Infinitesimal Calculus Introduction

The preface explicitly states the book's intention: to provide an introduction to calculus using the hyperreal number system and infinitesimals. This approach differs significantly from traditional calculus texts, which typically offer no logical justification for the real number system. Similarly, this text will not provide a rigorous justification for the hyperreal numbers, opting for a more intuitive and practical application-focused approach. The author acknowledges that while the book doesn't require prerequisites beyond basic algebra and trigonometry, a prior familiarity with calculus is strongly recommended due to the fast pace of the material. Readers seeking a deeper understanding of the foundational aspects of hyperreal numbers are directed to alternative resources such as Abraham Robinson's Non-standard Analysis and Robert Goldblatt's Lectures on the Hyperreals.

The text utilizes Zeno's paradoxes, particularly the arrow paradox, to illustrate fundamental concepts. One interpretation of Zeno's paradox suggests that since an arrow occupies a definite position at each instant, it doesn't move during those instants, leading to a velocity of 0. This introduces the problem of assigning meaning to ratios involving zero magnitudes. The text highlights the indeterminate nature of 0/0, explaining that 0 × c = 0 for all real numbers c, making division by zero undefined. Furthermore, the paradox also reveals the difficulty of understanding infinite sums of zero magnitudes and products of infinitesimal and infinite numbers; for example, the text raises questions surrounding the interpretation of ∞ × 0. The book hints at how infinitesimals resolve these ambiguities, forming the basis for differential calculus. The discussion concludes by noting the close relationship between division and multiplication, foreshadowing the importance of this connection to the fundamental theorem of calculus later in the text.

The section explains two approaches to defining velocity at an instant. The first, a standard approach since the mid-19th century, involves examining average velocities over increasingly smaller intervals (∆t approaching zero). The book's chosen method involves considering the situation when the start and end times (a and b) are immeasurably close, highlighting the utility of infinitesimals. A numerical example using a constant acceleration of -9.8 meters/second² is provided to illustrate the calculation of average velocity over a finite interval. This calculation is contrasted with the concept of the velocity at an instant using infinitesimals, introducing the idea of immeasurably small changes in time that are beyond the representation of real numbers. The concept of velocity serves as a particular example of a rate of change, extending the discussion to the general rate of change of a quantity y with respect to x, given a function y = f(x).

The text introduces hyperreal numbers (denoted by R) as a set of numbers that go beyond the real numbers. The set R is introduced as including numbers that are intuitively too small to measure, even theoretically, in addition to the real numbers. This explanation is not mathematically rigorous, avoiding technical details that are considered beyond the scope of the book. The discussion touches on the limitations of rational numbers (ratios of integers) in measuring geometric quantities, for example mentioning that the diagonal of a square with side length 1 is the irrational number √2. A concept of continuity from the right is introduced and defined for real numbers, using the concept of infinitesimals. The section briefly touches on the extreme-value and intermediate-value properties of continuous functions, emphasizing that their rigorous justification lies outside the scope of the text. The text also provides an example of how certain types of functions can be continuous with respect to positive but not negative infinitesimals.

This section contrasts two methods for determining instantaneous velocity. The traditional approach, prevalent since the mid-19th century, involves analyzing average velocities over progressively smaller time intervals (∆t), ultimately taking the limit as ∆t approaches zero. The book introduces an alternative method that employs infinitesimals. Instead of reducing ∆t to an arbitrarily small real number, this approach considers the scenario where the initial and final times are immeasurably close, exploiting the properties of infinitesimals. An example illustrating the calculation of average velocity over a finite interval with a constant acceleration of -9.8 meters/second² is provided. This example sets the stage for understanding how infinitesimals can provide a more intuitive framework for grasping instantaneous velocity, representing changes so minuscule they lie beyond the realm of standard real numbers. The concept extends beyond simple velocity, serving as a specific case of rate of change; the rate of change of an object's position concerning time.

Building upon the concept of velocity, the section expands the discussion to encompass rates of change more generally. Given any quantity y as a function of another quantity x (y = f(x)), the text explores how to determine the rate of change of y with respect to x. The average rate of change over an interval [x, x + ∆x] is initially discussed, represented as ∆y/∆x. The text then explains that for a straight line (f(x) = mx + b), this average rate of change simplifies to m, the slope of the line. This same result holds true even when ∆x is an infinitesimal. The section proceeds to explain that for non-linear functions, this average rate of change depends on both x and ∆x. However, the derivative, represented as dy/dx, depends only on x when ∆x is infinitesimal. Therefore, the derivative is interpreted as the slope of the curve y = f(x) at a specific point x, which unlike a straight line, will vary from point to point for other differentiable functions.

This section focuses on optimization problems, specifically finding maximum and minimum values of functions. The text states that for a continuous function f on a closed and bounded interval [a, b], the extreme value property ensures that both maximum and minimum values exist. If the function is differentiable at a point c within the interval (a, b) where a maximum or minimum occurs, then the derivative at that point must be zero (f'(c) = 0). This establishes the importance of stationary points (where f'(c) = 0) in optimization. The section also considers cases where the function might not be differentiable at the maximum or minimum, or where the maximum or minimum occurs at the boundaries of the interval (a or b). A case of a function decreasing before a point c and increasing after, leading to a minimum at c, is further described. Similarly, the text considers a function that increases before c and decreases after, indicating a maximum at c. The text also briefly considers continuous functions on intervals that aren't closed or bounded, acknowledging that the extreme-value property doesn't necessarily apply in these instances.

