Math Concepts

Calculus terms and Calculus symbols

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Table of Contents

1. Introduction
2. Calculus Terms
3. Calculus Symbols
4. Summary
5. FAQs

27 October  2020

Reading Time: 7 Minutes

Introduction

Calculus is all about changes

The word calculus is come from a Latin word, meaning originally "small pebble”

Calculus is also known as “ infinitesimals.

In mathematics, calculus represents courses of elementary and analysis which are mainly devoted to the study of function limits.

Image 1

For a proper understanding of the topic we must be well versed with the calculus terms and thus we have compiled a list of important basic calculus terms and also a table of the calculus symbols. 

This essentially is the calculus vocabulary including calculus terminology and calculus math symbols.

In this blog we have listed the calculus terms which are important with a brief description. To know more about any calculus term, you can click on it to get an in depth understanding. 

 

Also read:


Calculus Terms

The category is concerned with the calculus terms involved in topics under calculus like limits, integration, differentiation, etc. 

Calculus is a method using math to study changes in a system. Calculus is a path of measuring morals and ethics, like choosing the lesser of two evils.

Once in a time an apple bonked him on the head, Newton used calculus to fine-tune his Law of Gravity. Don’t break the law. 

Calculus played an important role In Medical fields calculus is a hard, crusty mass like a kidney stone; or tartar on a tooth. Calculus means it refers to the "small pebble used for counting."

So here are the calculus terms with links for their proper understanding.

Calculus Terms

Acceleration

Acceleration is also known as an increment in the rate of change.

Area

The measurement of a two-dimensional surface within a closed boundary is called the area.

Area under the curve

The area under a curve between two points can be found by doing a definite integral between the two points

Image 2

Asymptote

Asymptote refers to the straight line that gets very close but never touches the curve. It gives the limiting value of the curve. An asymptote is sometimes called a tangent.

Image 4
            

Calculus 

Calculus is a way of using math to study changes in a system and deals with limits, differentiation, and integration of functions.

Chain Rule 

The chain rule is used for finding the derivative of a composition of functions or multiple functions.

image 5


Since in general most of the used functions are compositions of multiple others this is very useful to us.

Concavity

There is a simple characterization of concavity in terms of the sign of the second derivative.

Image 6

            

Continuity

Continuity means uninterrupted. So, if there are not any breaks and everything goes on continuously, then there's continuity.

Continuous

Taking the function f which is continuous at x = a, then the value of f ( x )is equal to the value coming from the limit as x → a.

That is, Image 7

Definite integral

Definite integral is the integral of a function over actual defined limits or intervals.

Integral of f (x) is F(x) if dF / dx = f (x)

Derivative

The derivative of a function f is a function that measures the rate of change of the y-values of f with respect to change in the x-value. Another notation for the derivative is dy/dx.

Derivative image
            

Image 7

Differentiation

The process of obtaining the derivative of a function.

Example

If you see a differentiation between one brand of toothpaste and another, and you notice that, point out how they're distinct.

Discontinuous

Discontinuous is not continuing without interruption in time or space

Equation    

Equation is a mathematical statement where both sides of the equal signs are equal quantities.

Exponential function

Exponential function is a function that an independent variable appears as an exponent. And in mathematical form relation such that each element of a given set is associated with an element of another set.

Image 8


Extremum

The Extremum value is the point located farthest from the middle of something.      

First derivative

We can say that the first derivative is the result of differentiation the instant change of one quantity relative to another; df (x) / dx

Function

A function is the one word that gets used a lot and means lots of different things. A mathematical relation associating elements between sets. 

Infinite Limit

When we say a limit =∞, technically the limit doesn't exist.

Integral
Image 8                             


One of the important applications of the integral is in determining the area under or between curves.

Integrate

Integrate is calculating a summative operation in calculus. To integrate is to be part of something as a whole.

Integration

The word differentiate, means "set apart." Integrate is its opposite in the integration, an operation determining the area bounded by a function.

Interval

Interval is a set containing all points between two given endpoints. an interval is a done one utmost of time distance between two things.

Inverse function

The inverse function is a function obtained by expressing the dependent variable of one function as the independent variable of another. If and g are inverse functions if f (x) = y and g (y) = x.

Limit 

The limit of a function y = f (x) is similar to the extreme trend of the values of f (near x = a).
The limit represents, Image 9, stands for the phrase: “The values of getting closer and closer to y = L as the x-values approach a.”

Image 9


Linearity

Linearity is the property of having one dimension. They also tend to combine astounding logistical complexity with remarkable conceptual simplicity.

Logarithmic    

The logarithm number is equal to the exponent on a given base that would give that number.                 

