Friday, October 3, 2014

Calculus

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Calculus
1.
Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally considered to have been founded in the 17th century by Isaac Newton and Gottfried Leibniz, today calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.

Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (pluralcalculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.

2.
Calculus is the study of change, with the basic focus being on
  • Rate of change
  • Accumulation
What is Calculus ?
In both of these branches (Differential and Integral), the concepts learned in algebra and geometry are extended using the idea of limits. Limits allow us to study what happens when points on a graph get closer and closer together until their distance is infinitesimally small (almost zero). Once the idea of limits is applied to our Calculus problem, the techniques used in algebra and geometry can be implemented.

Differential Calculus
Algebra vs Calculus
In Algebra, we are interested in finding the slope of a line.
algebra line
The slope of the line is the same everywhere. The slope is constant and is found using
ΔyΔx
or
riserun
In Calculus, we are interested in find the slope of a curve
calculus slope of curve
The slope varies along the curve, so the slope at the red point is different from the slope at the blue point. We need Calculus to find the slope of the curve at these specific points
How does calculus help with curves???

To solve the question on the Calculus side for the red point, we will use the same formula that we used in Algebra - the slope formula:
ΔyΔx
However, we are going to make the blue and red points extremely close to each other. The key is, when the blue point is infinitesimally close to the red point, the curve becomes a straight line and
ΔyΔx
will then give us an accurate slope.

Differential Calculus in the Real World

This same idea can be applied to real world situations. Consider the distance versus time graph of a slowly moving car shown below.

Average Velocity during the first 3 seconds?
Instantaneous Velocity at 6 seconds ?
Calculus is not needed.

velocitydistanceΔtime=3−03−0=1ms
Calculus is needed.

We will need to use the method described above and try to bring two points infinitesimally close to each other.
Distance vs time
Integral Calculus
Algebra vs Calculus
Find the purple region

Area
Does not require Calculus. It is simply the area of a rectangle (base)(height).
Area = 2 × 3 = 6
Find the blue region

area under curve
To find the area of blue region, we need Calculus.
What can we do?

How does calculus help find the area under curves???
And the Answer is...
Calculus lets us break up the curved blue graph into shapes whose area we can calculate--rectangles or trapezoids. We find the area of each individual rectangle and add them all up. The key is : the more rectangles we use, the more accurate our answer becomes. When the width of each rectangle is infinitesimally small , then our answer is precise. See the example below:
10 rectangles
area under curve2
50 rectangles
area under curve 3
Integral Calculus in the Real World

This same idea can be applied to real world situations. Consider the velocity vs time graph of a person riding a bike.
Note: this is not the same graph that we looked at above. The first one that we looked at was distance vs time

Find the distance traveled during the first 3 seconds?
Find the distance traveled during the first 9 seconds?
Calculus is not needed.

Distance = (velocity)(time)
This is found, by looking at area under the velocity curve bounded by the x-axis. So we just have to find the area of the triangle from x=0 to x=3.
Calculus is needed.

We will need to use the method described above and find the area of infinitesimally small rectangles/trapezoids.
Distance vs timeIn later lessons,
  • We will learn how to systemically and practically solve these problems
  • How these basic principles apply to a wide array of real worlds problems dealing with physics, chemistry, biology, business, engineering, medicine, computer science, astronomy and other everyday problems that could not have been solved without Calculus.
  • How integral and differential Calculus are connected using the Fundamental Theorem of Calculus.

3.
What Is Calculus and Why do we Study it?
Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.

I have been around for a while, and know how things change, more or less. What can calculus add to that?
I am sure you know lots about how things change. And you have a qualitative notion of calculus. For example the concept of speed of motion is a notion straight from calculus, though it surely existed long before calculus did and you know lots about it.

So what does calculus add for me?
It provides a way for us to construct relatively simple quantitative models of change, and to deduce their consequences.

To what end?
With this you get the ability to find the effects of changing conditions on the system being investigated. By studying these, you can learn how to control the system to do make it do what you want it to do. Calculus, by giving engineers and you the ability to model and control systems gives them (and potentially you) extraordinary power over the material world.

The development of calculus and its applications to physics and engineering is probably the most significant factor in the development of modern science beyond where it was in the days of Archimedes. And this was responsible for the industrial revolution and everything that has followed from it including almost all the major advances of the last few centuries.

Are you trying to claim that I will know enough about calculus to model systems and deduce enough to control them?
If you had asked me this question ten years ago I would have said no. Now it is within the realm of possibility, for some non-trivial systems, with your use of your laptop or desk computer.

OK, but how does calculus models change? What is calculus like?
The fundamental idea of calculus is to study change by studying "instantaneous" change, by which we mean changes over tiny intervals of time.

