Basics of calculus
Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change.
~ source ncert
Introduction
you must read the introduction here…
Understanding Functions
In simple words, function is nothing but something in mathematics take takes in some values (inputs) and gives out a result (output).
This something or function has some well defined rules that is called the definition of the function.
The inputs that a function takes is nothing but a value on which the result is calculated, it maybe a number or a function itself.
For example : say, f(x) is a function which takes inputs as x, where x is nothing but a natural number (1,2,3…).
Also, f(x) is defined as : f(x) = 2x + 1
Now, when I replace x with different inputs (natural numbers), I get the corresponding result.
Like :
f(1) = 2(1) + 1 = 3
f(2) = 2(2) + 1 = 5
. . .
. .
.
You are already familiar of some famous functions,
like, trigonometric functions — sinx, cosx, tanx ; where x(input) is an angle.
We have a lot more already defined functions which helps us in physics and various other fields,
Logarithmic functions, Inverse Trigonometric Functions, Exponential Functions, etc.
Also, it is not always the case that a function takes a single variable as input.
Like : consider, g(x,y) = 2x+2y² ; where x and y are natural numbers
Understanding Limits
you can get a clear understanding of limits here…
Differential Calculus
In simple words, derivative of a function gives the rate of change of the function at a single point (input) with respect to a variable.
Rate of change of a function means that how is the value or result of a function changes with slight changes in input.
Changes like, does it increase? does it decrease? is it approaching zero? where is the result approaching?
It helps us answer such questions.
In other words, we can put it as the slope of a tangent at a certain point of a function is the derivative at that point of the function.
How to find derivative of a function?
Consider the function, f(x) = x³
the derivative of this function can be find using what we call the first principle of derivative.
if we apply the values and solve this limit, we will get the derivative of the f(x) to be f’(x) = 3x² (read as ‘f dash x’ is equal to 3x²).
Now, applying the first principle every time is a redundant task. Instead we have some general formulae which we use to find derivatives of simple functions.
You can find all the derivative formula list here. You can keep it for reference to use while you learn further.
[Note : while learning physics, you would most of the times will require only one or two formula]
Integral Calculus
We see from derivatives that we can find the rate of change of a function at a certain point. But what if we ask the other question that can we find the original function if we know its derivative?
Now, this original function is what we call an anti-derivative. And the formula that give these anti-derivatives is called the indefinite integral of the function and the process of finding the anti-derivative is called Integration.
Consider our previous derivative, f’(x) = 3x², if we find the original function (f(x) = x³) it will be the anti-derivative of it.
You can further read here….
