Mathematical Analysis Articles

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Solving Exponential Equations

  Equations consisting of exponents can be solved using the properties of exponents and logarithms. These properties are useful for evaluating arithmetic operations like addition, subtraction, multiplication and division. A very usual way of defining the exponents is: Any number raised to some power or the number of times it is multiplied. For example, 4c is an exponential expression whose base is equals to 4 and power equals to real number 'c'. The power is basically known as exponent in the exponential expression. Solving exponential equations means to solve for one or more variables in the base or exponent. Let us learn it with the help of some examples as follows:

Example: Suppose we have an exponential equation as: A (1 / 2) – A = 4. Find value of A?

Solution: Given equation can be solved by bringing the terms of A to the right side as:

A (1 / 2) – A = 4,

A (1 / 2) = A + 4,

To proceed, we 1st remove the fractional exponent from the equation by squaring both sides of the equation:

(A (1 / 2)) 2 = (A + 4) 2,

A = A2+ 16 + 8A,

Or A2+ 16 + 7A = 0,

We can get two possible values of A because the resulting equation after squaring is a quadratic equation and using the general formula we evaluate A as:

A = -7 ±√ (49 – 64) /2

We are getting imaginary roots for equation and so no real value as such is possible for original equation.

Instantaneous speed is referred to as the rate at a specific instance of time and is several times supposed to be the average speed. The latter one is different from former.

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repetition definition

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According to repetition definition, it is a process of iteration to approach a desired goal or result or target. During the repetition process each process is called an “iteration” and output of each iteration is used to find next iteration. In mathematics, this process is basically used in functions to find the next values of particular function where each output value works as input for next iteration.

It is used to check the behavior of complex functions by getting values at regular interval of time. Collatz Conjecture and Juggler Sequence are some of the problem where we mainly use the repetition process or iteration process. It is the repeated application for any process or function to minimize the complexity. There are lots of other problems like patterns in polynomials, candy for everyone, integer iterations on a circle, Mandelbrot and Julia sets, newton's method and the cantor set where we use repetition. To understand that more deeply we take an example of the Mandelbrot set. In this we have a function x2 + c where we denote x0 for real or complex number. By applying the function x0 we get the new number as:

x1 = (x0)2 + c

Now iterate the output result of previous computation then,

x2 = (x1)2 + c

Similarly we again iterate the output result of previous computation then

x3 = (x2)2 + c

Again we iterate the output result

x4 = (x3)2 + c

and finally we take another iteration,

x5 = (x4)2 + c

  In this process following outcomes which we get after the iteration are x0, x1, x2, x3, x4 and x5. We can also say that it is the orbit of x0 under the iteration of function x2 + c.

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partial fraction

In the previous post we have discussed about What is imaginary number and In today's session we are going to discuss about partial fraction.

Partial fraction is a technique that we use to simplify complex fractions. Let us learn how the approach of partial fraction works with help of an example.

Suppose we have a fraction of the form: (b3 + 4 b2 - 1) / (b2 - 4 b + 4).

Factorizing the denominator using mid – term splitting we get:

(b2 - 4 b + 4) = (b – 2)2,

Here we see that factorization of both numerator and denominator are not possible further. So, we directly use the technique of partial fraction as follows:

(b3 + 4 b2 - 1) / (b2 - 4 b + 4) = (b3 + 4 b2 - 1) / (b – 2)2,

= P / (b - 2) + Q / (b – 2),

= P (b – 2) + Q (b – 2),

= Pb – 2 P + Qb – 2Q,

(b3 + 4 b2 - 1) = b (P – Q) - 2 (P + Q),

Comparing LHS and RHS of above equation we get:

P + Q = 1 /2 and P – Q = 0,

Solving two equations in P and Q we get: P = Q = 1 /4

And our original fraction becomes:

(b3 + 4 b2 - 1) / (b2 - 4 b + 4) = 1 /4 (b – 2) + 1 /4 (b – 2) = 2 /4 (b – 2) = 1 /2 (b – 2).

Next we discuss the concept of noble gases. A noble gas configuration is considered to have a valence shell that is completely filled. Except for the element helium (He) which has only 2 electrons in its valence shell, all other elements have eight have 8 electrons in the outer shell of the atom.

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What is imaginary number

In the previous post we have discussed about What is Differential Calculus and In today's session we are going to discuss about What is imaginary number. In this blog we will see the Imaginary Number definition. Any numbers which is having iota (i) symbol with it are said to be imaginary numbers. Basically imaginary number is denoted as the symbol ‘i'. Now we will understand the concept of solving imaginary numbers. We will solve imaginary numbers step by step:

Step 1: First of all we have an expression which has imaginary values for solving imaginary numbers. Let we have an expression (7 – i) (4 + 9i).

