Mathematical Models with Applications

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Imaginary Number

In the previous post we have discussed about Define Mathematical Induction and In today's session we are going to discuss about Imaginary Number. 

Number that contains iota symbol is said to be imaginary numbers. For example: (12 + 31i), (50i – 5) these all numbers have iota symbol. In mathematics, imaginary number is represented as 'i'. Here value of 'i' is given as √ -1. This is all about Imaginary Number definition.

Let’s see how to calculate imaginary numbers. Steps to be followed to calculate imaginary numbers are given below:

Step 1: Take a set of imaginary or two imaginary number set for solving imaginary numbers. Suppose we have an expression (17 – i) (14 + 6i).

Step 2: As we see above imaginary number is present in expression. Here first multiply first digit of first set to every value of next set and now apply same method for second term of first set. So above number can be written as:

= (17 – i) (14 + 6i), on moving further we get:

= (17 * 14) + (17 * 6i) - (14 * i) – 6i2,

Step 3: Now solve above expression we calculate value of ‘i’. On further solving it can be written as:

= (17 * 14) + (17 * 6i) - (14 * i) – 6i2,

= 238 + 102i – 14i – 6i2, as we discussed above value of i2 is -1. Now put -1 in place of i2. So we can write above expression as:

= 238 + 102i – 14i - 6 (-1);

On solving further we get:

= 228 + 88i + 6.

This is how we solve imaginary numbers.

Now we will understand the concept of Liquid Chromatography;

It can be defined as collective term for combination of various types of laboratory techniques for separation of mixtures.

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Define Mathematical Induction

Mathematical Induction in mathematics is an approach that is generally used to prove some statement or an assumption factually by means of conflicting hypothesis when there is no such direct method. Its methodology is based on considering a hypothetical situation in the beginning of problem and then proving it to be incorrect at the extreme end of the solution and thereby proving original statement true. For proving a given statement as true, we must have at least one such value that satisfies it completely. Let us consider a statement a² <= 2 a where a = 2, 3, 4 and so on. To prove it true using mathematical induction, we use following steps:

At 1st we check the statement for a = 1, we see that statement is true. Next we suppose that for any a = b where b = 2, 3, 4 and so on, we have: b² <= 2 b.

If statement b² <= 2 b stays true for values of 'b', then it must be true for a = b + 1 also. In this step we finally prove our statement to be true by contradicting with our hypothesis. So we proceed as follows:

If a = b + 1, then (b + 1)² <= 2 (b + 1),

(b + 1)² <= 2 (b + 1),

b² + 2 b + 2 <= 2 b + 2,

We know for a = b if a² = 2 a, then b² = 2 b.

2 b + 2 b + 2 <= 2 b + 2,

We know that 2 b > 1 because b > 1,

2 b + 2 b + 1 < 2 b + 2,

Hence proved that:

Linear programming method of math is generally used to optimize the given quantity such that we get maximum or minimum value of it. Linear programming solver and mathematical induction are some of the important topics from the point of view of 10 th maths question paper. In the next session we will discuss about Imaginary Number.

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Differential equations of first order

In the previous post we have discussed about exponential growth and decay and In today's session we are going to discuss about Differential equations of first order.  We can define a Differential Equations of First Order as equations that can be written in the form: d k /d x +A (h) k = B (h). Where, “A” and “B” are continuous functions of a variable h. This is the standard form of a first order linear differential equation. The differentiation operation is involved in the equation. The first order differential equation can be linear equations, separable equations, exact equations etc. Suppose we have a first order differential equation as k’ +A (h) k = B (h). The integrating factor for this equation can be written as: c (h) = e∫A (h) d h. We calculate integrating factor to get the solution of the differential equation. Then the solution of the Differential equations of first order can be given as:

k e∫A (h) d h= ∫B (h) e∫A (h) d h + D,

Where, D is any constant of integration. Let us consider an example of a first order linear differentiation equation: h k’ – 2 k = h2

Writing the standard form of first order differential equation:

k’ + A (h) k = B (h)

k’ + 2 / h (k) = h

Thus we have A (h) = 2 /h.