Magnitude

The magnitude is the value or dimensions or distance from a reference point like origin. 

Example :- Earthquakes have great magnitude in that they are powerful            

Image 10
        

Maxima

The highest point on the curve which gives the largest output. 

Minima

The minimum is the point on a curve where the tangent changes. The minimum is the smallest possible outcome. 

Monotonic

Monotonic is a sequence that always increases or always decreases in value. 

Second Derivative

Also the acceleration, the second derivative is the rate of change of the rate of change of a point at a graph.

Series

The Sum of the sequencing of numbers is called a Series.

Image 15
          

Slope

The slope is equal to the value of dy/dx that you get at each point. It is the deviation from the horizontal

Image 12


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Substitution

A substitution is an event in which one thing is replaced by another. It means "putting in place of another."

Summation

The definition for summation is the process of adding things together, or the final sum or total.

Tangent

Tangent is mainly a mathematical term, meaning a line or plane that intersects a curved surface at exactly one point. 

Theorem

A theorem is a widely followed true mathematical statement.
It is important  to understand apply theorems like:

  • Extreme Value Theorem (EVT)
  • Mean value theorem (MVT)

Velocity

Velocity is the rate of change of position that is derivative of position. 
And acceleration is the derivative of velocity. Or we can say that acceleration is the second derivative of position.

Velocity image

 
Volume

Volume refers to the amount of 3-dimensional space occupied by an object. 

Volume image
        


Calculus Symbols

As important as it is to know the calculus terms, it is also important to know how to interpret calculus symbols.

Calculus & analysis math symbols table below mentioned

Symbol Symbol Name Meaning/definition

Example

\lim_{x\to x0}f(x) Limit  The limit value of a function   
ε epsilon represents a very small number, near-zero ε → 0
 e e constant / Euler's number e = 2.718281828... e = lim (1+1/x)x , x→∞
y ' derivative derivative - Lagrange's notation (3x3)' = 9x2
y '' second derivative derivative of derivative (3x3)'' = 18x
y(n) nth derivative n times derivation (3x3)(3) = 18
\frac{dy}{dx}
 
derivative derivative - Leibniz's notation d(3x3)/dx = 9x2
\frac{d^2y}{dx^2}
 
second derivative derivative of derivative d2(3x3)/dx2 = 18x
\frac{d^ny}{dx^n} nth derivative n times derivation  
\dot{y} time derivative derivative by time - Newton's notation  
Math expression time second derivative derivative of derivative  
Dx y derivative derivative - Euler's notation  
Dx2y second derivative derivative of derivative  
\frac{\partial f(x,y)}{\partial x} partial derivative   ∂(x2+y2)/∂x = 2x
integral opposite to derivation  
double integral integration of function of 2 variables  
triple integral integration of function of 3 variables  
closed contour / line integral    
closed surface integral    
closed volume integral    
[a,b] closed interval    
(a,b) open interval    
i imaginary unit i ≡ √-1 z = 3 + 2i
z* complex conjugate z = a+bi → z*=a-bi z* = 3 + 2i
z complex conjugate z = a+bi → z = a-bi z = 3 + 2i
Re(z) real part of a complex number z = a+bi → Re(z)=a Re(3 - 2i) = 3
Im(z) imaginary part of a complex number z = a+bi → Im(z)=b Im(3 - 2i) = -2
| z | absolute value/magnitude of a complex number |z| = |a+bi| = √(a2+b2) |3 - 2i| = √13
arg(z) argument of a complex number Angle of the radius in the plane complex arg(3 + 2i) = 33.7°
nabla / del gradient / divergence operator ∇f (x,y,z)
Math expression vector    
Math expression unit vector    
x * y convolution y(t) = x(t) * h(t)  
Math expression Laplace transform F(s) = {f (t)}  
Math expression Fourier transform X(ω) = {f (t)}  
δ delta function    
lemniscate infinity symbol  

Summary

Calculus is one of the greatest intellectual achievements of humankind. It allows us to solve mathematical problems that cannot be solved by others.

We can Calculus divided into two main parts which is below mentioned :- Differential calculus and Integral calculus.

Differential calculus studies the derivative and integral calculus studies the integral.

In this blog we covered both the calculus terms and calculus symbols. 

Written by Neha Tyagi


Frequent Ask Questions (FAQ’s)

What is calculus in layman terms?

Calculus means Calculation.

What is the meaning of algebraic expression ?

A combination of numbers and letters equivalent to a phrase in language e.g. x2 + 3x – 4.

What calculus of variations ?

An extension of calculus used to search for a function which minimizes a certain functional (a functional is a function of a function).

What is the meaning of derivatives ?

A measure of how a function or curve changes as its input changes.


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