Calculus

Functions represent the relationship between two variables, which are the independent variable and the dependent variable. Let’s consider the following diagram.

There is an INPUT, a black box, and an OUTPUT. For example, suppose we want to make a pizza. We would need the following basic ingredients.

• Pizza Base
• Pizza Sauce
• Cheese
• Seasoning

The above real-life example can be represented in the form of a function as explained below,

The taste of our pizza depends on the quality of the ingredients. Let’s take another example

Suppose that: y = x²

Using the diagram given above, we get:

We can see that the value of y depends on the value of x.  We can conclude that

• INPUT is independent of the OUTPUT
• OUTPUT depends on the INPUT
• Black Box is responsible for the transformation of the INPUT to the OUTPUT

In calculus,

• INPUT is an independent variable 
• OUTPUT is a dependent variable 
• Black Box is a function

Differential calculus focuses on solving the problems of finding the rate of change of a function with respect to the other variables. To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Differential helps in the study of the limit of a quotient, dealing with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The notations dy and dx are known as differentials. The process used to find the derivatives is called differentiation. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x). 

Limits

Limit helps in calculating the degree of closeness to any value or the approaching term. A limit is normally expressed using the limit formula as,

lim_x→cf(x) = A

This expression is read as “the limit of f of x as x approaches c equals A”.

Derivatives

Derivatives represent the instantaneous rate of change of a quantity with respect to the other. The derivative of a function is represented as:

lim_x→h[f(x + h) − f(x)]/h = A

Continuity

A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied –

• f(a) is defined
• lim_x→af(x) exists
• lim_x→a⁻ f(x) = lim_x→a⁺ f(x) = f(a)

Continuity and Differentiability

A function is always continuous if it is differentiable at any point, whereas the vice-versa for this condition is not always true.

Integral calculus is the study of integrals and the properties associated to them. It is helpful in:

• calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval.
• calculating the area under a curve for any function.

Integration

Integration is the reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as a collection of small parts in order to form a whole. It is generally used for calculating areas.

Definite Integral

A definite integral has a specific boundary or limit for calculation of the function. The upper and lower limits of the independent variable of a function are specified. A definite integral is given mathematically as,

∫_a^b f(x).dx = F(x)

Indefinite Integral

An indefinite integral does not have a specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:

∫ f(x).dx = F(x) + C

Calculus is a very important branch, mathematical model that helps in:

• Analyzing a system to find an optimal solution to predict the future of any given condition for a function.
• Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system affect us, etc.
• Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, Architects and engineers use different concepts of calculus in determining the size and shape of the construction structures. 
• Calculus is used in modelling concepts like birth and death rates, radioactive decay, reaction rates, heat and light, motion, electricity, etc. 

What is Calculus?

Calculus is one of the most important branches of mathematics, that deals with continuous change. Calculus is also referred to as infinitesimal calculus or “the calculus of infinitesimals”. Infinitesimal numbers are the quantities that have value nearly equal to zero, but not exactly zero.

What is Differential Calculus?

Differential calculus focuses on solving the problems of finding the rate of change of a function with respect to the other variables. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x). 

What do you Use Calculus for in Real Life?

Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, the safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system that affect us, etc. Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, architects and engineers use different concepts of calculus in determining the size and shape of the construction structures. 

How to Find the Maxima and Minima of a Function?

To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Maxima and minima are the highest and lowest points of a function respectively, which could be determined by finding the derivative of the function.

What is Integral Calculus?

Integral calculus is the study of integrals and the properties associated with them. It is helpful in calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval. It enables us to calculate the area under a curve for any function.