proof of mean value theorem using infinitesimals

In spite of this, however, it is still worth doing.Ĭalculus 1, Lecture 16B: Infinitesimal Calculus for d(sin(x)) and d(cos(x)), Product Rule and Applications In fact, it is, in part, meant to be mildly humorous. Of course, none of this is rigorous mathematics. This is how the error goes to zero “very fast” when described in terms of infinitesimals. Should we describe as unbelievably small? Inconceivably small? Unspeakably small? I chose to use “unspeakably small” in later lectures. If is an infinitesimally small, then is even “more” infinitesimally small. Then we can think of the error as a function of and write. In the previous example, let (imagine is “infinitesimally close” to 3). It turns out that errors can also be thought of in terms of infinitesimal calculus. The error function goes to zero much faster as than (orange) does. The error function (green) in using the red line to approximate the blue curve is a parabola with a vertex at. The function outputs are very close to zero when is close to 3. The error function has a graph which is a parabola with a vertex at. This is fast enough to call the tangent line approximation “good”. In other words, goes to zero significantly faster than does. The tangent line (linear) approximation of near is. In other words, the top of this fraction goes to zero significantly faster than the bottom does.īut what is the error? In applied mathematics, the error in an approximation is always defined to be. This, in turn, is defined by requiring that as. īy “rapidly”, we mean that the error goes to zero faster than does as.

It means the error in the approximation goes to zero “rapidly” as approaches. What does it mean for a linear approximation to be a “good” approximation for a nonlinear function near ? While the blue curve grows by 100% for every unit of time of time that goes by, its tangent line approximations grow by about 69.3% for every unit of time that goes by. This is described visually in the lecture embedded above and shown in the figure below. Hence, the relative change along the tangent line when is. Then, at any moment in time, the tangent line approximation to at the point gives. If the growth continued along a straight line rather than a concave up exponential growth curve, it would grow by about 69.3% in one year. This is an instantaneous relative (percent) rate of growth. The quantity is the continuous growth rate in this situation. In this situation, if $1000 is invested at time, then the investment’s value at an arbitrary time is. The only situation where this might be somewhat realistic is for a newly-formed company whose value skyrockets in its first few years. In other words, the value of the investment doubles every year. I start by considering a situation where an investment grows by 100% for every year that goes by. These benefits are not something that come so naturally to people trained to be pure mathematicians.Ĭalculus 1, Lecture 16A: Continuous Growth Rates, Errors for Tangent Line Approximations, and Newton’s Method Continuous (Instantaneous) Relative Growth Rates Indeed, this approach results in some of the main benefits that scientists, engineers, economists, etc… get out of learning calculus. I discussed this at the end of my previous blog post, “Differentiable Functions and Local Linearity”.Ī more serious benefit of Approach #2 is that it can give you insight into many applications of calculus. You could call this approach Calculus Sans Limits. Part of the fun that arises from this approach is that calculus formulas can be derived without resorting to the use of limits. Approach #2 also has the benefit of being a lot of fun! - once you get used to it, at least. Most people don’t have the stomach for approach #1. They can be kept at an intuitive, but non-rigorous, level.They can be made rigorous through the arduous process of studying the subject of non-standard analysis.So if there are no such real numbers, how can they possibly be used? There are two approaches to the answer this question. Moreover, if, then, , etc… can be “much smaller than” itself (by many “orders of magnitude”). Given any real number, the number, but. There is also no smallest positive real number! īut, first things first: there are no such real numbers! You cannot divide by infinity! In a sense, you can think of them as quantities of the form. These are quantities so small that they are smaller than any positive real number.

It is based on the concept of infinitesimal quantities, or just “infinitesimals”, for short. Today, this intuitive method is called infinitesimal calculus. Instead, they approached calculus in an intuitive way. Did you know that Newton and Leibniz did not know the precise definition of a limit?

Proof of mean value theorem using infinitesimals