical meaning of \(C'(r)\) is that it indicates how much the total cost of the car loan will increase (or decrease) for each one percentage point increase in the interest rate. It helps in understanding the sensitivity of the cost to changes in the interest rate.

Step 5: Determining the sign of the derivative

Typically, as interest rates increase, it causes the overall cost of paying off a loan to increase as well. Thus, \(C'(r)\) is generally positive, representing an increase in total cost with an increase in interest rate.

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. For functions involving variables such as interest rate and cost, the derivative gives insight into how one quantity affects another. When we talk about the derivative of a cost function, denoted as \(C'(r)\), it specifically tells us how the cost of a car loan changes as the interest rate, \(r\), changes.
To compute a derivative, we observe the change in the function's value (here, the cost \(C\)) with a small change in its input (the interest rate \(r\)). This is mathematically expressed as the limit of the average rate of change as the change in \(r\) approaches zero. The primary purpose of evaluating a derivative in this context is to understand the loan's cost sensitivity to different interest rates.

• Measures rate of change: How fast the cost function responds to interest rate changes.
• Units of measure: Here, units are dollars per percentage point, indicating cost change per each 1% interest rate change.

In practical terms, knowing \(C'(r)\) equips a borrower with the knowledge of what to expect financially as market interest rates fluctuate.

Interest Rate

An interest rate is a crucial factor in financial calculations, especially when dealing with loans and investments. It represents the cost of borrowing money or the reward for saving. In the context of this exercise, interest rate \(r\) is expressed as a percentage. It directly impacts the total cost \(C(r)\) of a loan. The relationship between interest rates and loan costs is usually positive, meaning that as interest rates rise, the total cost of borrowing increases. This is because interest is part of the repayment amount on the original loan balance.Understanding how changes in interest rates affect your financial obligations can help you make more informed decisions.

• Influences loan repayment amounts over time.
• Typically quoted yearly, affecting monthly payments.
• Higher rates lead to higher costs if all other factors remain constant.

Borrowers should be attentive to interest rates to manage their financial commitments better.

Cost Analysis

Cost analysis in finance involves evaluating the total cost incurred over an investment or a loan. It's a vital process in decision-making, especially when choosing among different financing options. For our discussion, \(C(r)\), the cost function, embodies the total expenses associated with a car loan based on the interest rate.By examining \(C(r)\), and intuitively understanding its derivative \(C'(r)\), borrowers can grasp how sensitive the cost is to changes in the interest rate.
This sensitivity analysis is crucial for budgeting and financial planning. If \(C'(r)\) is positive, then any increase in interest rates will lead to increased loan costs. This might affect spending and saving strategies.

• Helps in forecasting future financial requirements.
• Supports identifying cost-effective loan options.
• Aids in understanding the economic impact of interest rate adjustments.

Drilling into cost analysis allows borrowers to foresee potential financial challenges and opportunities, optimizing loan management.