vertex of parabola

In the previous post we have discussed about Permutation and Combination and In today's session we are going to discuss about vertex of parabola.


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In mathematics, parabolas are basically graphs which have equations of the form:

                                                F(x) = ax² + bx + c                             where a≠ 0,

In parabolas, there are both highest and lowest points depending upon where they are going to open up or down. These points are called vertex. Parabolas are basically used to justify many real life situations like height above ground of an object thrown upward, to check period of time after some time, to check the area of rectangle with a particular width, to check the maximum height of any object etc. In all these situations we need to find some co-ordinates which we get with the help of parabolas. In parabolas there is vertical line of symmetry which passes through its vertex. This line of symmetry helps us to find out co-ordinates. To understand the vertex of a parabola we take an example where we need to examine line of symmetry also. So we have an equation which is

                                                                F(x) = x² – 4x - 5

And we have to find out the vertex of parabola. Now to get the roots of 'x' we use the quadratic formula which is:

                                                                X = [-b ±√(b² – 4ac)]/ 2a,

Where the values of a, b and c are 1, -4 and 5 in the equation. Now we put these values in above equation. So

                                                                X= - ( - 4)±√((-4)² – 4(1) (-5) ) / 2(1),

After solving further we get         x = (4 ± 6) / 2 = 2 ± 3,

These value is also called as x- intercept. Above point shows that it starts from 2 and add 3 to get one intercept, and when we subtract 3 from 2 we get another intercept. So 2 is the midpoint of two intercepts. Finally, we have intercept points which is 5 and -1. Now if we find 'y' value which passes through 'x' value then we put 'x' value in the given equation so

                                                                F(2) = (2)² – 4(2) – 5 = -9,

So the vertex of parabola is (2, -9).

Next we will discuss How to Calculate Molar Mass.

Cbse sample papers 2013 are available online.

Permutation and Combination

 In the previous post we have discussed about parametric equation and In today's session we are going to discuss about Permutation and Combination.

Collection of different objects and symbols in a particular sequence or particular order is known as permutation.

Collection of different objects is called as combination here order doesn't matters. In other words it is an unordered collection of a unique size.

Now we will talk about the formula for finding the Permutation and Combination. 

Formula to find the permutation is given as:

Permutation = ⁿp_r = n! / (n – r)!,

And formula to find the combination is given as:

Combination = ⁿC_r = ⁿp_r / r!;

Here, value of ‘n’ and ‘r’ indicate the non- negative integers and also r ≤ n value.

Value of ‘r’ indicates the size of each permutation.

Value of ‘n’ indicates the size of set from which element are permuted.

‘!’ indicate the factorial operator.

Now we will see example of combination and permutation.

Example: Calculate the number of permutation and combination where value of ‘n’ is 7 and the value of ‘r’ is 4?

Solution: We know that formula for permutation and combination is:

Permutation = ⁿp_r = n! / (n – r)!,

Combination = ⁿC_r = ⁿp_r / r!;

Given, n = 7 and r = 4.

First we will find the factorial of 7. The factorial of 8 is = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040.

Now find the factorial of (7 – 4);

The factorial of (7 – 4) is = (7 – 4)! = 3!,

So the factorial of 3 is = 3 * 2 * 1 = 6;

Now divide 5040 by 6;

Permutation = 5040 / 6 = 840;

Now find the factorial of 4.

The factorial of 4 is = 4 * 3 * 2 * 1 = 24;

Now divide 840 by 24.

Combination = 840 / 24 = 35.

We will see How to Find the Volume of a Cube in the next session.

