Lec 36 | MIT 18.01 Single Variable Calculus, Fall 2007

This paragraph introduces the topic of dealing with infinity in mathematical contexts, specifically within the realms of limits and integrals. The professor begins by discussing the importance of MIT OpenCourseWare and its reliance on donations to provide free educational resources. The main focus then shifts to L'Hopital's Rule, which is a method for finding limits, particularly in cases where the limit results in infinity over infinity. The professor clarifies the hypotheses and the conclusion of the rule, emphasizing that it applies when both functions involved tend towards infinity and the derivative of the ratio tends to a limit L. An interesting point is made regarding the rule's applicability even when the limit L and the variable a are infinite. The paragraph concludes with an introduction to the concept of comparing growth rates of functions, introducing a notation to denote when one function grows significantly slower than another in the limit as x approaches infinity.

The second paragraph delves into the comparison of rates of growth and decay of functions. The professor discusses how different functions approach infinity at varying rates, using the natural logarithm, exponential functions, and powers of x as examples. A similar approach is taken for rates of decay, which are the speeds at which functions approach zero. The reciprocals of these functions are used to illustrate decay rates, with the professor emphasizing the importance of understanding these rates in relation to improper integrals. The concept of improper integrals is introduced, with a focus on integrals that extend to infinity and the conditions under which they converge or diverge. The professor also touches on the geometric interpretation of convergence, relating it to the finiteness of the area under the curve. The paragraph concludes with a discussion of the integral from a to infinity of f(x) dx, highlighting the importance of determining whether this integral converges to a finite value or diverges to infinity.

In this paragraph, the professor provides a detailed exploration of improper integrals, focusing on their convergence properties. The integral from 0 to infinity of e^(-kx) dx is examined, where k is a positive constant. The antiderivative of this function is calculated, and the limit as n approaches infinity is taken to determine the convergence of the integral. The professor uses this example to illustrate the concept of convergence and divergence in the context of improper integrals. A shorthand method for calculating such integrals is introduced, emphasizing the importance of paying attention to the signs and the behavior of the function as the limit is approached. The paragraph also addresses a common question about the use of infinity as a limit and the conditions under which the integral converges to a finite value. The professor encourages students to practice these techniques, as they are applicable to various problems in mathematics.

The fourth paragraph discusses a physical interpretation of the integrals covered in the lecture, specifically relating to radioactive decay. The professor explains how the integral from 0 to infinity of e^(-kx) dx can be used to model the average number of particles that decay in a radioactive substance over time. The limit as the time T approaches infinity is considered, and it is shown that this integral equals the total number of particles in the substance. The professor emphasizes that while waiting for an infinite amount of time is not realistic, the theoretical concept is useful for normalization and understanding the total quantity of a substance. The simplicity of the resulting formula is highlighted, showing that the limit process can lead to more elegant expressions, which is a recurring theme in mathematics.

This paragraph highlights the importance of understanding the tails of probability distributions, particularly in the context of probability and financial mathematics. The professor discusses how the tails represent the likelihood of unlikely events, which can have significant impacts, as seen in financial scandals like the mortgage crisis. The focus is on the need for mathematicians to determine which parts of a distribution can be considered negligible in finite calculations and which require theoretical reasoning. The concept of 'fat tails' is introduced, referring to events with low probability but high impact. The professor emphasizes the importance of getting the mathematics right in these areas, as they can have serious real-world consequences.

The sixth paragraph focuses on analyzing borderline cases of convergence and divergence in improper integrals. The integral from 1 to infinity of 1/x^p dx is examined, with p being a variable power. The professor demonstrates that when p equals 1, the integral diverges, serving as a borderline case between convergence and divergence. A systematic approach is taken to evaluate the integral for different powers p, showing that the integral diverges if p is less than or equal to 1 and converges if p is greater than 1. The antiderivative is used to determine the behavior of the integral, and the professor emphasizes the importance of careful calculation and understanding the mathematical properties at play.

In this paragraph, the professor introduces the limit comparison test, a method for determining the convergence of improper integrals when the integral itself cannot be easily evaluated. The test is based on comparing the given integral with another whose convergence properties are already known. If the ratio of the two functions tends to 1 as x approaches infinity, then both integrals converge or diverge simultaneously. The professor provides examples to illustrate the application of the limit comparison test, showing how it can be used to quickly determine the convergence of an integral without needing to compute it. The examples demonstrate the power of this method in analyzing the behavior of integrals at infinity.

The eighth paragraph continues the discussion on assessing the convergence of improper integrals, focusing on the use of comparison and limit comparison methods. The professor presents an example involving the integral from 0 to infinity of dx/(sqrt(x^2+10)), which cannot be easily evaluated. By comparing this integral to a known convergent integral, the professor demonstrates how to determine its convergence properties. The importance of avoiding the origin when making these comparisons is emphasized, as the behavior of the functions at the origin can be misleading. The professor also addresses a student's question about the choice of changing the limits of integration and explains the rationale behind it.

The final paragraph of the script addresses the challenges associated with improper integrals that have singularities, a second type of improper integral. The professor warns of the potential for incorrect results when calculating these integrals without considering their convergence. An example is given where the integral from -1 to 1 of dx/x^2 results in a seemingly nonsensical negative value, highlighting the importance of understanding the nature of the integral before attempting to calculate it. The professor emphasizes the need to be cautious and to consider the entire behavior of the function, especially near points of discontinuity or infinity, to avoid falling into such traps.