Step 2: In above given expression imaginary number is present. Now multiply first term of first set to each value of the next set and we have to follow same procedure for the second term of first set. On multiplying above number can be written as:

= (7 – i) (4 + 9i),

= (7 * 4) + (7 * 9i) - (4 * i) – 9i2,

Step 3: Then simply multiply the above and also find value of ‘i’. On further solving we get:

= (7 * 4) + (7 * 9i) - (4 * i) – 9i2,

= 28 + 63i – 4i – 9i2, as we know that the value of i2 is -1. Put -1 in place of i2. So it can be written as:

= 28 + 63i – 4i - 9 (-1); on further solving we get:

= 28 + 59i + 9. In this way we can solve the imaginary expression.

For imaginary number one condition is given as:

Value of i = √-1, Here if we use iota in the expression then we can write the square root of negative numbers. For example: (√ a (√ b)), it can also be written as √ ab.

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What is Differential Calculus

In the previous post we have discussed about Calculating p Values and In today's session we are going to discuss about What is Differential Calculus. Hi friends, in mathematics, we will study different types of differential derivative. In this blog we will discuss about differential calculus. Differential Calculus can be defined as the rate of change of a quantity, integral calculus that calculates the quantity where the rate of change is given.

The elementary aim of study in differential calculus are the derivative of a function, that is related assumption such that the differential and their properties. The derivative of a function at a decided input value defined the rate of change of the function which is near to the input value. The action of calculating a derivative is said to be differentiation. Mathematically, the derivative we get at a point same as the slope of the tangent line to the graph of the function at that point. If we take the case of real - valued function of a single real variable then the derivative of a function at a point commonly defined the best linear approximation to the function at that given point.

As we see that the differential calculus and integral calculus are joined by the basic theorem of calculus in which the differentiation is the opposite case to integration.

We also know that, differentiation is having the properties which is closely to all quantitative disciplines. For example, if we see the case of differential calculus in physics then the derivative we get for the displacement of a moving body with respect to time is the velocity of the body, and the derivative we have for the velocity with respect to time is acceleration. According to the Newton's second law of motion the derivative of the momentum of a objects is same as the force given to the body. This is all about differential calculus.

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Calculating p Values

Before learning how to calculate p value, let us first understand what is the p value?

First of all P- value stands for probability value and it is nothing but a statistical measure which helps the scientists in ensuring or you can say determining whether their assumptions or hypotheses are correct. So we can now say that p value is used for measuring the importance or significance of any hypothesis test.

The p value is generally used in statistics. Now coming back to the question: how to calculate p value, we have a level called significance level which helps the scientists in determining whether the data from the research is statistically correct or not and p value is directly related to that level. We have to find out so many other statistical values like chi-square etc first then only we can calculate the p value. And we can use the table for finding it having other statistical values.

In the question: how to calculate p value, we have more to elaborate like in any hypothesis test there are two types of hypotheses: null hypothesis and alternative hypothesis. First one sees no difference between the two populations and the second one says that there is a difference between two populations.

You must be thinking that how can we determine that which value of p can be termed as statistically significant. When we have very small values of p, we can just reject the null hypothesis but we cannot say much about the alternative hypothesis. So we can say either when the p values are less than the significance level α which is 0.05 or 0.01 or the null hypothesis is being rejected, the result is said to be statistically significant.

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Mathematical Induction

In the previous post we have discussed about First Order Differential Equation and In today's session we are going to discuss about Mathematical Induction. Whenever we have any expression to solve then Mathematical Induction is define as the process of solving or proving the given expression. We can describe the process of mathematical induction as falling of dominoes.

In this process first of all prove the first statement true and then on the basis of first statement all the further statements are proved that may be infinite in numbers. It is explained as if the first statement is true in among infinite number of statements then further statements are automatically proved true.

In the process of mathematical induction there is basically three steps defined that are basis steps, next is inductive step and last statement is for all numbers satisfied the expression.

In basis step the given statement is proved true for n = 1 or n =0 means expression is solved for the lowest term. (know more about Mathematical Induction , here)

In next step that is known as inductive step expression is solved for n +1 or it will also be described as if basis step is solved value n = k then in inductive step it will proved for

n = k + 1.

In this process the hypothesis also known as inductive hypothesis that is proved for several values of n.

And at last the given statement is proved true for all the values of 'n' that define giving hypothesis true for the different values of n.