Cbse class 12 board papers can be downloaded from CBSE board website.

parametric equation

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Parametric equation is used to define a relation with help of parameters. Parameter equation is given as: x = f (c) and y = g (c). Now we will understand how to find the second derivative of this equation. Second derivative of a parametric equation is given as:

⇒ d²y / dx² = d / dx (dy / dx), it can also be written as:

⇒ d²y / dx² = d / dc (dy / dx).dc / dx, we can also write it as:

⇒ d²y / dx² = d / dc (y / x) 1 / x;

⇒ d²y / dx² = xy – yx / x³;

Now we will understand this derivative with help of an example:

Suppose we have x (c) = 8c² and y (c) = 5c, then first derivative of this equation is given as:

⇒ [(dy / dc) / (dx / dc)] = y (c) / x (c),

Here the value of x (c) indicates the derivative of ‘x’ with respect to ‘c’. If derivative of an equation is given in this way then we will use chain rule for calculating the equation. So on moving further we get:

⇒ dy / dx = [dy / dc . dc / dx] by the definition of chain rule:

⇒ dy / dc = dy / dx . dx / dc,

Differentiate both given function with respect to’ c’ and we get:

⇒ dx / dc = 16 c and dy / dp = 5,  if we put these values in the equation then we get:

⇒ dy / dx = y / x = 5 / 16c, which is the required solution.

Median of a Triangle can be defined as a line segment which joins a vertex to the midpoint of the opposing side.

Icse 2013 solved papers can be downloaded from different websites. In the next session we will discuss about Permutation and Combination. 

Definition of complex number

In the previous post we have discussed about 11th Grade Math and In today's session we are going to discuss about  Definition of complex number.

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Here we are going to discuss an important concept that is complex number. Complex number can be defined as any number that can be written in the form of p + iq, here value of ‘p’ and ‘q’ stands for real number and ‘i’ stands for imaginary unit. Here value of ‘i’ is given as √ -1. In above given form ‘p’ is denoted as real part and ‘q’ is denoted as imaginary part of complex number. Complex number p + iq can also be described using the point (p, q). If its real part is given as zero then it is called as pure imaginary and if its imaginary part is zero then it is known as real number. Now we will discuss some rules of complex number.

 i = √ -1, the value of 'i' is represented as √ -1. Here if we use iota symbol in the equations then we can write the square root of negative numbers. For example: (√ x (√ y)), we can also write this given value as √ xy.

Let’s discuss some rules that are used for solve imaginary numbers. As we discussed above value of 'i' is given as √ -1, on squaring the iota we get:

= i² = √ -1 * √ -1, we get the value of i² = -1.

In case of i³ it can also write this equation in squaring form:

= i³ = i² * i; (we know that value of i² is -1).

= -1 * i = -i.

In case of i⁴, first write it in the squaring form,

= i⁴ = i² * i², put the value of i² we get:

= -1 * -1 = 1. This is all about complex number definition.

There are different types of Multiplication Properties which are identity, commutative and so on.

Iit jee syllabus can be downloaded online.

11th Grade Math

Mathematics is an interesting subject, which needs innovation and practice to prove yourself. When we look at the contents taught to students till class 10, we observe that they are quite different from what child studies in grade 11. Now when we talk about topics of grade 10th, it includes the basic concepts of mathematics which includes arithmetic, geometry at basic stage, trigonometric basics too.

But moving ahead, we observe that 11th Grade Math includes the topics which are quite new for the students and are of different interest. When we learn about the integration and differentiation in beginning it looks as it we are learning some research topics. The high level mathematics helps the child to relate some of the topics of mathematics with physics. Even learning Physics becomes easy for the students of mathematics in grade 11, as many of the proofs and solving equations are based on the concepts which the child is learning in math class. Besides this we have the topic of mathematical induction, permutation and combination along with the higher level of probability, which helps the child to develop the logical skills and the reasoning concepts.

Learning mathematics in grade 11 has its own thrill, but the child need to be in the regular practice of the subject. We often observe that the science student who has opted for the mathematics in grade 11 is busy in solving the mathematical problems in about 45 – 60 % of his academics studies, and the left over subject is distributed evenly among all other subjects. Mathematics needs involvement and interest of the child to perform well in the subject.

Ordinary Differential Equations is one of the topic taught in class 11. Icse Syllabus For Class 8 is also available online and papers of different subjects can be downloaded. 

