Infinitesimal

Definition

An infinitesimal is an extremely small quantity, one that is smaller than any finite quantity but not zero. It is a fundamental concept in calculus that allows for the precise mathematical treatment of continuous change and the notion of a limit.

5 Must Know Facts For Your Next Test

1. Infinitesimals are essential in the definition of the definite integral, which represents the accumulation of a function over an interval.
2. The definite integral is defined as the limit of a sum of infinitesimal areas under the curve of the function over the interval.
3. Infinitesimals allow for the precise mathematical treatment of continuous change, which is fundamental to the study of calculus.
4. The concept of infinitesimals was historically controversial, but it was ultimately formalized and rigorously defined using the concept of limits.
5. Infinitesimals are used in the definition and calculation of other important calculus concepts, such as derivatives and integrals.

Review Questions

• Explain how infinitesimals are used in the definition of the definite integral.
  • The definite integral is defined as the limit of a sum of infinitesimal areas under the curve of the function over the interval. Infinitesimals allow for the precise mathematical treatment of the accumulation of the function, where the interval is divided into an infinite number of infinitesimally small subintervals, and the sum of the areas of these infinitesimal subintervals approaches the total area under the curve as the subintervals become smaller and smaller.
• Describe the role of infinitesimals in the concept of limits and their importance in calculus.
  • Infinitesimals are fundamental to the concept of limits, which is a central idea in calculus. Limits allow for the precise mathematical treatment of continuous change, and infinitesimals are used to define and analyze limits. The ability to work with infinitesimally small quantities enables calculus to model and describe the behavior of functions and their rates of change, which is essential for understanding concepts like derivatives and integrals.
• Analyze how the use of infinitesimals in calculus has evolved over time and how it has contributed to the development of the field.
  • The concept of infinitesimals was historically controversial, as it challenged the traditional understanding of mathematical quantities. However, through the work of mathematicians like Cauchy and Weierstrass, infinitesimals were ultimately formalized and rigorously defined using the concept of limits. This allowed for the development of a sound mathematical foundation for calculus, enabling the precise treatment of continuous change and the analysis of functions and their properties. The incorporation of infinitesimals has been a crucial step in the evolution of calculus, contributing to its power and widespread applications in various scientific and engineering disciplines.

Related terms

Limit: The limit of a function is the value that the function approaches as the input approaches a particular point. Infinitesimals are used to define and analyze limits in calculus.

Derivative: The derivative of a function measures the rate of change of the function at a particular point. Infinitesimals are used to define and calculate derivatives in calculus.

Integral: The integral of a function represents the accumulation of the function over an interval. Infinitesimals are used to define and calculate integrals in calculus.