Infinitesimal

Definition

An infinitesimal is a quantity that is extremely small, approaching zero but never quite reaching it. It plays a crucial role in calculus and analysis, particularly in the development of limits, derivatives, and integrals. Infinitesimals provide a way to rigorously handle concepts of continuity and instantaneous change, which are fundamental to understanding rates of change in mathematical functions.

5 Must Know Facts For Your Next Test

1. Infinitesimals were originally used by Newton and Leibniz as foundational concepts in their respective calculus formulations, leading to the development of their theories of motion.
2. In the context of Newton's fluxions, infinitesimals are used to describe instantaneous rates of change, highlighting how quantities evolve over time.
3. Leibniz's differentials utilize infinitesimals to express changes in variables, allowing for the derivation of rules for integration and differentiation.
4. The formalization of infinitesimals faced criticism due to its non-rigorous nature until the introduction of non-standard analysis in the 20th century provided a solid mathematical framework.
5. Infinitesimals help bridge the gap between algebra and calculus by enabling mathematicians to understand concepts like area under curves and slopes at points on graphs.

Review Questions

• How do infinitesimals contribute to understanding instantaneous rates of change in calculus?
  • Infinitesimals allow mathematicians to conceptualize instantaneous rates of change by considering the behavior of functions at an infinitely small scale. When examining a function's behavior as it approaches a point, infinitesimals help define derivatives by capturing how much a function changes with an infinitesimally small change in input. This approach provides a rigorous foundation for analyzing motion and dynamics in calculus.
• Discuss how Newton's fluxions and Leibniz's differentials differ in their treatment of infinitesimals.
  • Newton's fluxions focus on the concept of instantaneous velocity as derived from infinitesimal changes in position over time. His approach treats these small changes as fundamental to understanding motion. In contrast, Leibniz's differentials emphasize the relationship between differentials of variables, providing a systematic method for integrating and differentiating functions. While both utilize infinitesimals, their frameworks reflect different perspectives on motion and change.
• Evaluate the significance of the development and acceptance of non-standard analysis for the rigorous treatment of infinitesimals.
  • The acceptance of non-standard analysis marked a pivotal moment for the treatment of infinitesimals in mathematics. It provided a rigorous framework that legitimized the use of infinitesimals by establishing their properties within an axiomatic structure. This development helped resolve long-standing debates about the validity of infinitesimals, allowing mathematicians to confidently apply them in calculus and related fields, ultimately enriching mathematical analysis.

Related terms

Limit: A limit is a value that a function approaches as the input approaches some value, fundamental in defining derivatives and integrals.

Derivative: A derivative represents the rate of change of a function with respect to a variable, typically defined as the limit of the difference quotient.

Differential: A differential is an infinitesimal change in a variable, often used to represent how a function changes with respect to changes in its input.