 

What is graphing exponential functions

As we all are very well aware about the concept of exponent that describe a value to the power of any number through which any number can multiplied by itself. In the same aspect exponential function can also be describe as a function which can be represented as the power of x of exponential function. This can be represented as a^x. In the section, the discussion held on the topic of graphing exponential functions. To understand the methodology of plotting a exponential function on to a graph we first need to understand why we use exponential function. The basic reason behind using a exponential function is that it is capable of modeling or representing the rate of change between the independent and dependent variable.
The exponential function is a kind of function which can be represented in the form of f (x) = a^x. In this exponential function ‘a’ is a variable which include any positive value but greater than 1. The concept of graphing an exponential function can be categorized into two categories on the basis of their shapes. When the value of variable ‘a’ of exponential function is positive but less than one then it generate a curve from top left side to right down side. On other hand when value of variable ‘a’ of exponential function is greater than one then it generate a curve of graph left down side to right top side. This function have their application in various field of real world like estimating population growth, compound interest, bacterial calculation and radioactive growth. Some time the concept of exponential function also be describe as xy^a where x is a positive value and y is also a positive value but not equal to 1.
In chemistry the concept of Properties of Acids can be describe as characteristics which is able to define the feature of acids. Indian certificate of secondary education having the syllabus of all subjects for any particular class, which is popularly known as icse syllabus 2013.

solve using matrices

In the previous post we have discussed about 11th Grade Math and In today's session we are going to discuss about solve using matrices. Matrices are defined as the set of objects or set of numbers or may be a set of characters in the form of a particular format of rows and columns. These rows and columns are defined for all matrices and a matrix is also known as two dimensional arrays. Various operations can be performed over matrices that are mathematical operations such as addition, subtraction, multiplication and inverse of a matrix can also be determined. Matrices are defined in the order “m x n” where m is the number of rows and n is the number of columns. An equation can be represented in the form of matrix, as left hand side of equals to symbol is represented in the matrix form to the left side only and similarly the right side is represented to the right side in the matrix form. Let us consider an example with the equations x + y = 5 and 3x – y = -1. These equations can be represented in the form of matrix and we will solve using matrices. Representing these equations in the form of matrix we get coefficients of x and y in one matrix and x and y in other matrix. And the constants will be represented to the right hand side in the form of matrix.

Potential energy is defined as the energy which is in the form of stored energy. Potential energy formula is given as potential energy is equals to the product of mass of object to the acceleration of gravity to the height of object. In a mathematical form it is represented as P.E= m x g x h where m is the mass of object, g is the acceleration due to gravity and h is the height of object. Icse board is similar to cbse board which runs all over India. It appears to give more practical knowledge and icse 2013 board papers can be easily discovered on net. It provides help to the students by giving an idea of questions.

11th Grade Math

In the previous post we have discussed about Standard Deviation Calculator and In today's session we are going to discuss about 11th Grade Math. The grade 11th is one of the turning points of our life that decide our professional future in a particular field. When students came in 11th standard then they need to choose particular subject like science, commerce or arts stream. In today’s life science - math is one of the popular streams which are chosen by the most of students. At the time when students came into 11th grade then they noticed that course of math in 11th grade in harder or not easy to tackle in comparison of previous year.

The interest towards the math of students gets higher when they select math stream in grade 11th. Here in this section discussion held on the topic of 11th Grade Math that helps the students when they are discussing about which stream they need to select or syllabus of math in grade 11th. The 11th Grade Math’s we generally deals with advance concept of algebra which includes some topic that are real numbers and their algebraic expressions. It also include problems that are related with 1st degree of inequalities and polynomial equations. In the calculative part of mathematics, math includes other important topic like slope of a line and their rate of change, calculative logarithm functions and matrix analysis and their equations, last but not least the concept of rational root theorem.

In the syllabus of math of 11th grade students can also understand the concept of pre calculus that deals with sequences and series, trigonometry functions and their inverse, concept of conic section, graphs and their sinusoidal functions and so on.

