Unit II: Lec 2 | MIT Calculus Revisited: Single Variable Calculus

The script begins with an introduction to the concept of infinitesimals, a fundamental idea in calculus, and discusses the importance of approximations. It sets the stage for the lecture by referencing a common problem in algebra involving distance, rate, and time, and how calculus extends this concept. The professor emphasizes the need to understand the rate of change when it's not constant, which is where infinitesimals come into play. A geometric interpretation of the tangent line and its slope as a derivative is introduced, highlighting the difference between 'delta y' (change in y along the curve) and 'delta y-tan' (change along the tangent line).

This paragraph delves deeper into the approximation of 'delta y' using the tangent line. It uses the specific example of cubing the number 4.01 and compares the actual value to the value obtained by approximating along the tangent line at x=4. The professor illustrates how the tangent line serves as a good approximation near the point of tangency and explains the mathematical process behind this approximation, including the calculation of the derivative and its role in estimating 'delta y-tan'.

The script continues by examining the error involved in using the tangent line as an approximation for the curve. It discusses the binomial theorem and how it can be used to expand and understand the difference between 'delta y' and 'delta y-tan'. The concept of an infinitesimal is introduced, explaining that the error term 'k delta x' is an infinitesimal because it approaches zero faster than 'delta x' as 'delta x' gets smaller.

This section presents a more abstract and mathematical discussion of the approximation error. It introduces the idea of taking the limit of the difference between the slope of the tangent line and the average change of 'y' with respect to 'x' along the curve, and how this leads to the definition of an infinitesimal. The professor emphasizes the importance of understanding the rate at which the error term approaches zero.

The script explains the profound significance of infinitesimals in calculus, particularly in the context of the 0/0 indeterminate form. It discusses how the error term 'k delta x', being an infinitesimal, allows for the simplification of expressions involving derivatives when taking limits. The professor uses the example of differentiating a composite function to illustrate how the error term drops out, leaving a clean expression for the derivative.

This paragraph introduces the concept of differentials, 'dy' and 'dx', as a way to formalize the language of infinitesimals. The professor explains the notation 'dy/dx' and how it can be thought of as a fraction, with 'dy' representing the change in 'y' along the tangent line and 'dx' representing the change in 'x'. The script also humorously touches on the idea of canceling common factors in fractions and the importance of not misapplying this concept in calculus.

The final paragraph wraps up the lecture by summarizing the key points discussed about infinitesimals and approximations. It emphasizes the importance of understanding the underlying concepts rather than just the approximations themselves. The professor also previews the next lecture, which will delve into the derivative of composite functions and the role of differentials in calculus. The script concludes with acknowledgment of funding and a call to support MIT OpenCourseWare.