In chemistry basically there are three State Of Matter that are solid, liquid and gas and these all are changes from one to another.

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First Order Differential Equation

In the previous post we have discussed about differentiate calculator and In today's session we are going to discuss aboutHi friends, we will study different order of equation in blogs. Here we will see the First Order Differential Equation. Differential equation can be defined as an equation that accommodatederivatives and those derivatives are ordinary derivatives or partial derivatives. The ratio defined for changing the function to change in its autonomous variable is said to be differentiations.

In mathematics, a differential equation is known by two names that is ordinary and partial differentiation equation. It is depends on the conditions. There are so many types of differential equations. Some of the differential equations are said to be first order differentiation and some of these are known as second order differentiation equation. Here we will talk about first order differential equations. In first order differential equations many equations are involved such as linear equation, separable equation, exact equation, Bernoulli differential equation, substitution equation and Euler’s method. Let’s have introduction about all first order differential equations.

Linear equation: Linear equation is used to characterized and find the linear first order differential equations.

Separable equation: Separable equation is used to describe and calculate separable first order differential equations.

Exact equation: It is used to characterized and evaluate the exact differential equation.

Bernoulli differential equation: In mathematics, one of the best equation in differential is Bernoulli differential equation. Bernoulli differential equation substitution method is used to evaluate the differential equations. (know more about Differential Equation, here)

Substitution: This equation is also used to define the differential equation.

Euler’s method: It is also one of the best methods in the differentials that is used to calculate the approximation of solutions of differential equations. These all equations are included in case of first order differential equations.

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differentiate calculator

Differentiation is the process of finding the rate of change for a given value. This rate of change is calculated on the basis of some other variable. Differentiation is calculated by using the rules of differentiation that gives the derivative of the particular function. Derivatives are basically calculated for three types of functions and that are:

Algebraic function: These functions are describe as f (x>n) that is equal to n (x > n-1).

Trigonometric function: These functions are expressed by using some of trigonometric functions as f (sin x) that is define as cos x.

 Exponential function: These functions are described in the form of exponent as d (e>x) and it will solve as e>x.

For calculation of derivative of such functions, there is an online tool that is known as differentiate calculator. This calculator helps in calculations of derivative faster and in more efficient way. In this calculator there is one text box in which user enter the required expression for which they want to find the derivative and then click on the submit button. After clicking on the submit button it will give the appropriate answer in very short period of time. There is one other text box that is used for enter the variable on which whole differentiation is based means rate of change of the given function is calculated on the rate of change of that variable. If in a given function there are more than one variable than there are more than one derivatives of the function varied on the basis of variable on which differentiation is done.

Third Law of Thermodynamics that is providing the information about the entropy of a matter according to this law for a perfect crystal at the absolute zero temperature entropy of that crystal is also zero. 

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Horizontal Asymptote

In the previous post we have discussed about Horizontal Asymptote Rules and In today's session we are going to discuss aboutAsymptote of a curve is defined as a line in such a way that the distance between curve and line reaches almost to the value zero as they go to the infinity distance. In coordinate geometry generally there are two type of asymptote.

-> Horizontal asymptote: Horizontal Asymptotes are the horizontal lines in which graph of function tend to a → + ∞.

The horizontal line is given by:

⇒a = c; the given equation is a Horizontal Asymptotes of a function a = f (t); If it satisfy the given equation.

⇒lim t → - ∞ f (t) = c

Or, we can write it as:

 ⇒lim t→ + ∞ f (t) = c.

Now we will see how to solve horizontal asymptotes? Take an example and see how to solve horizontal asymptote?

Let f (t) = (2t – 1) (t + 3) / [t (t – 2)]

As we know that the given function is in factor form, first we will convert the given function in the standard form. To find standard form we have to multiply the given values.

So, the standard form of the equation is:

⇒f (t) = 2t2 + 5t – 3 / t2 – 2t

From the above mention equation avoid every value except the largest exponents of ‘t’ that is present in numerator and denominator. (know more about Horizontal Asymptote, here)

So we can write it as:

⇒f (t) = 2t2 / t2

On solving we get 2.

So the horizontal asymptote for the horizontal line y = 2. 

-> Vertical asymptote: The equation of vertical line is given by:

⇒x = t;

Given equation is a vertical asymptotes of a graph which has a function y = f (t); this function is applicable when one of the given condition is true.

The two conditions are mention below:

1. lim t → a- f (t) = + ∞;

2. lim t → a+ f (t) = + ∞;

This is a brief introduction of horizontal asymptote definition and vertical asymptote.

