calculus for dummies pdf
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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 in 1990 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.
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.
functionsFunctions are the relation between two variables : one is dependent variable and other is independent variable.
Limit definitionLimit is the value of a function when its variable approaches another number. It can be defined using inequality . First continuity of function needs to be defined. A function is continuous when the following condition holds:
So for every value of δ there is always an infinitesimal ε such that absolute value |f(x)-L| is always <= ε . This condition must always hold true.
If this is the case then the function f(x) is continuous and differentiable at the point a.
So continuity is the sole condition for the existence of limit. This condition is the same as existence of both side limits :
The left hand limit and right hand limit must exist and equal at the point where the function is continuous . What we have learned is that limit and continuity is the two sides of the same coin. Limit exist when continuity exists and vice versa. The summary of the above discussion can be put like this :
A cartoon can make it much clear:
DifferentiationDifferentiation is defined as the limit of a fraction. It is rate of change of the function with respect the dependent variable. The mathematical expression is:
The geometrical interpretation is :
Product rule of differentiationWhen we differentiate product of two functions we will follow this rule
Some difficult differentiation problems.
PiHow many ways π can be written ? π is a mysterious number. It can be formulated in various ways. The value of π is exactly the ratio between the circumference of a circle to its diameter.
functionalFunctional is function of a function. When we differentiate the functional a different method is applied.
Example can be given :
Vectors and geometric calculusThe equation given below is basically Stoke's theorem. It says that integral of some differential form F over the boundary dM of manifold is equal to integral of the exterior derivative over the whole of the manifold M. In language of calculus it is written as :
The integration of differential form is defined in the following way:
Exact definition of differential form is unavailable at the moment. But it relates the multivariable calculus. One varraibe dx is known as 1-form. Two variable expression dx1.dx2 is known as 2-form and so on. The calculus involving the differential forms are known as exterior calculus.
The definition of integral is :
Which is the same thing as this expression below:
Liebniz integral lawLiebniz defined differentiation of a integral as follows:
The caculation of volume of an object involves this formula.
Differentiation vs integralDifferentiation is the inverse process of integration. So if we differentiate the integral we get back the original function.
Differential equations solution methodsA simple method for solving second order linear differential equation is given here.
Important calculus formulaSome geometric and algebraic formulas are related to calculus intimately. These can be mentioned.
Some important rules for integration are :
Some more formulas are :
The equations that form parabolic trajectory are :
Finding of area under the curve
Trigonometric formula cheat sheet
Food for thoughtsThere is a paradox named "Zeno's paradox' which goes like this :
What could be the resolution of this paradox?
Fractional order calculusFractional order calculus can be important sometimes. It is the differentiation process where we differentiate with respect to fractional power of dependent variable. As example :
Example can be given
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