The most common practical use of calculus is when plotting graphs of certain formulae or functions. Using methods such as the first derivative and the second derivative, a graph and its dimensions can be accurately estimated.
These 2 derivatives are used to predict how a graph may look like, the direction that it is taking on a specific point, the shape of the graph at a specific point if concave or convex , just to name a few. When do you use calculus in the real world? In fact, you can use calculus in a lot of ways and applications. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine.
It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. In fact, even advanced physics concepts including electromagnetism and Einstein's theory of relativity use calculus. In the field of chemistry, calculus can be used to predict functions such as reaction rates and radioactive decay.
Meanwhile, in biology, it is utilized to formulate rates such as birth and death rates. In economics, calculus is used to compute marginal cost and marginal revenue, enabling economists to predict maximum profit in a specific setting.
The branch of philosophy known as determinism takes free-will a step further to explain why man chooses to act in one way over another. When man is confronted with a situation, the line of action he decided on taking is based on experience, personal interests and preferences.
The decisions we make are thus entirely influenced by our past experiences. The will is not free to behave on its own. For example my decision to write this book was not based on some impulsive instinct but on a culmination of circumstances I was exposed to and reacted against. Any action from the most random to the most perverse can be explained by the set of situations, experiences, thoughts and feelings that preceded that action.
Essentially determinism says that our lives are predetermined to the extent that regardless of how we live we can never change our course of life. As a French thinker once said , " We change yet we remain the same ". The logic behind this statement is that if our past can never be changed then our future will always remain the same.
Each action is dictated by the previous one. It is at this point where science and art diverge. For the scientist, determinism is an accurate enough explanation of human life, as it says that everything that occurs, occurs for a set of reasons. Observing and understanding these set of reasons becomes the work of the scientist. The artist, however, interprets determinism as saying, 'Since everything in life is predetermined then life is meaningless'.
Life may have no meaning but it is the goal of the artist to question that statement by exploring the mysterious depths of human nature and the heart. Perhaps it is the utter randomness of life that causes us to ignore our unchanging destiny. No human being has any control over actions that he or she will be exposed to. Fate begins to lose meaning as one never knows what will happen to oneself. It takes the risks of the artist to create chaos from which wisdom evolves.
On the other hand, the study of nature is more precise and is less likely to be influenced by a wide variety of unrelated factors. Within natures all actions, occurrences, or changes are dependent on a few factors that can be carefully isolated and studied individually. Science is specifically about analyzing these interacting systems and then forming hypothesis that can accurately explain them.
What makes observing phenomena in nature so interesting is that they always occur in a closed setting where external factors can easily be removed to leave behind just a few interacting objects. It is these objects along with their properties, that become the focus of study. Any attempt to logically explain their unique interaction must come from the objects themselves and not from imaginary external factors.
Through reasoning and observation, nature can be understood, such that the future can be determined from the present. As Sherlock Holmes would often warn Watson, " You see but you do not observe! Often enough, human beings fail to grasp this simple rule of nature by ignorantly attributing any naturally occurring phenomena to the Gods, heavens, or some mysterious substance with superpowers.
Understanding and accepting the truth requires an open and critical mind. Charlotte Bronte humorously wrote about this ironic flaw of human nature in her popular novel, Shirley. Note well! Whenever you present the actual, simple, truth, it is somehow always denounced as a lie: they disown it, cast it off, throw it on the parish; whereas the product of your own imagination, the mere figment, the sheer fiction, is adapted, termed pretty, proper, sweetly natural: the little spurious wretch gets all the comfits - the honest, lawful bantling all the cuffs.
Such is the way of the world Not only is science stymied by ignorance and deception, but it is also muddled by the work of the pseudo-scientist. The pseudo-scientist is described by the Spanish philosopher, Jose Ortega, in his stunning book on modern western civilization, The Revolt of the Masses: " By a third generation takes command in the intellectual world, and we find a type of scientist without precedent in history.
He is a person who knows, of all that a routinely dutiful man must know, only something of one specific science; even of this science, he is well informed only within that limited area in which he is an active researcher. He may even go so far as to claim he has an advantage in not cultivating what lies outside his own narrow field, and he may declare that curiosity about general knowledge is the sign of the amateur, the dilettante.
Immured within his small area, he succeeds in discovering new facts, advances the science which he scarcely knows, and increases perforce the encyclopedia of knowledge of which he is conscientiously ignorant Most of the scientists' work fell into one of these categories: Completely ununderstandable Vague and indefinite Something correct that is obvious and self-evident, worked out by a long and difficult analysis and presented as an important discovery A claim based on the stupidity of the author that some obvious and correct thing accepted and checked for years is in fact false An attempt to do something probably impossible but of certainly no utility which it is finally revealed at the end fails Just plain wrong.
Most of the scientists' work fell into one of these categories: Completely ununderstandable Vague and indefinite Something correct that is obvious and self-evident, worked out by a long and difficult analysis and presented as an important discovery A claim based on the stupidity of the author that some obvious and correct thing accepted and checked for years is in fact false An attempt to do something probably impossible but of certainly no utility which it is finally revealed at the end fails Just plain wrong One of the remarkable masterpieces of the mind is the science of mathematics, often called the science of deductive reasoning.
While science is a logical system of thought used to study the natural world; mathematics is the precise language of science. It is the form of communication for scientific analysis.
Number and symbols are nothing more than vague abstractions unless they refer to something specific. Before mathematics can exist there must be a situation to give it meaning.
It is scientific analysis that determines the structure of mathematics. Through mathematics we are able to define the present. The present is only dependent on the conditions that exist within the short frame of time that it occupies. Quickly it vanishes before our eyes, becoming a memory. The goal of science is to define the objective world in terms of existing quantifiable conditions expressed by mathematics.
Our dimensions or properties remain fixed and do not change. It is when our dimensions change that our study becomes a bit more complicated and calculus arises. But first, what is meant by change? To understand change we need to explain the concept of time. By definition, time is a passage of events, such that for time to pass, something must change with respect to itself.
For example a moving object implies a changing distance covered from a reference point. This comprises an event that defines time. Or a rising temperature implies that the temperature is changing , thus occupying time. Changes are the results of actions that comprise a situation. While calculus is the study of mathematically defined change, it is not necessarily the study of time alone.
Now it is within the realm of possibility, for some non-trivial systems, with your use of your laptop or desk computer. The fundamental idea of calculus is to study change by studying "instantaneous " change, by which we mean changes over tiny intervals of time. 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.
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 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. How to find the instantaneous change called the "derivative" of various functions. The process of doing so is called "differentiation". How to go back from the derivative of a function to the function itself.
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