Some portion of 11th grade math’s include the study of statistics to handle the real world problem very easily.

Protons and Neutrons are the part of any molecules that can be consider as a nucleons which are attracted with each other by nuclei force. In school examination, for student preparation various websites pro vide 10th maths question paper to perform better in exam.  

Standard Deviation Calculator

In the previous post we have discussed about sample standard deviation calculator and In today's session we are going to discuss about Standard Deviation Calculator. Hi friends, in this blog we will discuss the definition and calculation of Standard Deviation Calculator. In mathematics, Standard Deviation can be defined as the variation exists from the given mean value or in other word expected value  or average value. Generally it is known by symbol sigma as (σ). Now we will understand the concept Standard Deviation Calculator. It is a online tool or machine which is used to find the value of standard deviation. Using this online machine we can get the vlaue of standard deviation within a second. In other word this online machine will gives us the value of standard deviation of given set of data in one click. To find the value of deviation with the help of calculator we have to understand some steps so that we can easily the answer. (know more about Standard Deviation Calculator, here)
Steps to follow to find the value of standard deviation are mention below:

Step 1: Using this online machine we can easily find the vlaue of standard deviation. so we use the formula as ∑ = √[(1 / N) Σi (xi - μ) 2].

Step 2: Here the value of ‘σ’ shows standard deviation, vlaue of ‘N’ shows the number of data Points and value of ‘Xi’ shows the Random Variable in which the value ‘i’ lies between 1 to N.

Step 3: In above formula ‘μ’ shows the mean value of the given data set. At last we have to calculate the exact value of σ. This above formula is predefined in this online machine. We have to put the data values in the text box and press the solve button to get the answer. It is very easy to use.

Specific Heat of Iron can be defined as the amount of heat it takes to raise iron one degree. Before entering in the examination hall please prefer all cbse paper for class 9. It is very useful for 9th class students.

sample standard deviation calculator

In the previous post we have discussed about How To Find The Range Of A Function and In today's session we are going to discuss about sample standard deviation calculator. In mathematics, sample standard deviation calculator is a online machine that is basically used to solve the standard deviation of the given data values form its mean. Those students are not know the concept of standard deviation can also use it very comfortably. This machine make the calculation so easy. In other word we can easily solve any standard problem using this calculator. Let’s understand some steps to solve standard deviation problem. (know more about standard deviation , here)

Step 1: First enter the data values in the text box.

Step 2: Then press the solve button to get the result.

In standard deviation calculator the formula defined to find standard deviation is give as:

σ= √ [(1 /N) ∑i (xi – μ)²

Step 3: ‘μ’ symbol shows mean values of data set. At last we have to write the final answer for standard deviation.

Let’s understand it with the help of small example:

Find the standard deviation for the given data values 10, 20, 30, 40.

Solution: To solve it we need to follow the above steps so that we can easily solve the standard deviation.
Step 1 : Formula to find standard deviation is given as σ = √[(1 / N) Σi (xi - μ) ²], here value of N = 4, so we find mean of given data values.

Mean = μ = 10 + 20 + 30 + 40;

μ = 100 / 4, μ = 25.
Step 2: Then find the value of (xi - μ)², xi = the random variable, i.e. 10, 20, 30, 40;

(10 - 25)² = 225,

(20 - 25)² = 25,

(30 - 25)² = 25,

(40 - 25)² = 225,

-------------------------

                 = 500,

--------------------------

Σ ( xi - μ )² = 500,

Step 3 : Then the variance is given as: Variance = Σ ( xi - μ )² / N,

Variance = 500 / 4, Variance = 125;
Standard deviation = σ = √[(1 / N) Σi (xi - μ)²]; put given values in formula to get answer. σ = √ [(1 / 4) 125],

At last we get the value of standard deviation is 11.180.

Specific Heat of Ice is given as  2.108 kJ/kgK. icse syllabus for class 1 is important for class first student.

