What is the basis and types of calculus?

In summary, What is the basis and types of calculus? More specifically, it’s how we can reason about infinity, and something infinitely small. It just so happens that reasoning about the infinitely small is equivalent to reasoning about change, which is what calculus tends to focus on, because of the numerous real-life applications to working with change.

When I say it works with infinity, it was developed to answer the question, what is ∞∞, which is undefined, but with enough starting information (like how quickly you are approaching infinity from the numerator and denominator), it may converge to a known value.

What is the basis and types of calculus?

It works with the multiplicative inverse of infinity, called an infinitesimal, which is similar to 1∞, but defined a bit more rigorously using limits (the concept of approach that I just mentioned).

In terms of integration, it is the basis for studying infinite sums, and more so, sums of terms getting “closer and closer” to each other.

More generally, the work with the infinitesimal concept, is the study of a region of a surface, progressive smaller and smaller (toward infinitely small), and because of this, the analysis begins the discussion of a new mathematical topic: continuous measures, which are basically “ways to get a distance” which allow us to perform calculus (actually analysis) on something that admits measure at increasingly smaller regions (infinitesimal regions).

Moving beyond calculus, we learn that infinity can be more rigorously defined, and there are many types of infinity.

The basis of calculus is to study changes over very minute intervals of time. Like if i were to ask you to find the acceleration of a care at one millionth of a second after the car is put into drive, calculus would have to be used as the basic s= d/t simply will not do. Calculus is the study of how things change. It provides a fundamental fabric for modeling systems in which there is change, and a way to deduce the predictions of such models.

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 extraordinary power over the material world.

What are the types of calculus?

The word Calculus is actually misunderstood to deal with infinitesimal quantities and limits. This is but one type of calculus, called Infinitesimal Calculus. There are numerous calculi, because “a calculus” (from the Latin “calculus” which means “pebble”) is a set of computational rules on a mathematical structure:

• Logic
• Propositional calculus and predicate calculus are commonly used computational rule sets for logic
• Cirquent Calculus is a computational rule set for Linear Logic, which is often used to reason about resources
• Computer Science
• Lambda calculus and SKI combinator calculus are commonly used computational rule sets for functions/programming
• Pi-calculus and join-calculus are computational rule sets for programming parallel processes and communication channels
• Relational Calculus and Tuple-Relational Calculus for relational databases
• Event Calculus for reasoning about events
• Modal Mu-Calculus for reasoning about recursive structures
• Infinitesimal
• Differential Calculus for differentiation
• Integral Calculus for integration
• Calculus of Variations for a more general evaluation of the maxima and minima of functions
• Tensor Calculus for calculus on tensors
• On Sums and Differences
• Calculus of sums and differences for operations using the difference operator
• Umbral Calculus for combinatorics on particular classes of polynomial functions
• And there are so many more that aren’t listed here.

What does it mean to differentiate in calculus?

It’s to compute the slope of a thing that doesn’t obviously have a slope.

Let me back up a bit.

What’s the slope of a line? That’s a concept you probably understand. Depending on how I give you the line, you can probably compute its slope in a few different ways. If I show you a graph, you can pick two random points and compute “rise over run.” If I give you an equation like 2y+3=3x+2, you can manipulate it into y=mx+b form and read off the slope from the m coefficient.

Fine. Lines have slopes.

But what if I ask you for the slope of a parabola? If you haven’t thought about it, it’s not clear what that even means.

Differential calculus is about giving meaning to that question and answering it.

I’ll even sketch the meaning and the answer:

First, what does it mean? Well, with lines, slope measures the rate of change. With more general functions, you can still ask for the rate of change. Except this time, the rate of change isn’t just a constant… the rate of change can itself change depending on the where you’re asking about. For example, consider a function like this:

(This is a Logistic function, if you’re curious)

Even if you don’t know any formulas or how to compute anything, you can see from the graph that at x=4, the function is very close to 1. At x=6, it’s even closer to 1. But the change isn’t that much from 4 to 6. But you can see that at x=0, the function is 1/2, and at x=2 the function changes to what appears to be about .8 or .85. It’s a much bigger jump, even though we incremented x by 2. The same increment (x to x+2) gives a different change for this function.

Well, we can take smaller steps. Instead of incrementing x by 2, we can increment by 1. Or by .1. Or by .0000001.

If you think about that for a moment, you might realize that incrementing by something really really small (like .0000001) starts to look like a rate of change of the function associated at a single point. And moreover, if you think of “zooming in” on the graph of any function to a great enough degree, you might realize the graph starts to look a lot like a line. And now we’re back into “plain slope” territory.

Calculus is about formalizing the process of zooming in, in a systematic way. The end result is that I can say “what’s the slope of the function f(x)=x2?” and you can answer “it’s g(x)=2x,” in a way that’s logically rigorous.

I guess we could say that a basis for calculus is analysis. In particular real analysis is the very basis of what we learn in the usual calculus courses taught at university.

In particular classical analysis covers topics such as limits and continuity of real valued frunctions, differentiation and integration of functions, usually all in the context of functions of the form

f:Rn→Rm 

The basic concepts from where qe draw power in classical analysis is the distance between mathematical objects. This is pretty clear when we formulate a limit in the  ϵδ  formalism. This is generalized when we start our study in modern analysis, where one can start by studying metric spaces, which are arbitrary non-empty sets where se can define a distance or metric between two elements of the set. We say that a function  d:X×X→X  is a metric if

d(x,y)≥0 

d(x,y)=0⟺x=y 

d(x,y)=d(y,x) 

d(x,y)≤d(x,z)+d(z,y) 

This properties define a distance function in an arbitrary non-empty set. This extends (in a way) the nice behaviour of  R  used to describe single and multivariable calculus in the Euclidean space  Rn  to more arbitrary and abstract spaces.

The notion of distance is one of the most fundamental in analysis, and it forms the basis for the calculus we use. From here we define the limit as getting arbitrarily close to a point without ever touching it, the derivative as aproximating a arbitrary function to a linear one in a small neighborhood of a point, and the integral as finding the given area or volume formed by a set.

I hope i was helpfull.

Without using math, what is calculus?

I’ll try to give you as much flavor as I can in ten points, without using any equations

• It concerns two things, rates of change and areas
• In some simple cases you can solve these without it
• You don’t need calculus to get the area of a 10 x20 cm rectangle, or, say, that it takes you one hour to travel 60 km at a constant speed of 60kph
• But it also solves more general cases where things get very very complicated
• The relationship of two variables is called a function
• The two variables can be used as the horizontal and vertical axes and you can draw a graph of the function
• The instantaneous rate of change of a function is the slope of the tangent of its graph, this is calculated mathematically as something called the derivative of the function, and that process is called differentiating
• The area under the graph of a function is calculated by mathematically taking what is called the integral of the function, called integrating
• The fundamental theorem of calculus essentially states that differentiation and integration are opposites
• The fundamental theorem means then that the rate of growth of the area under the graph of a function is that function itself!