Calculus

Calculus is all about the comparison of quantities which vary in a one-liner way. The calculus concepts are covered in class 12 math subjects and Science & Engineering applications. Calculus is the branch of mathematics that deals with continuous change. In this article, let us discuss the calculus definition, problems and the application of calculus in detail. Also, we are going to discuss what is advanced calculus along with the concepts involved in the basic calculus.

Table of Contents:

Calculus Definition

Calculus, a branch of Mathematics, developed by Newton and Leibniz, deals with the study of the rate of change. Calculus Math is generally used in Mathematical models to obtain optimal solutions. It helps us to understand the changes between the values which are related by a function. Calculus Math mainly focused on some important topics such as differentiation, integration, limits, functions, and so on.

Calculus Mathematics is broadly classified into two different such as:

  • Differential Calculus
  • Integral Calculus

Both the differential and integral calculus deals with the impact on the function of a slight change in the independent variable as it leads to zero. Both differential and integral calculus serves as a foundation for the higher branch of Mathematics known as “Analysis”. Calculus Mathematics plays a vital role in modern Physics as well as in Science and technology.

Basic Calculus

Basic Calculus is the study of differentiation and integration. Both concepts are based on the idea of limits and functions. Some concepts like continuity, exponents are the foundation of the advanced calculus. Basic calculus explains about the two different types of calculus called “Differential Calculus” and “Integral Calculus”. Differential Calculus helps to find the rate of change of a quantity whereas integral calculus helps to find the quantity when the rate of change is known.

Important Calculus Topics:

Differential Calculus Basics

Differential Calculus is concerned with the problems of finding the rate of change of a function with respect to the other variables. To get the optimal solution, derivatives are used to find the maxima and minima values of a function. Differential calculus arises from the study of the limit of a quotient. It deals with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The symbol dy and dx are called differentials. The process of finding the derivatives is called differentiation. The derivative of a function is represented by dy/dx or f'(x). It means that the function is the derivative of y with respect to the variable x. Let us discuss some of the important topics covered in the basic differential calculus.

Limits

The degree of closeness to any value or the approaching term. A limit is normally expressed using the limit formula as-

\(\lim_{x \rightarrow c}f(x)=A\)

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

Derivatives

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

\(\lim_{x \to h}\frac{f(x+h)-f(x)}{h}=A\)

Go through the links given below for Calculus Class 11 and class 12 concepts and get a detailed knowledge of the basic calculus.

To learn more on calculus class 11 and calculus class 12, visit our BYJU’S page to get a proper definition with examples.

Continuity

A function is said to be continuous at a particular point if the following three conditions are satisfied-

  • f(a) is defined
  • \(lim_{x \to a}f(x)\) exists
  • \(lim_{x \to a^-}f(x) = lim_{x \to a^+}f(x) =f(a)\)

Continuity and Differentiability

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

Quotient Rule

The Quotient rule is a method for determining the derivative (differentiation) of a function which is in fractional form.

Chain Rule

The rule applied for finding the derivative of the composition of a function is basically known as the chain rule.

Integral Calculus Basics

Study about integrals and their properties. It is mostly useful for the following two purposes:

  • To calculate f from f’(i.e from its derivative). If a function f is differentiable in the interval of consideration, then f’ is defined in that interval.
  • To calculate the area under a curve.

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 area.

Definite Integral

A definite integral has a specific boundary within which function needs to be calculated. The lower limit and upper limit of the independent variable of a function is specified, its integration is described using definite integrals. A definite integral is denoted as:

\(\int_a^b\) 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:

\(\int\) f(x).dx = F(x) + C

Advanced Calculus

Advanced Calculus includes some topics such as infinite series, power series, and so on which are all just the application of the principles of some basic calculus topics such as differentiation, derivatives, rate of change and o on. The important areas which are necessary for advanced calculus are vector spaces, matrices, linear transformation. Advanced Calculus helps us to gain knowledge on a few important concepts such as

  • Quadratic forms
  • Generalized Stokes theorem
  • Vector fields as derivatives
  • Integration of forms
  • Multilinear algebra
  • Continuous differentiability
  • Tangent space and normal space via gradients
  • Dual space and dual basis
  • Critical point analysis for multivariate functions, etc.

Applications of Calculus

Calculus is a Mathematical model, that helps us to analyse a system to find an optimal solution o predict the future. In real life, concepts of calculus play a major role either it is related to solving area of complicated shapes, safety of vehicles, to evaluate survey data for business planning, credit cards payment records, or to find how the changing conditions of a system affect us, etc. Calculus is a language of physicians, economists, biologists, architects, medical experts, statisticians and it is often used by them. For example, Architects and engineers use concepts of calculus to determine the size and shape of the curves to design bridges, roads and tunnels etc. Using Calculus, some of the concepts are beautifully modelled such as birth and death rates, radioactive decay, reaction rates, heat and light, motion, electricity, etc.

Calculus Problems 

Go through the below calculus problems to understand the process of differentiation and integration.

Problem 1: Let f(y) = ey and g(y) = 10y. Use the chain rule to calculate h′(y) where h(y) = f(g(y)).

Solution: Given,

f(y) = ey and

g(y) = 10y

First derivative above functions are

f'(y) = ey and

g'(y) = 10

To find: h′(y)

Now, h(y) = f(g(y))

h'(y) = f'(g(y))g'(y)

h'(y) = f'(10y)10

By substituting the values.

h'(y) = e10y x 10

or h'(y) = 10 e10y

Problem 2: Integrate sin 3x + 2x with respect to x.

Solution: Given instructions can be written as:

∫ sin 3x + 2x dx

Use the sum rule, which implies

∫  sin 3x dx+ ∫ int2x dx ……… Equation 1

Solve ∫  sin 3x dx first.

use substitution method,

let 3x = u => 3 dx = du (after derivation)

or dx = 1/3 du

=> ∫  sin 3x dx turned as∫ sin u  X 1/3 du

or 1/3 ∫  sin u du

which is 1/3 (-cos u) + C, where C= constant of integration

Substituting values again, we get

∫  sin 3x dx= -cos(3x)/3 + C ……… Equation 2

Solve∫ 2x dx

∫ 2x dx = 2∫  x dx = 2 * x2/2 + C = x2 + C ……. Equation 3

Equation (1) => ∫  sin 3x dx+ ∫ 2x dx

= -cos(3x)/3 + x2 + C

Practice more questions on calculus class 11 and calculus class 12 topics such as limits and derivatives, integration using different methods, continuity etc. To learn more about calculus in maths, register with BYJU’S – The Learning App today.

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