This chapter introduces the unit step function (or Heaviside function) and the Dirac delta function so that students will be comfortable with these functions before the Laplace transform is introduced in Chap. 11 . This is to avoid students having to grasp and assimilate new information while at the same time learning about Laplace transform methods. Moreover, it is a continuation of the modeling in Chap. 6 of the balance of a deposit account, such as a savings or money market account, where the accrued interest is credited to the account periodically. It is shown how to set up a one-compartment model of a loan, such as a car loan or a home mortgage, in order to model the amortization of the loan with a differential equation using the Dirac delta function. The chapter begins with the narrative of a first-time gambler who has just arrived at a casino to try her luck at the slot machines. She has access to only $100 while at the casino. A function called $x(t)$ is defined to be her “net worth” at time t, where $t=0$ represents the moment she sits down at a slot machine to play. Thus, $x(0)=100$ dollars. It is shown how to express $x(t)$ in terms of unit step functions to reflect the abrupt changes in her net worth due to any winnings and losses while gambling. Other examples are also presented illustrating how unit step functions can be used to model abrupt changes, which are inherent features of sawtooth voltage pulses, impulsive forces, rectangular wave functions, piecewise-defined functions, just to name a few. The casino scenario is also used to introduce the Dirac delta function. This is achieved by formally differentiating the net worth function $x(t)$ , which results in a differential equation with terms involving derivatives of the unit step function. It is shown that if the derivative of the unit step function is defined to have certain properties, such as the sifting property, then the net worth function can be recovered from the differential equation. This derivative is known as the Dirac delta function. Despite its name, the Dirac delta function is not really a function—at least not in the classical sense. Nonetheless, because of its unusual properties, it can be used to model situations involving abrupt changes, such as the impulsive force acting on a pitched baseball when it comes into contact with a bat and the sudden drops or increases in the population of an organism.

Modeling Abrupt Changes

章节
发表日期：

2026年

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Leigh C. Becker

摘要

This chapter introduces the unit step function (or Heaviside function) and the Dirac delta function so that students will be comfortable with these functions before the Laplace transform is introduced in Chap. 11 . This is to avoid students having to grasp and assimilate new information while at the same time learning about Laplace transform methods. Moreover, it is a continuation of the modeling in Chap. 6 of the balance of a deposit account, such as a savings or money market account, where the accrued interest is credited to the account periodically. It is shown how to set up a one-compartment model of a loan, such as a car loan or a home mortgage, in order to model the amortization of the loan with a differential equation using the Dirac delta function. The chapter begins with the narrative of a first-time gambler who has just arrived at a casino to try her luck at the slot machines. She has access to only $100 while at the casino. A function called $x(t)$ is defined to be her “net worth” at time t, where $t=0$ represents the moment she sits down at a slot machine to play. Thus, $x(0)=100$ dollars. It is shown how to express $x(t)$ in terms of unit step functions to reflect the abrupt changes in her net worth due to any winnings and losses while gambling. Other examples are also presented illustrating how unit step functions can be used to model abrupt changes, which are inherent features of sawtooth voltage pulses, impulsive forces, rectangular wave functions, piecewise-defined functions, just to name a few. The casino scenario is also used to introduce the Dirac delta function. This is achieved by formally differentiating the net worth function $x(t)$ , which results in a differential equation with terms involving derivatives of the unit step function. It is shown that if the derivative of the unit step function is defined to have certain properties, such as the sifting property, then the net worth function can be recovered from the differential equation. This derivative is known as the Dirac delta function. Despite its name, the Dirac delta function is not really a function—at least not in the classical sense. Nonetheless, because of its unusual properties, it can be used to model situations involving abrupt changes, such as the impulsive force acting on a pitched baseball when it comes into contact with a bat and the sudden drops or increases in the population of an organism.