How To Find The Range Of A Function

Hi friends, in this blog we will see How To Find The Range Of A Function. Function is taken to show a association between set of inputs values and set of outputs values in such a way that every value of input is associated to exactly one value of output. Let we have a function f (r) = r / 6 (f of ‘r’ is ‘r’ divided by the value 6") is a function. So if we are going to put the value of ‘r’ as 6 then we get the value of function as 1, it can be written as: f (6) = 6 / 6 = 1,

Now we will understand how to calculate range of a function. In function, all the ‘y’ coordinate values are said to be range of a function. In the same way we can also find out the domain of a function, all the possible terms of ‘x’ coordinate are known to be domain of a function. Let we have given some values (12, -18), (-71, 84), (53, -42), (-25, 17), then range of function is all the ‘y’ coordinate terms.

Domain = 12, -71, 53, -25.

Range is all ‘y’ coordinate values,

Range = -18, 84, -42, 17.

Let's understand that how to calculate the range of a function. Some steps are taken to calculate the range of a given function which are shown below:

Step1: To calculate the range of a function first have to take a function which contains ‘x’ and ‘y’ coordinates.

Step2: As we discuss above the range of a function is all ‘y’ coordinates values.

Step3: In above function the terms of ‘x’ and ‘y’ coordinate are there then we can easily calculate the range and domain of a function.

In this way we can easily find out the range of a function.

Now we will see Properties of Acids.

Acids are sour in taste. It is also turn blue litmus paper to red. Free download cbse books to get more information about properties of acids.

 

How to Find the Domain of a Function

Hi friends, we will study different types of functions such as linear function, quadratic function and so on. Here we will see How to Find the Domain of a Function. Function can be defined as a tool used to confirm the relationship between the given values. Now we will see how to find the domain of a function. Generally, functions are defined as f (u) here ‘u’ is the value that we given it. Like, f (s) = s / 2 ('f' of 'p' is divided by 2) is a function. Here we can find out different values on putting the different value of variable s.

For calculating the domain of function first we need to discuss about what the domain of a function is. If we select the entire values of x - coordinates in the given function, then these x- coordinates values are said to be the domain of a function. In same way we can also find the range of a function, all possible ‘y’ coordinate values are known as range of a function. Let we have some values (4, -6), (-3, 7), (15, -9), (-19, 8), then domain of function can be obtained as:

The domain of a given function = 4, -3, 15, -19.

Range is all ‘y’ coordinate values,

Range = -6, 7, -9, 8. Now we will understand that how to find domain of a function in details. We will see some steps to calculate domain of a function:

Step 1: To find the domain of a function first we assume a function which contains ‘x’ and ‘y’ coordinates.

Step 2: As we discuss above the domain of a function is all ‘x’ coordinates values.

So if we have values of ‘x’ and ‘y’ coordinates then we easily find the domain and range of a function. This is all about the domain of a function.

Product Differentiation is used to make a product more attractive by contrasting its unique. cbse syllabus for class 9th 2013 is useful for class 9 th student.

graphing exponential functions

In the previous post we have discussed about How To Find The Range Of A Function and In today's session we are going to discuss about graphing exponential functions. Exponential functions are the kind of functions that can be represented in ex. Here the variable e can be defined as a number that has approx value equal to 2.7183. The basic reason behind the popularity of exponential functions is that it is a function that models a relationship between constant change and a proportional change of independent variables. The most basic exponential function’s notation is given below:

y = e^x

In the above given notation the value of y depends on the exponential value of x. In the same aspect, the value of get increases faster when the value of x increases. Here we are going to discussing how to Graphing Exponential Functions. In this concept we study how to plot a exponential functions on a graph. Before discussing about this topic we need to clear one thing that we required a little bit knowledge about a graph that helps us on plotting exponential function’s value on graph. The graph of exponential function’s always lies above the x – axis and some time get very closer to the x – axis at the negative side of x but in both cases it do not touch the x – axis. So, finally at the time of graphing exponential function we need to follow some steps that are given below: (know more about graphing exponential functions  , here)

I ) From starting a table that records all the values of y which is calculated on the basis of y.

