Problem 9 - 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

Units: dollars/%. Meaning: Cost change per % interest change. Sign: Positive.

Step 1: Identify the Function and Derivative

We are given the function \(C(r)\), which represents the total cost of paying off a car loan with interest at rate \(r\). \(C'(r)\) is the derivative of \(C(r)\) with respect to \(r\), representing the rate at which the total cost changes as the interest rate changes.

Step 2: Determine the Units of C'(r)

\(C(r)\) is measured in monetary units (e.g., dollars), and \(r\) is measured as a percentage. Thus, \(C'(r)\) has units of dollars per percentage (%).

Step 3: Interpret the Practical Meaning of C'(r)

The derivative \(C'(r)\) indicates how the total cost of the loan changes with a change in interest rate. Practically, it tells us how much the cost will increase or decrease for each percentage point change in the interest rate.

Step 4: Determine the Sign of C'(r)

Generally, the cost of a loan increases with a higher interest rate, implying \(C(r)\) is an increasing function. Therefore, \(C'(r)\) is expected to be positive, as costs typically rise with increasing interest rates.

Key Concepts

Interest Rate

The term "interest rate" refers to the percentage charged by a lender to a borrower for the use of money. Interest rates are typically expressed as annual percentages. In the context of loans, the interest rate is crucial as it directly influences the total cost of borrowing. A higher interest rate means a higher cost for the borrower. Therefore, understanding interest rates is vital for financial planning and decision-making.

• Interest affects the amount of money to be repaid over the life of a loan.
• It's essential to compare different loan offers based on their interest rates.

Borrowers should be aware that even a small percentage increase in the interest rate can significantly impact the total loan cost. Consideration of current interest trends and predictions can further guide whether it is preferable to lock in a rate or wait for a better opportunity.

Monetary Units

Monetary units are the standard of money used to quantify the cost or value of economic transactions. Commonly, these units are in a currency such as dollars, euros, or yen. In the context of a loan, monetary units communicate the amount a borrower is expected to repay.

• Monetary units allow for the clear communication of financial obligations.
• They help individuals assess the affordability of loans and payments.

It's crucial to understand how monetary units relate to both personal and loan finances. They give borrowers a tangible understanding of their repayments, helping them to accordingly budget and manage their financial responsibilities.

Rate of Change

The rate of change in calculus describes how a quantity changes in response to a change in another quantity. In the context of our exercise, it specifically refers to how the cost of a car loan changes as the interest rate changes. Mathematically, it is represented as a derivative, in this case, noted by \( C'(r) \).

• This derivative represents the sensitivity of the loan cost to changes in interest rate.
• If \( C'(r) \) is positive, it indicates that an increase in the interest rate will increase the cost.

Rate of change provides insights into dynamic relationships in financial contexts. Being able to interpret this can help both lenders and borrowers predict how fluctuations in interest rates might affect loan costs.

Loan Cost Analysis

Loan cost analysis involves examining the complete financial implications of taking out a loan. It includes understanding not only the principal amount and interest rate but also how changes in these factors affect the total repayment amount. This analysis often employs derivatives like \( C'(r) \) to understand the impact of varying interest rates.

• Analyzing loan costs aids in comparing different loan offers on the market.
• It allows borrowers to make more informed decisions regarding which loans to pursue.

In essence, thorough loan cost analysis can prevent borrowers from incurring unexpected costs due to interest rate changes. It empowers individuals with the foresight needed to manage their financial future effectively.