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Horizontal Asymptote Rules

In the previous post we have discussed about horizontal asymptote and In today's session we are going to discuss about Horizontal Asymptote Rules. The horizontal asymptote of any function may be defined as that value on the vertical axis or the y axis of the coordinate system that the function wants to achieve but actually it does not reaches there. So here we will discuss some of the important horizontal asymptote rules which help us in finding the horizontal asymptotes of any kind of the function.

Generally there are just two very simple horizontal asymptote rules or we can say the steps which we have to follow to get the horizontal asymptotes of any function. (know more about Asymptote, here)

The first and the most important rule is that the function or the equation which is being given to us or for which we have to find the horizontal asymptotes should be converted into the form that is standard for it because we can apply the further operations only on the standard form of the function to get the horizontal asymptotes. Then the second and the final step is that we have to vanish every single term from the place of the denominator and the place of the numerator but leave those terms in which the exponents of the x are the greatest.

Now let us take an example in which we will take use of these rules of the horizontal asymptote. Suppose we have to find the horizontal asymptote of the function y = ( 7x2 + 8 ) / ( 2x2 + 5 ). According to the rules first we have to convert the function into the standard form but we see that it is already in standard form. Thus the horizontal asymptote is y = 7x2 / 2x2 that is y = 7 / 2.

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horizontal asymptote

In mathematics, asymptote of a curve can be defined as a line in such a way that the distance between curve and line reaches to value zero as they tends to infinity.

Horizontal asymptote: Horizontal Asymptotes are the horizontal lines in which graph of function tend to a → + ∞. The horizontal line is given by: s = u; the given equation is a Horizontal Asymptotes of a function s = f (t); If it satisfies the given equation

⇒lim t → - ∞ f (t) = u, we can also write it as:

 ⇒lim t→ + ∞ f (t) = u. Let’s discuss how to solve horizontal asymptote? Take an example and see how to solve horizontal asymptote?

Let f (p) = (2p – 1) (p + 3) / p (p – 2). As we know that the given function is in factor form, first we will convert the given function in the standard form. To find standard form we have to multiply the given values. So, the standard form of the equation is:

⇒f (p) = (2p2 + 5p – 3) / (p2 – 2p); in the given equation we avoid value excluding the largest exponents of‘t’. So we can write it as:

⇒f (p) = 2p2 / p2, on further solving we get 2. So the horizontal asymptote for the horizontal line y = 2. Now see small introduction about vertical asymptote. (want to Learn more about horizontal asymptote, click here),

Vertical asymptote: The equation of vertical line is defined as: x = p. It is vertical asymptotes equation that has a function y = f (p); this function is applicable when one of the given condition is true. The two conditions for vertical asymptote are:

1. lima t → a- f (p) = + ∞;

2. lima t → a+ f (p) = + ∞;

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How to Tackle Asymptote

The term asymptote is actually used in the analytical type of the geometry. For any type of the curve we can define the asymptotes as those lines such that the distance from them to the curve approaches zero when the both are tending near the infinity. But sometimes another condition is also added to the definition of the asymptotes that the curve may not cross the line very frequent infinitely although it is not seen much in the modern descriptions of the asymptotes. (know more about Asymptote, here)

The term asymptote is not just defined in one way. It has also got one definition according to the concept of the algebraic type of the geometry. On the basis of this theory, the asymptotes can be defined as those lines which are the tangent to the curve near the infinity.   

Now let us discuss that from where the word asymptote has been derived. This word has been derived from the Greek word which means “ not falling together “.

The asymptotes are also of the different kinds. Commonly we have three kinds of the asymptotes which are the vertical asymptotes, the horizontal asymptotes and the oblique type of the asymptotes.

Suppose any curve that is being given by the graph of a function y = g ( x ) then in that condition the horizontal asymptotes are those horizontal lines that the graph of the function tends to approach if x is tending toward the negative or the positive infinity. While the vertical asymptotes are those vertical lines close to which the function tries to grow without having any bound. We should know that the rational function always has a minimum of one horizontal or oblique asymptote.  Multiple Linear Regression is also an important topic of mathematics which is also comes in IIT JEE exams.In order to get IIT JEE Papers visit online educational portals and In the next session we will discuss about horizontal asymptote. 

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Asymptotes

Asymptote is a term of analytical geometry. Asymptotes of any curve are lines with condition that the distance between the line and the curve tends to approach 0 when they tends to infinity. However some of the definitions include one more condition that curve might not intersect the line very often infinitely but this is not common in the modern definitions.