II ) To calculate the value of y we required to use the properties of exponents .

III) When above given both step completed then plot the points on graph.

IV ) After that draw a line according to the plotted points.

The Specific Heat Capacity of Water is 1 calorie/gram in degree Celsius which is equal to 4.186 joule per gram °C that has higher capacity in comparison of any other common substance. Students who are appearing in icse board exam 2013 and they want to check their exam preparation can use icse sample papers 2013 for checking their performance. 

How To Find The Range Of A Function

Range is the difference between greatest data value and least data value.
Suppose there is two set of data value first  one is X( a,b,c,d) and second one is Y(p,q,r,s).
If a function ‘f’is defined from set X to set Y then for f:X->Y , set x is called the domain of function f and set y is called co-domain function of f. The set of f images of the elements of x is called the range of function .
So in this case
Range = p,q,r

How to find the range of a function
#The easiest way to the range is on the graph. The range of the function is the range of y values it enclose.
#If domain is given, the range is the range of y values  corresponding to x values in the domain.
# check if function repeats. Any function which repeats along the x-axis will have the same range for the entire function.
Example: sin(x) has a range of -1
#The domain of a function’s inverse function is equal to that function’s range.
#Take a derivative of the graph. Find the y values at these points and the ends of the domain and take the most extreme ones as the boundary of the range. (know more about Range, here)

 Example: find the range of the function : g(x)= x/3+5 if the domain is -6,-3,0,3,6
Solution: we know that domain is the values x takes. Range is the corresponding values of the function takes.
Here g(x)= x/3+5.
For example when x=-6,we get -6/3+5=3
When x=-3, we get -3/3+5=4
When x=0, we get 0+5=5
When x=3, we get 3/3+5=6
When x=6, we get 6/3+5=7
So the range is (3,4,5,6,7)
In cbse syllabus for class 9th 2013 , specific heat equation is given by
Q = mc delta T
Where Q is amount of heat needed
M is the mass
C is specific heat capacity
Delta t = temperature difference.

Cartesian Product

In the previous post we have discussed about Differential Calculus and In today's session we are going to discuss about Cartesian Product. An ordered pair, usually denoted by (x , y) is pair of elements x and y of some sets. Like the Cartesian product of two sets A and B is set of those pairs whose first co coordinate is an element of A and second co-ordinate is an element B. The set is denoted by A x B and is read as ‘A cross B’ or ‘product set of A and B’. Basically these sets are work as ordered pair usually denoted by (x, y) is a pair of element x and y of some sets, which is ordered in the sense that (x, y) ≠ (y, x) whenever x≠y. Here x is called first co-ordinate and y is called second co-ordinate of the ordered pair (x, y), for example ordered pair (1, 2) and (2, 1) consist of same element 1 and 2 but they are different because they represent different points in the co-ordinate plane. (know more about Cartesian Product, here)

Now we use this concept in Cartesian product as following manner:
A x B = (x, y): xϵ A  É… yϵ B, A and B are different sets which are multiplying with each other.
Let the value of set  A = 1, 2, 3 and the value set  B=3, 5;
A x B=1, 2, 3 x 3, 5;
=(1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5) (Here we get many ordered pair of A x B  which are called Cartesian product value of A x B),
And B x A=(3, 1), (3, 2), (3, 3), (5, 1), (5, 2), (5, 3) (Here we get many ordered pair of B x A which are called Cartesian product value of B x A ),
A x B ≠ B x A (it means both product are not same).
Similarly, we can define Cartesian product for n sets A1, A2, A3 …….An
A1, A2, A3 …….An  = ( x1, x2, x3 …….xn  )  x1 Ïµ A1 É…  x2ϵ A2 É…  x3ϵ A3 ……………. É… xnϵ An
The element (x1, x2, x3 …….xn  ) is called as n-type elements.
In this way we can solve the all Cartesian product problem.
Now we discuss what is Logarithmic Differentiation? We know exponent expression is defined as in the power expression like if y = ax, where x is logarithmic of y to the base a, in this case we will use the logarithmic differentiation. ISEET Physics syllabus helps students for their better study.