And so our integrating factor is c (h) = e∫A (h) d h= 1 /h2,

We multiply both the sides of the equation by 1 /h2 to get:

k’ /h2 + 2 k /h3 = 1 /h,

On solving it using differentiation and integration we get:

d /d h [k /h2] = 1 /h,

[k /h2] = ò(1 /h) d h,

[k /h2] = ln │h│+ C,

k = h2 (ln │h│+ C),

Where, 'C' is the constant of integration.

Let us now discus the next concept related to potential energy diagram. The potential energy diagram can be drawn as follows:

In the above diagram we discuss about the rate of change of the potential energy that arises during an organic reaction where student can read cbse syllabus for class 11. 

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exponential growth and decay

In the previous post we have discussed about First Order Differential Equation and In today's session we are going to discuss about Hi friends, in this blog we will study about exponential growth and decay. Exponential Growth and Decay can be defined as a real world problems that involves all the problems which increases or decreases exponential. Commonly the exponential growth of any mathematical function comes if growth rate of value of given mathematical function is proportional to actual value of function. If we talk about some important cases then it is also said to be geometric growth. Formula to find the exponential growth for a variable P at the rate of growth “s” for a specific time interval “t” in Integer form can be written as:

Pt = P0 (1 + S) t, here ‘Pt’ show value at time ‘t’ and ‘P0’ shows value of the function at initial time t = 0.

Now we will understand it with the help of example:

If increase in population of any country and decay of any platenum particle because of some reasons, function of exponential as x = 2(5 / 2)t.

So if growth or decay value of any function comes in some of exponential cases then this type of the growth is known as exponential growth. In the same way when there is some cases of decay in function then this types of decay is said to be exponential decay. The formula to find exponential function can be given as: Pt = P0 (1 - s) t

Here ‘Pt’ shows the value of function at time ‘t’ and ‘A0’ shows value of function at initial time as t = 0. This types of model of exponential growth is also called as Malthusian growth model. (know more about exponential growth, here)

Here we get the growth and decays values are the rates in such a way that they display change in some quantity of a particle as per the time goes up.

Third Law of Thermodynamics states that the entropy of a perfect crystal substance at absolute zero value is exactly equal to zero. It is define systematically in cbse syllabus for class 9th 2013.

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First Order Differential Equation

In the previous post we have discussed about second order differential equation and In today's session we are going to discuss about First Order Differential Equation.  Hi 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 accommodate derivatives 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. (know more about First Order Differential Equation, here)

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.

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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second order differential equation

In the previous post we have discussed about calculus worksheets and In today's session we are going to discuss about second order differential equation. A second order differential equation can be defined as an equation that include the unknown function 'p' and its derivative p' and p”, and the variable 'q'. Second order differential equation can be written as: d2p / dq2 = f (p, q, q'). let's discuss which equations are involved in the second order differential equation. In the second order differential equations many equations are involved that are mention below:

We can find the real root of differential with the help of second order differential equations. Using second order differential equations we can also calculate the complex roots of an equation. (know more about differential equation , here)

If the differential equation has repeated roots then it also finds the repeated roots of differential equation.

In brief we can say that with the help of second order differential equations we find the roots of equations.

Now discuss on Laplace transformation because it also very important part of differential equation. Suppose that f (u) is any continuous function then the Laplace transform of the function f (u) is denoted as ∑f (u) and it is defined as:

⇒ ∑f (u) = ∫∞ e-su f (u) dt;

For finding the differential equations we can also use inverse Laplace transformation. Higher order differential also a part of differential equations:

There are many higher order differential equations which are given below:

Nth order differential equation is used to calculate the nth order of an equation.