The term asymptote can also be defined in theory of algebraic geometry. According to this theory, an asymptote can be defined as the line that is tangent to curve in region of infinity.

Word asymptote is derived from Greek word 'asymptotos' that means ‘not falling together’. Now let us discuss briefly about different types of asymptotes. Generally there are 3 types of asymptotes that is horizontal asymptote, vertical asymptotes and the oblique asymptotes.

If we consider the curves which are given by graph of any function say y = f (x) then horizontal asymptote is horizontal line which graph of that function tries to approach when 'x' tends to positive infinity or negative infinity whereas vertical asymptote is the vertical line near which that function just grows without any bound.

Now let us discuss about how the asymptotes of any function which is rational occur. Any function which is rational possesses at least 1 asymptote which is horizontal or which is oblique and it is possible that it has many asymptotes which are vertical.

In order to get more help on the topics: Asymptotes, How to Find the Volume of a Sphere, cbse biotechnology syllabus, you can visit our next article.

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inverse of a function

Inverse of a function is expressed as the reverse process of doing something that will take to the starting point from where we have started previously. It will be explained in simple words as undo process in which all the work done is going to its starting position. If we have some values that are given into the specific function and produce some output then inverse of function is define as a function in which we will enter the coming output and it gives the initial values.

We can express it in form of two expression as if there are two functions named as A and B that take an input c as A (c) then B (c) is inverse of function if A B (c) = c and B A (c) = x .

It is a process in which we find inverse value of the function and by putting one of value we get another value.

According to the definition of inverse of a function it is a set of ordered pairs and by interchanging the elements as (first element with second and vice versa) , get that set of ordered pair. If a function F (a) then its inverse function is expressed as F - 1 (a).

We can explain it as if a function F (a) = (a , b) , (c , d) , (e , f) then inverse of a function is denoted as F -1 (a) = (b , a) , (d , c) , (f , e).

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limit calculator

In the previous post we have discussed about How to Use Implicit Differentiation Calculator and In today's session we are going to discuss about limit calculator,  In calculus, there is need to find the limits of function for calculation and sometimes it is very difficult to calculate the limits for the function. So for ease of calculation of limits we can use the limit calculator that is an online tool .This tool computes the limits for a function on the basis of the variable that is given into the text box of limit calculator.

This tool also generate the limits that are one sided or two sided that are support by the function given into the text box of the limit calculator. In this calculator user can enter the point of limit just by entering the number or may enter the expression from which expression or function is calculated.

We can define it as if there is an expression and we want to calculate it up to infinity then we have to enter infinity into the text box and it automatically generate the expression that is describes the calculation up to the infinity.

 The direction of the calculation that is expressed as negative, positive or both (two sided) that means if we select the negative side calculation it calculate the limit in negative direction or less than zero or when select the positive side it calculates the limits greater than zero means in positive direction and when select both that means positive and negative both limits are calculated by the limit calculator.

Limit of the function is defined as when a function is equal to any constant value and its defined through a variable for which there is no predefined value then that constant value is the limit of function and it tends to the value of variable for which that constant value occurred.

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How to Use Implicit Differentiation Calculator

In the previous post we have discussed about Measures of Central Tendency and In today's session we are going to discuss about Implicit Differentiation Calculator, In calculus when we talk about the differentiation it define as the way of finding the derivative with respect to some variable. Some times in mathematics there are some equation in form of p and q that are define as one function in form of other means function q = f (p). In this expression function q defined in terms of function that is based on the variable p. This function are known as implicit function and when the function q defined in terms of p is differentiated then this technique is known as the implicit differentiation. This differentiation is expressed as find the derivative of function q in form of p and there is no need of solving the equation of function q. This is expressed in the chain rule that defines a function may be expressed in terms of other function.

There is one online tool that is known as Implicit Differentiation Calculator that helps in finding the implicit differentiation of given function by applying the chain rule functionality. there is a text box in the tool in which we have to enter the implicit function and submit it for implicit differentiation. Implicit Differentiation Calculator calculate the derivative of the entered implicit function accurately and efficiently.

We can define it as if there is function p(q (x)) then according to the chain rule first differentiate the outer function and then inner function. It will expressed as d p(q (x)) = p' (q (x)) q' (x) and implicit differentiation is the special category of the chain rule differentiation.

Partial derivative calculator is an onine tool that help in calculation of the partial derivative of the given function just by defining the variable on which function is differentiated. ICSE board stands for Indian certificate of secondary education that conducts the examination for the students and also provide the books, papers .model papers etc to the students.