Unit V: Lec 1 | MIT Calculus Revisited: Single Variable Calculus

The professor begins by acknowledging MIT OpenCourseWare's commitment to providing free educational resources, supported by donations. The lecture is titled 'Logarithms Without Exponents,' aiming to explore the concept of logarithms beyond their traditional association with exponents. The professor reviews the basics of calculus and expresses the intention to apply these principles to special functions, starting with the logarithm. A physical principle, the rule of compound interest, is introduced as an analogy to explain the logarithmic function without referencing exponents. The lecture aims to answer how to find a function whose derivative is the inverse of its argument, setting the stage for a deeper exploration of logarithms.

The professor delves into the challenge of finding a function, denoted as 'L of x', whose derivative is '1 over x'. This quest stems from the inability to integrate '1 over x' using standard recipes. The discussion highlights the limitations of the integral formula for 'x to the n' with respect to 'x', particularly when 'n' equals -1, which leads to a 0 denominator issue. The professor uses both differential and integral calculus to hypothesize the form of 'L of x', suggesting it belongs to a family of curves that are always rising but 'spilling water', indicating concavity. The domain of 'L' is restricted to positive numbers, reflecting the physical interpretation of the principle of compound interest.

The lecture continues with an exploration of the area under the curve 'y equals 1 over t' from a fixed positive number 'a' to 'x', drawing parallels with the function 'L of x'. The professor explains how this area can be viewed as 'L of x' and how changing 'a' affects the area by a constant, akin to an indefinite integral. A brief digression into the definition of a logarithmic function is presented, highlighting the key property that the logarithm of a product is the sum of the logarithms. This leads to the deduction of other familiar logarithmic properties, such as the logarithm of 1 being 0 and the logarithm of the reciprocal being the negative of the original logarithm.

The professor introduces the concept of the natural logarithm, 'ln of x', as a member of the 'L of x' family that passes through the point (1,0). This function is characterized by its derivative being '1 over x' and its logarithmic property. The lecture demonstrates how to use calculus to verify whether two functions are equal by taking derivatives and comparing them. The natural logarithm is shown to satisfy the logarithmic property when 'L of 1' is set to 0, thus defining the natural logarithm without reference to exponents.

Building on the properties of the natural logarithm, the professor rederives the product rule of differentiation using logarithmic differentiation. By taking the natural log of a product of two differentiable functions and applying the logarithmic property, the professor shows that the derivative of the product is the sum of the derivatives of the individual functions multiplied by the other function. This creative approach reinforces the utility of logarithms in calculus and their integral role in deriving fundamental rules.

The professor discusses the geometric and integral interpretations of the number 'e', which is defined as the base of the natural logarithm where 'ln of e' equals 1. Two methods for identifying 'e' are presented: one geometric, involving the intersection of the line 'y equals 1' with the curve 'y equals ln x', and the other integral, involving the area under the curve 'y equals 1 over t' from 1 to 'e'. The lecture also provides an estimation technique for 'e' using inscribed and circumscribed rectangles to show that 'e' lies between 2 and 4.

In conclusion, the professor summarizes the lecture by emphasizing the physical motivation for inventing the natural logarithm function, which is to solve differential equations where the rate of change is proportional to the amount present. The natural logarithm is shown to be a one-to-one function, allowing for the discussion of its inverse. The lecture ends with a preview of future topics, including the exploration of the inverse logarithm function, and a reflection on the acceleration of the course's pace as it builds upon previously introduced concepts.