Differential Calculus

In the previous post we have discussed about limit laws and In today's session we are going to discuss about Differential Calculus. Dear students in today’s class we will study a very important and very interesting topic that is calculus. It is a branch of mathematics. We will take a brief idea about this topic.“Calculus” is a Latin word which means a tiny stone that is used for counting. It is a branch of mathematics which is basically based on functions, limit, differentiation and  integration. It constitutes of two major branches. One of which is differential calculus and another is integral calculus. Differential calculus is the study of change of limit, function and derivative . Differential calculus has vast  applications in the field of science, engineering and  sometime in economics also.  (know more about Differential Calculus, here)
It is also called that differential calculus is a very important method or system of calculation which is explained by the representative manipulation of expressions. There are many examples of differential calculus. Differential Calculus was the first success of modern mathematics and it is very tough to avoid its importance.  It only clears more efficiently the system of mathematical analysis.
Calculus is generally developed by manipulating very small quantities. Calculus has been used in every branch of the computer science, physical science, statistic, engineering, economics, business,etc. Calculus can also be used in combination with other mathematical branches.
Lets take a example: it can be used in theory of probability to settle on the probability of a nonstop arbitrary variable from an assumed density function. it can also be used with linear algebra to determine  the "best fit" linear approximation.
Calculus is also applied to evaluate estimated solutions to equations.  It is the very important way to solve differential equations in most applications.
So, we can say that calculus has very wide applications in the field of mathematics and even imagine of mathematics is not possible without “Calculus”.
There are different types of carbohydrates. It can be seen in icse syllabus 2013. 

limit laws

Limit is used to define the value in which the function is approaches as an input. Limit is also used in calculus and also used to define the continuity function. Let’s talk about the limit laws. In mathematics, there are different types of laws which are given below. Now see all the limit laws one by one.
1.      Addition law: - The additive law is given by: Let the lim _{y ⇥b} f (y) and lim _{y ⇥b} g (y) both are exist then
lim _{y ⇥b} f (y) + lim g (y) = lim _{y ⇥b} f (y) + lim _{y ⇥b} g (y).
2.      Subtraction law: - The subtraction law of limit is given by: Let the lim _{y ⇥b} f (y) and
lim _{y ⇥b} g (y) both are exist then
lim _{y ⇥b} f (y) - lim g (y) = lim _{y ⇥b} f (y) - lim _{y ⇥b} g (y).
3.    Constant law: - The constant law of limit is given as: Let ‘c’ is a constant, and the limit lim _{y ⇥b} f (y) exist, the constant law can be written as: lim _{y ⇥b} c. f (y) = c. lim _{y ⇥b} f (y).
4.      Multiplication law: - The multiplication law is given by: Let the lim _{y ⇥b} f (y) and
lim _{y ⇥b} g (y) both are exist then
lim _{y ⇥b} f (y) . lim g (y) = lim _{y ⇥b} f (y) . lim _{y ⇥b} g (y). (know more about limit laws, here)
5.      Division laws: - The division law of limit is given by: Let the lim _{y ⇥b} f (y) and
lim _{y ⇥b} g (y) both are exist and lim _{y ⇥b} g (y) ≠ 0, then the division law can be written as: lim _{y ⇥b} f (y) / g (y) = lim _{y ⇥b} f (y) / lim _{y ⇥b} g (y). This is all about the limit laws. Now we will see the Triangle Inequality Theorem. It says that the sum of two sides of a triangle is always larger than the third side. To get more information then prefer online tutorial of cbse syllabus for class 9 and In the next session we will discuss about Differential Calculus.