Linear homogeneous differential equation: it is used to find the second order differential equation to the higher order. Laplace transformation is also included in higher order differential equation. System of differential is also included in higher order differential equation. So there are many differential equations.

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calculus worksheets

In the previous post we have discussed about Vertical Asymptote and In today's session we are going to discuss about calculus worksheets. Calculus is a mathematical concept that helps in solving problems that are related to the integrals, functions, limits, derivatives and so on. In mathematics, the concept of calculus can be understand by categorizing it into two categories first is differential calculus and second is integral calculus. The both concept helps in understanding the fundamental theory of calculus. Here we are going to discuss about calculus worksheets. Calculus worksheets can be consider as a collection of several queries or questions that helps in solving problems related to the concepts of calculus. (know more about calculus worksheets , here)

To find out the various solution of calculus worksheet we need to understand the concept of calculus. It is just a theory that helps in performing the study of change for functions and limits. As we describe that calculus are performed in differential calculus and in integral calculus. These concepts are studying in geometry and algebra by using the concept of limits. Limit is also another concept of mathematics that helps in studying different cases between different distances of two points on graph. So, by whole study we can say that the concept of limit with algebra and geometry enables to solve calculus problem.

In comparison of algebra differential calculus are most popularly used in mathematics because in algebra we focused on getting a slope point on line. On the other hand, the concepts of calculus concentrate on finding slope of a curve.  If we want to define differential calculus then we can say that it a process to find the derivative of a function.

If we want to discuss about the integral calculus then it helps in calculating the area of any region and which has no specified shape. Integral calculus can be define as describing a relationship between definite and indefinite integral. Sometime a question arises that What are Supplementary Angles then we can say that a pair of two angles that forms 180 degree of angles. In board examination, icse syllabus helps the students for guiding in each and every subject for proper study. 

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Vertical Asymptote

In the previous post we have discussed about Imaginary Number and In today's session we are going to discuss about Vertical Asymptote. We can have 3 types of the asymptotes for any kind of the function that is the horizontal asymptote, the vertical asymptote or the oblique asymptote. So we will discuss here some of the crucial facts about the vertical asymptotes.

The vertical asymptote of any function which is rational can be defined as the vertical line that corresponds to the 0’s in the denominator. (know more about Asymptote, here)

For example we have an equation say y = ( x2 + 5x – 3 ) / ( x2 – 7x + 6 ). We know that it is a rational kind of the function. Also it can be seen that it is in the form of a fraction. Now we will see that whether we can have any 0 in the denominator position of the fraction or not. Therefore if we equate the denominator of this fraction to the 0 and then try to solve it, it will give us the values which x should not be. So let us equate it to 0 and solve.

x2 – 7x + 6 = 0

( x – 1 )( x – 6 ) = 0

So x = 1 or x = 6   

Thus x should not be equal to 1 or 6 as we would then have to divide by the 0.

If we draw the graph of this function then we will see that it will avoid the vertical lines x = 1 and x = 6. It is due to the reason that the value of the x should not be 1 or 6. Hence we can say that those values which are just restricted in the domain of the function would be the vertical asymptotes of the function.

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Imaginary Number

In mathematics there are various types of numbers on our real number line like natural numbers, rational numbers, irrational numbers, whole numbers, prime numbers, integers, even and odd numbers, etc and we can see all these numbers on the coordinate plane  but there are numbers like complex numbers and imaginary numbers which are not visible in the coordinate plane. Complex numbers have their own plane called complex plane where they can be located.

An imaginary number is occurs when squared gives a negative result or less than zero and sometimes it can also be a zero. (know more about Imaginary number, here)

We can also say that an imaginary number is nothing but a real number multiplied with the imaginary unit denoted by ‘i’ because according to the property i squared is equal to -1.

For instance, √-4 is an imaginary number as the square of this number is negative that is -4 but as we know that the i2 = -1, therefore, i = √-1, so one can just take out the number from the root and replace the √-1 with the imaginary unit ‘i’. Hence, we can also write √-4 as 2i.

