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Chapter 9: Problem 13

The monthly payments, \(P\) dollars, on a mortgage in which \(A\) dollars were borrowed at an annual interest rate of \(r \%\) for \(t\) years is given by \(P=f(A, r, t)\). Is \(f\) an increasing or decreasing function of \(A\) ? Of \(r\) ? Of \(t\) ?

Short Answer

Expert verified
\( f \) is increasing in \( A \) and \( r \), but decreasing in \( t \).

Step by step solution

01

Understanding the Mortgage Formula

The monthly payment, \( P \), on a mortgage is determined by the formula \( P = f(A, r, t) \). This function depends on three variables: the amount borrowed (\( A \)), the annual interest rate (\( r \)), and the number of years (\( t \)) over which the loan is to be repaid.
02

Analyzing \( P \) as a Function of \( A \)

When the amount borrowed \( A \) increases, the total amount to be repaid also increases. Since the monthly payment \( P \) is directly related to \( A \), \( P \) is an increasing function of \( A \). This means as \( A \) increases, \( P \) also increases.
03

Analyzing \( P \) as a Function of \( r \)

As the interest rate \( r \) increases, the cost of borrowing the same amount \( A \) increases. Hence, the monthly payment \( P \) will also increase. Therefore, \( P \) is an increasing function of \( r \).
04

Analyzing \( P \) as a Function of \( t \)

When the loan term \( t \) increases, the amount borrowed is spread out over a longer period, generally resulting in smaller monthly payments. However, the interest paid over a longer term increases. From the borrower鈥檚 perspective, \( P \) generally decreases as \( t \) increases, making \( P \) a decreasing function of \( t \) for fixed \( A \) and \( r \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mortgage Payment Formula
The mortgage payment formula is a crucial tool for anyone entering the world of home loans and financing. It serves to calculate the monthly payment, denoted by \( P \), required to pay off a mortgage. This payment depends on several factors, namely the amount borrowed (\( A \)), the annual interest rate (\( r \)), and the loan term (\( t \)), which is the duration over which the loan is to be repaid. The general form of this formula can be written as \( P = f(A, r, t) \).
When applying this formula, it's important to keep these variables in mind:
  • Amount Borrowed (\( A \)): This is the principal amount of the loan. The larger \( A \) is, the higher the monthly payment will generally be.
  • Interest Rate (\( r \)): Expressed as a percentage, this rate determines how much additional money you'll pay back on top of the principal.
  • Loan Term (\( t \)): Measured in years, it's the period over which you agree to repay the loan.
Increasing and Decreasing Functions
In calculus, determining whether a function is increasing or decreasing is vital to understanding how the variables affect outcomes. For the mortgage payment function \( P = f(A, r, t) \), we must understand how changes in \( A \), \( r \), and \( t \) impact \( P \).
  • Function of \( A \): P is an increasing function of \( A \). This means that as more money is borrowed, the monthly payment increases. This correlation is direct because the principal amount is being amortized over the loan term.
  • Function of \( r \): Similarly, \( P \) is an increasing function of \( r \). Higher interest rates increase the cost of borrowing and, consequently, the monthly payment.
  • Function of \( t \): In contrast, \( P \) is a decreasing function of \( t \). Extending the repayment period generally lowers monthly payments, as they are spread out over a longer time. However, this can lead to more total interest paid over time.
Interest Rate Impact
Interest rates play a significant role in calculating mortgage payments. They determine how much you pay on top of your loan's principal. Here's why interest rates are crucial:
  • Cost of Borrowing: Higher rates mean higher payments because the lender charges more for providing the loan. This effect makes \( P \) an increasing function of \( r \).
  • Variable Rates: Be aware that some mortgages have variable rates, which can change. This potential variation can lead to increased costs over time.
  • Long-Term Impact: Even small changes in interest rates can have a significant impact due to the compounding effect over the loan term.
Managing and understanding interest rates can save borrowers a lot of money in the long run. It pays to shop around for the best rate.
Loan Term
The loan term is the period over which the mortgage is repaid. It has a direct impact on the size of the monthly payment and the total interest paid.
  • Longer Loan Terms: These often come with lower monthly payments because the principal is spread over more months. However, they usually result in higher total interest payments over the life of the loan.
  • Shorter Loan Terms: These generally lead to higher monthly payments but less interest paid in total. If you can afford higher payments, this option can save money in the long run.
Balancing between a comfortable monthly payment and minimizing total interest paid is key to choosing the right loan term. Calculating different scenarios using the mortgage payment formula can help in making the best decision for your financial situation.

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Most popular questions from this chapter

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(\frac{\partial P}{\partial r}\) if \(P=100 e^{r t}\)

The quantity \(Q\) (in pounds) of beef that a certain community buys during a week is a function \(Q=f(b, c)\) of the prices of beef, \(b\), and chicken, \(c\), during the week. Do you expect \(\partial Q / \partial b\) to be positive or negative? What about \(\partial Q / \partial c\) ?

People commuting to a city can choose to go either by bus or by train. The number of people who choose either method depends in part upon the price of each. Let \(f\left(P_{1}, P_{2}\right)\) be the number of people who take the bus when \(P_{1}\) is the price of a bus ride and \(P_{2}\) is the price of a train ride. What can you say about the signs of \(\partial f / \partial P_{1}\) and \(\partial f / \partial P_{2}\) ? Explain your answers.

The cardiac output, represented by \(c\), is the volume of blood flowing through a person's heart, per unit time. The systemic vascular resistance (SVR), represented by \(s\), is the resistance to blood flowing through veins and arteries. Let \(p\) be a person's blood pressure. Then \(p\) is a function of \(c\) and \(s\), so \(p=f(c, s)\). (a) What does \(\partial p / \partial c\) represent? Suppose now that \(p=k c s\), where \(k\) is a constant. (b) Sketch the level curves of \(p\). What do they represent? Label your axes. (c) For a person with a weak heart, it is desirable to have the heart pumping against less resistance, while maintaining the same blood pressure. Such a person may be given the drug nitroglycerine to decrease the SVR and the drug Dopamine to increase the cardiac output. Represent this on a graph showing level curves. Put a point \(A\) on the graph representing the person's state before drugs are given and a point \(B\) for after. (d) Right after a heart attack, a patient's cardiac output drops, thereby causing the blood pressure to drop. A common mistake made by medical residents is to get the patient's blood pressure back to normal by using drugs to increase the SVR, rather than by increasing the cardiac output. On a graph of the level curves of \(p\), put a point \(D\) representing the patient before the heart attack, a point \(E\) representing the patient right after the heart attack, and a third point \(F\) representing the patient after the resident has given the drugs to increase the SVR.

If \(f(u, v)=5 u v^{2}\), find \(f(3,1), f_{u}(3,1)\), and \(f_{v}(3,1)\).
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