And what good is that?
It turns out that such changes tend to be lots simpler than changes over finite intervals of time. This means they are lots easier to model. In fact calculus was invented by Newton, who discovered that acceleration, which means change of speed of objects could be modeled by his relatively simple laws of motion.

And so?
This leaves us with the problem of deducing information about the motion of objects from information about their speed or acceleration. And the details of calculus involve the interrelations between the concepts exemplified by speed and acceleration and that represented by position.

So what does one study in learning about calculus?
To begin with you have to have a framework for describing such notions as position speed and acceleration.

Single variable calculus, which is what we begin with, can deal with motion of an object along a fixed path. The more general problem, when motion can take place on a surface, or in space, can be handled by multivariable calculus. We study this latter subject by finding clever tricks for using the one dimensional ideas and methods to handle the more general problems. So single variable calculus is the key to the general problem as well.

When we deal with an object moving along a path, its position varies with time we can describe its position at any time by a single number, which can be the distance in some units from some fixed point on that path, called the "origin" of our coordinate system. (We add a sign to this distance, which will be negative if the object is behind the origin.)

The motion of the object is then characterized by the set of its numerical positions at relevant points in time.

The set of positions and times that we use to describe motion is what we call a function. And similar functions are used to describe the quantities of interest in all the systems to which calculus is applied.

The course here starts with a review of numbers and functions and their properties. You are undoubtedly familiar with much of this, so we have attempted to add unfamiliar material to keep your attention while looking at it.

I will get bogged down if I read about such stuff. Must I?
I would love to have you look at it, since I wrote it, but if you prefer not to, you could undoubtedly get by skipping it, and referring back to it when or if you need to do so. However you will miss the new information, and doing so could blight you forever. (Though I doubt it.)

And what comes after numbers and functions?
A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the "derivative") of various functions. (The process of doing so is called "differentiation".)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called "integration".)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.
There are a few other standard topics in such a course. These include description of functions in terms of power series, and the study of when an infinite series "converges" to a number.
So where does this empower me to do what?
It doesn't really do so. The problem is that such courses were first designed centuries ago, and they were aimed not at empowerment (at that time utterly impossible) but at familiarizing their audience with ideas and concepts and notations which allow understanding of more advanced work. Mathematicians and scientists and engineers use concepts of calculus in all sorts of contexts and use jargon and notations that, without your learning about calculus, would be completely inscrutable to you. The study of calculus is normally aimed at giving you the "mathematical sophistication" to relate to such more advanced work.

So why this nonsense about empowerment?
This course will try to be different and to aim at empowerment as well as the other usual goals. It may not succeed, but at least will try.

And how will it try to perform this wonder?
Traditional calculus courses emphasize algebraic methods for performing differentiating and integrating. We will describe such methods, but also show how you can perform differentiation and integration (and also solution of ordinary differential equations) on a computer spreadsheet with a tolerable amount of effort. We will also supply applets which do the same automatically with even less effort. With these applets, or a spreadsheet, you can apply the tools of calculus with greater ease and flexibility than has been possible before. (There are more advanced programs that are often available, such as MAPLE and Mathematica, which allow you to do much more with similar ease.) With them you can deduce the consequences of models of various kinds in a wide variety of contexts.

Also, we will put much greater emphasis on modeling systems. With ideas on modeling and methods for solving the differential equations they lead to, you can achieve the empowerment we have claimed.

And I will be able to use this to some worthwhile end?
Okay, probably not. But you might. And also you might be provoked to learn more about the systems you want to study or about mathematics, to improve your chances to do so. Also you might be able to understand the probable consequences of models a little better than you do now.

Well, what is in the introductory chapter on numbers?
We start with the natural numbers (1,2,3,...,) and note how the operations of subtraction, division and taking the square root lead us to extending our number system to include negative numbers, fractions (called rational numbers) and complex numbers. We also describe decimal expansions and examine the notion of countability.

And in the chapter about functions?
We start with an abstract definition of a function (as a set of argument-value pairs) and then describe the standard functions. These are those obtained by starting with the identity function (value=argument) and the exponential function, and using various operations on them.

Operations, what operations?
These are addition, subtraction, multiplication, division, substitution and inversion.

But what is the exponential function, and what are substitution and inversion?
The exponential function is mysteriously defined using calculus: it is the function that is its own derivative, defined to have the value 1 at argument 0. It turns out, however, to be something you have seen before. And it turns out to bear a close relation to the sine function of trigonometry.
Substitution of one function f into another g produces a new function, the function defined to have, at argument x, the value of f at an argument which is the value of g at argument x. This is simpler than it sounds.

An inverse of a function is a function obtained by switching its values with its arguments. For example the square function, usually written as x2 has the square root function as an inverse.

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