Complex numbers and imaginary numbers are related as when an imaginary number bi is added with a real number a, it becomes a complex number : a + bi, where a is called as the real part of the complex number and bi is the imaginary part of the complex number. And a and b are real numbers.

So we can say that an imaginary number can be defined as a complex number whose real part is zero.

Imaginary numbers play a very significant role in science, physics, engineering and many other important areas.

In order to get help in understanding the topics: imaginary number, upper limits and cbse syllabus for class 10, you can just visit our next page and In the next session we will discuss about Vertical Asymptote. 

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Rotational Symmetry

By Rotational Symmetry, we mean to know that how many times does a particular 2 dimensional figure appears to be  exactly same as it was at the original position till it completes one round of 360 degree.  For different figures, we have different rotational symmetries. Some figures have infinite rotational symmetries and some figures do not have even one rotational symmetry, which means that on completing one round of rotation, the figure will appear exactly same as in the original form. Such figures are said to have the order of rotational symmetry as 1.

 Let us first take a circle. We rotate the circle by any angle from its center of axis. Now we will observe that this figure rotated by any angle of rotation will appear rotationally symmetrical. Thus we say that all the angles of the circle brings the figure at the rotational symmetry. Thus we come to the conclusion that the circle has infinite order of rotational symmetry and all its rotational angles are  the angle of rotation.

Now we take into consideration a scalene triangle. We will start rotating this figure and observe that it does not come in the rotational symmetry until it comes to its original position , i.e. at the rotation of 360 degrees. So we concluded that the order of rotational symmetry of scalene triangle is 1 and the angle of rotation is 360 degrees. Thus we further find that all the irregular polygons are of the order of rotation 1, which means that they only look same after the rotation of the figure by the angle of 360 degrees. (know more about Rotational symmetry, here)

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Inverse Function

In mathematics, a function is a set of order pair in which two elements in a pair should be different. Let we have a function f (a) = (a, b): (1, 2), (4, 3), (7, 8), (6, 11); If we want to calculate the inverse of the given function then we get: f -1 (a) = (a, b): (2, 1), (3, 4), (7, 6), (10, 5). Let’s discuss some of the terminology related to a function: Suppose we have R = F (P); the meaning of the given function is resistance and it is a function of temperature (P). Then the inverse of the given function is: We can write the Inverse Function as: P = f-1 (R); here the temperature (P) is the inverse function of resistance (R). Now we will discuss the inverse functions with the help of example. Let we have given f (s) = s3; and f-1 (s) = 3√ s; If we put the value‘s’ is 5 then we get: = f (5) = 125. And if we find the inverse of the given function then we get: f-1 (125) = 5. Inverse of above mention function is 5. (know more about Inverse Function, here)

Thus, f (f-1 (125) = 125; and f-1 (f (5)) = 5. In general f (f-1 (n)) = n and f-1 (f (n)) = n, here both the function f and f-1 solve the value ‘n’. Now let’s discuss the domain and range of the inverse function. Let we have a function ‘f’, then the domain of the function ‘f’ is range of the function f-1. And the range of the function ‘f’ is the domain of the function f-1. This is all about the composition of inverse functions. In mathematics probability formula helps us to find the probability of any given data. Before entering in the examination hall prefer the icse sample papers. It is very helpful for exam point of view and In the next session we will discuss about Rotational Symmetry.

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inclination definition

In mathematics, inclination is a degree of deviation measure from horizontal and vertical axis or a degree of deviation from a given plane is said to be inclination definition. Inclination is an angle that the path of the plane makes with the axis(horizontal and vertical axis). The minimum positive angle plot form the positive x – axis to the line is said to be inclination of a Line.