The final subsection introduces implicit differentiation as a technique to find the slope of curves defined by equations like f(x, y) = c. This technique is useful when y cannot be easily expressed explicitly as a function of x. The method involves differentiating both sides of the equation with respect to x and then solving for dy/dx, assuming y is differentiable (which is often true but the details aren't fully explored here). Implicit differentiation is further presented as useful for determining rates of change in variables linked by an equation. The text provides example exercises: finding the tangent line to an ellipse and calculating the rate of change of distance between a point and a moving plane, as well as finding the rate of change of area of a rectangle whose length and width are changing over time. These examples illustrate the practical application of this powerful calculus technique in addressing diverse problems involving related rates.

This section introduces the concept of an integral. It begins by posing the inverse problem to differentiation: given a function representing the velocity of an object moving along a straight line, determine the object's position at a given time. The text defines an integral of a function f, denoted as F, as a function whose derivative is f (F'(x) = f(x)). It emphasizes that an integral isn't unique; if F(x) is an integral of f(x), then F(x) + c (where c is any constant) is also an integral. This is demonstrated with the example of f(x) = 3x², showing that both x³ and x³ + c are integrals for any constant c. This highlights the fact that integrals differ by a constant term. The section uses the mean value theorem to demonstrate that if the derivative of a function H(x) is 0 over an open interval (a, b), then the function H(x) must be a constant over that interval. This property of integrals is crucial for later developments within integral calculus.

The section moves on to discuss methods for finding the position of a moving object given its velocity, which serves as a crucial example for understanding integration. The text contrasts two approaches. The first method approximates the position by assuming a roughly constant velocity over short time intervals, using the formula x(t) = x(a) + r(t - a), where r is an assumed constant velocity. This method gives a good approximation as long as v(t) doesn't change significantly. However, if v(t) changes drastically, the text proposes dividing the entire time interval into smaller intervals, applying the approximation over each sub-interval, which then leads to a Riemann sum approximation. The second approach, leading to an exact result, involves dividing the interval [a, b] into infinitely many equal subintervals of infinitesimal length (dx) using an infinitely large integer N. This method uses Riemann sums with the limit as N goes to infinity, which represents the definite integral. The text notes that for a continuous function on a closed bounded interval, the Riemann sum converges and produces a unique definite integral; however, the complete justification for this is beyond the book's scope.

Having introduced definite integrals, the section explores their properties. The text states that the definite integral of a function over an interval [a, b] can be divided into subintervals; for example, the integral from a to b is equal to the sum of integrals from a to c and from c to b. This is intuitively explained with respect to the total change in the position of a moving object. This property is crucial for the development of the fundamental theorem of calculus, which is mentioned but not fully proven, as the full proof requires deeper properties of continuous functions. The section further highlights the application of definite integrals through several examples, hinting at the significance of the fundamental theorem of calculus in evaluating definite integrals, and setting the stage for later sections that delve deeper into the various applications of definite integrals, including calculating the area between curves.

This section addresses optimization problems, focusing on finding maximum and minimum values of continuous functions defined on closed and bounded intervals [a, b]. The extreme-value property guarantees the existence of both maximum and minimum values within such intervals. The text highlights that if a maximum or minimum occurs at a point c within the open interval (a, b), and the function is differentiable at c, then the derivative at that point must be zero (f'(c) = 0). Points where the derivative vanishes, known as stationary points, are thus crucial for optimization. The section notes that maxima or minima can also occur at points where the function is non-differentiable or at the endpoints of the interval (a or b). Specific conditions for identifying minima and maxima based on the behavior of the derivative around a stationary point are described. The text further notes that for functions defined on intervals that aren't closed or bounded, guaranteeing the existence of extreme values requires a more careful analysis.

The discussion extends to optimization problems involving continuous functions on intervals that are either not closed or not bounded. In these cases, the extreme-value property, which guarantees the existence of maximum and minimum values on closed bounded intervals, does not directly apply. Consequently, finding extreme values requires a more thorough analysis than for closed, bounded intervals. The text emphasizes that there's no guarantee of the existence of extreme values in these cases. An example is provided illustrating a function with a minimum value but no maximum value on an unbounded interval, highlighting the complexities of optimization on non-closed or unbounded intervals and the need for case-specific considerations.

This section applies calculus to model real-world phenomena. The focus is on two key examples: population growth and radioactive decay. Both are modeled using differential equations where the rate of change is proportional to the current quantity (population size or radioactive material amount). The text introduces the differential equation dy/dt = ky, where k is a constant that reflects growth (k > 0) or decay (k < 0) rate. For the case k = 1, the solution is the exponential function y = eᵗ. The text then explains how this basic model can be used to describe radioactive decay, introducing the concept of half-life, which is the time required for half the material to decay. The half-life (T) relates to the decay rate (k) by the equation k = -log₂(T). Carbon-14 dating is mentioned as a real-world application of this modeling technique, using the known half-life of carbon-14 (5730 years) to estimate the age of organic remains. The text further explores how this type of differential equation describes population growth without limiting factors, and shows how to modify this basic model (creating a logistic growth model) to account for limiting factors such as available space and resources. Specific exercises are provided to reinforce these modeling techniques using different model parameters.