# Calculus

© 2017 Kevan Hashemi, Brandeis University, BNDHEP

## Contents

Introduction
The Infinitesmal Limit
The Slope of a Graph
Loan Repayment

## Introduction

An introduction to calculus in the form of several example problems.

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## Loan Repayment

When we borrow money from a bank, we pay the bank interest, which is like rent for money. The more money we borrow, the more interest we must pay. As we pay off the loan, the amount of money we still owe the bank is what we call the principle of the loan. The interest rate is the interest we must pay per year for each dollar of principle that remains. If our principle is \$100,000 and our interest rate is 3%/yr, we owe \$3,000 interest per year. If we pay only \$3,000/yr to the bank, we will be paying off only the interest, and the principle will remain \$100,000. If we pay \$10,000 in the first year, we will pay off \$7,000 of the loan, leaving \$93,000, and the next year we will owe less interest. Suppose we want to pay off the entire loan in 10 years, making small, frequent payments. What will our total annual payment be?

We need to solve a differential equation to obtain the annual payment. In the following derivation, we imagine that we are paying continuously in infinitesmal amounts for each infinitesmal payment period.

Figure: Differential Equation and Shape of the Solution.

Equation (1) relates dP/dt to P. It is a differential equation. The differential equation itself does not tell us the value of P at time t. We assume, however, that there exists some equation that will give us the value of P in terms of t. We now guess what this equation will look like, and then test our guess to see if it satisfies the differential equation. For brevity's sake, our first guess is correct. Any other guess would result in a contradiction when we tested it against Equation (1).

Figure: Solution to the Equation and Calculation of Payment Rate.

Our value of M is the fixed annual payment rate that repays the entire loan with interest in time T. The time T is the loan term. We divide M into twelve parts to make monthly payments. The following table gives the payment rate for various interest rates, loan terms, and amount borrowed. We also give the total amount paid over the course of the loan.

Figure: Examples of Payment Rates for Various Loan Terms. Money in thousands of dollars.

In the case of a 30-year mortgage at 3%, we end up paying the bank a total of 50% more than the initial loan amount. At 10%, we pay triple the loan amount over thirty years.