The angle that lies between a line and x – axis is called angle of inclination of a line. This angle always lies between 0 degree to180 degrees. The line lies in the horizontal part are always having zero degree angle of inclination. And in case of vertical part, it has angle of inclination equals to 90 degree. (know more about inclination, here)

And also the slope of a line is given by the angle of inclination. Any number which shows the steepness of a line is called a slope of a line. Slope of a line is denoted by ‘m’. The equation of line is given by:

Let ‘m’ is the slope of the line and the coordinates of line is u1, v1 and u2, v2 then the equation of line is:

Slope m = rise = v2 – v1,

                  run      u2 – u1

The slope ‘m’ stands for the horizontal line is zero, the slope ‘m’ for the vertical line is undefined and the slope of parallel lines is same. Let for two perpendicular lines u1 and u2 then u1 u12= -1;

This can also be written as: u1 = -1

u2

This is all about the definition of inclination. If we solve math questions and answers regularly then we achieve good score in the exam. It is very helpful for exam point of view. To get more information than prefer on-line tutor of icse syllabus for class 8 and In the next session we will discuss about Inverse Function. 

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Learn Z-Transform

In the previous post we have discussed about How to Tackle Fourier Transform Problems and In today's session we are going to discuss about Z-Transform, When we want to compare the z-transform with other transform then Laplace transform is a perfect mathematical tool which also performs the task of analysis, monitoring and design of system. Z-Transform is the improvise version of Laplace transform that has discrete time counter and generalization of the Fourier transformed of signal.

According to standard definition we can say that Z Transform performs the task of converting the sequence of real or complex numbers into a complex frequency signal.  Z-transform manipulates the discrete data sequence and acquired a new significance that helps in analysis and formation of discrete time system. To understand the Z-transform and their application, function notation for sequences are very helpful.

Suppose there is a function F(x) which has x >= 0 that can be consider as a sample at time x = t, 2t, 3t and so on. Here ‘t’ can be consider as a sampling rate. When we want to define the z-transform then we need express an infinite series of the reciprocals like z-n.

The Z Transform performs like a integral transform which can be expressed as either a one sided or two sided transform. Here we represents the functioning of  z-transform in two sided form that has discrete time signal f(n) in the formal power series F(z) that are defined below:

F(z) = Z [f(n)] =∑n=-∞∞ f(n)z-n

In the above given notation we define the z-transform where n is an integer and z can be consider as a complex number. Let’s show below how:

z = A (cosɸ +j sin ɸ)

In the above given notation the represented variable A can be consider as a magnitude of z, where else j is the imaginary unit and the given symbol ɸ represents complex value in form of radian. Quotient Rule Derivative is a rule for calculus which act as a method for finding the derivative for a particular function which also a quotient of two other function that already have their derivative. Indian science engineering eligibility test provide the ISEET Books which is very helpful to the students to prepare for exam according to ISEET norms. 

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How to Tackle Fourier Transform Problems

Most of us would have studied about the Fourier series in the subject of the math but many of us are not even familiar with the term Fourier transform. So in this article we will discuss some of the useful facts about the Fourier transform.

The Fourier transform is basically named after the name of the great mathematician Joseph Fourier. This transform is a type of the mathematical transform and is very useful in many different areas. Generally this type of the transform represents any function of the time which is mathematical in the terms of a function of the frequency which is also called as its spectrum of the frequency. The integral theorem of the Fourier gives the details of this relation.

We can now take an example to understand about the Fourier transform more precisely. For that consider the transform which is of a musical chord and is made only of those notes which are pure that is without any kinds of the overtones and which is expressed as the amplitude in the form of any function of the time and thus is such a representation of the amplitudes which is mathematical and also of the phases of the notes which are individual and make it.

 In the transform described in the above paragraph the function of the time is generally known as the representation of the domain of the time whereas the spectrum of the frequency is known as the representation of the domain of the frequency.

Now let us discuss something about the inverse of the Fourier transform. The inverse of this transform is used to express a function of the domain of the frequency in the domain of the time.

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Define Rolle s Theorem

There are many theorems which we usually study under the theory of the calculus in the subject of the math and one of those theorems which is very important and very useful in the theory of the calculus is the Rolle S theorem. Therefore in this article we will discuss some of the important facts about the Rolle S theorem.

Let us first start with the statement of the Rolle S theorem. This theorem of the calculus says that a function which is differentiable and which achieves the same value at the two different points should possess such a kind of the point anywhere between those two points where the 1st derivative of the function is exactly equal to zero. In the above statement of the Rolle S theorem, by the 1st derivative we mean the slope of the line which is tangent to the graph of that function. 

We have now discussed the statement of the Rolle S theorem in the last paragraph so let us now understand this theorem mathematically in this paragraph to have a precise knowledge about it. If a function g which is real valued and is continuous over a interval which is closed represented by [ a, b ] and is also differentiable on the interval which is open represented by ( a, b ) and it is also given that the value of the function g at a is equal to the value of the function g at b that is g ( a ) = g ( b ) then there will be a point c which will lie in the open interval shown by ( a, b ) which will follow the equation which is given below.

g’ ( c ) = 0

In order to get more help on topics: Rolle s Theorem, Partial Derivatives and ISEET Books, you can visit various Online Portals and In the next session we will discuss about How to Tackle Fourier Transform Problems.

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How to Learn Product rule

In the previous post we have discussed about Learn Parabola and In today's session we are going to discuss about How to Learn Product rule. We generally come across many types of the rules in the math related to the differentiation and the integration and one of such rules is the most common, simple and a very easy rule that we all use very often while solving the problems of the differentiation and the integration and that is the product rule so in this article we will discuss about the product rule and the different kinds of the forms in which it can be written.

We generally take more use of the product rule for getting the derivatives of those terms which are made by the multiplication of more than one function. We can explain the use of the product rule by writing the formula which is given as follows. (know more about Product rule, here)

( m . n ) ‘ = m . n ‘ + n . m ‘

The form of the product rule which is given above is the simplest form and also very easy to understand the rules of the differentiation when we come across any function of such kind which itself is made by the multiplication of the two functions. But if we have to make such a form of the product rule which is like the notation of the Leibniz then the product rule will appear like in the line given below.

d / dx ( m . n ) = m . dn / dx + n . dm / dx

where m and n are the functions in x that is the variable in these functions is x, dm / dx is the differentiation of the function  with respect to the x and dn / dx is the differentiation of the function n with respect to the x.

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Learn Parabola

In the previous post we have discussed about differentiate calculator and In today's session we are going to discuss about Learn Parabola. Parabola which is a mathematical term and studied under conic section is nothing but a U-shaped curve with certain specific properties. Parabola can be created from the intersection of any right circular conical surface and a plane which is parallel to a generating straight line of that particular surface. Formally, another way of defining or generating a parabola is as follows: for a given point called as the focus and a given line, which is called the directrix, on a plane, parabola is the locus of points in that plane such that the distance to the focus or point equals the distance to the directrix or line.

Note that any U-shaped curve is not a parabola, it should also satisfy the above conditions and a parabola always has a quadratic equation. (know more about Parabola, here)

The line which divides or splits the parabola from the middle by passing through the focus and is also perpendicular to the directrix is called the axis of symmetry. There is a point on the parabola which intersects the axis of symmetry of the parabola is called the vertex. And the vertex is the only point on the parabola where the curvature is the greatest.

Parabola can be classified into two types: horizontal parabola and vertical parabola. Vertical parabola can open up or down whereas horizontal parabola can open left or right, or it can be in any other arbitrary direction. We can say that all the parabolas are similar as every parabola can be repositioned and rescaled to fit exactly into any other parabola.

There are many applications of parabolas; they are used frequently in engineering, physics, construction purpose and many other areas. Other important applications are in designing of ballistic missiles and automobile headlight reflectors.

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