In previous article, one of the important topics of mathematics that is Quadratic functions was explored and in today's article we are going to move further in grade XII to learn Algebra online syllabus with Exponents and exponential functions. This topic has very short introduction of itself but various situations and applications of it, tends it to cover a wide portion of Algebra unit.
Let us first start with the general description of exponent after that we will move to the main topic of Exponential functions with their growth and decay property. Before coming into 12 standard students learn almost all the aspects of logarithm and in respect of those exponents as well. Whenever an exponent is described it is clear that presence of Euler constant 'e' is surely there in the function. The main reason to use exponent in any function is that the corresponding constant change in any variable in parallel with other independent variable is measured with the help of graph. For more on exponents visit here.
If any function is described as f(x) = yp then 'y' is called base and 'p' is its relative exponent. There are some pre-defined properties for exponential function as follows:
Suppose two integer variables are m and n, then
• yn = y × y × · · · × y with n factors
• y1/n means the nth root of a. That is, a1/n is that positive number which satisfies
(y1/n ) × (y1/n )) × · · · × (y1/n ) = y
Where there are n factors on the left hand side.
• ym/n = (y1/n ) × (y1/n ) × · · · × (y1/n ) (y1/n )
• y−n = 1/ yn
The above situations are used when x is an integer but if ‘x’ is a real number then:
Suppose ‘p’ is a rational number
p=m/n; here m, n are integers
Now assume ‘x’ as a real number that cannot be written in a form of rational number. Two common examples of this type or irrational numbers are x = √2 and x = π.
For other possible real numbers that can be easily converted into fraction of rational form use decimal to fraction conversion for example:
x = 3.914712334317 . . .
Then, if we are working to 3 decimal points we would write
x ≈ 3.915
and this number can certainly be expressed as a rational number:
x ≈ 3.915 = 3915/1000
So, in this case
yx = y3.914712... ≈ y3.915 = y 3915/1000
and the final term: y 3915/1000 can be determined in the usual way by using your calculator.
This is all about Exponents; now let us move towards the next part of this session that includes the description of Exponential functions.
Exponential Functions: - suppose a function is as f(x) = yx
Now for this function the value of ‘x’ clearly varies according to a fixed base value and its appropriate exponent value.
So the functions (yx) (as different values are chosen for a) are called exponential functions. Let us explain in more practical manner:
If y>z> 0 then yx > zx only if x>0
And if x<0 then yx < zx
Euler constant ’e’ is the most important and widely used exponential function whose particular base value is 2.7182818 . . . ,
e = 2.7182818 . . .
The number ‘e’ has its own importance in mathematics like number π and, like π it is also an irrational. So ‘e’ also cannot be written as the quotient of two integers. The value of ‘e’ is stored in most of the science calculators to sort out exponential function.
There are various ways of calculating the value of ‘e’. For example, ‘e’ represents the end point of sequence of numbers as follows:
(2/1)1 . (3/2)2 . (4/3)3 . ...... (18/17)17..... .. (65/64)64.....
Decimal form of these ( up to 6 decimal points) is as: 2.000000, 2.250000, 2.370370, ..., 2.637929, . . . , 2.697345, ...
This slowly converging unit can represent as single unit as ‘e’s expression as:
e= Lim m-> infinity [( m + 1) /m]m
There is one more way to calculate the value of ‘e’ and that is infinite series:
e = 1 + 1/1! + ½! + 1/3! + ¼! .........+ 1/m! + .....
Here students know that
m! = m × (m − 1) × (m − 2) × . . . (3) × (2) × (1)
According to the standard analysis it can be shown that first eleven terms of this series represents the value of ‘e’ with almost negligible error of 3 × 10−8.
In general all functions of the form of ax are called exponential functions but still to be more specific we refer to ex as
The exponential function.
Exponential growth:-
If s > 1 then it can be easily represented no matter how large K is:
sx/xk → ∞ as x → ∞
It means if K is a fixed value and can be chosen as large as wanted, still if the value of ‘x’ increases then sx will also increase and being ahead of it always. It will happen until the value of‘s’ is greater than 1. This phenomenon of sx growth as x increases is called exponential growth.
Exponential decay
As explained in previous point that the behavior of ‘ex’ as x → ∞ is known as exponential growth. Just in similar passion,
if we explain or characterize the expansion of e–x then that procedure will called as Exponential decay.
This is all about Exponential functions and their relative properties. As the definition of exponential function defines that in it we observe the relative change in any variable in respect of any other variable. This variation may occur in two different styles as Direct and Indirect.
Let us first talk about the direct variations in exponential function. If two variables are said to be direct variant of each other then it means that if one of them get doubles then other one also changes in same ratio, similarly if the second one is reduced in some amount then first one is also decreased in same ratio. For example: assume a function as y = x + 1
If value of ‘x’ grows up then at the same time value of ‘y’ also gets increased.
Now let’s see “what is Inverse variation?”, this type of variation causes opposite change in any value whenever the related value has increased or decreased. For example, if x and y are two values that are related to each other with Inverse relation then if value if x increases as it doubles then at the same time the value of y gets decreased by half of its value. This type of change gets complicated when more than two independent variables are included in the function. Let us take example of Newton’s kepler theorem:
Formula is as F = Gm/r2
So it represents that whenever mass of the body gets doubled then the strength of force also gets doubled and when distance is increased then the strength of the force gets fall in its fourth.
The thing is that both variations: direct and Inverse, are occurring together so determination of the value gets complicated.
This is all about Exponents, Exponential function with their growth and decay functionality, and types of variations in exponents; all of these grade XII Algebra topics comes as a single chapter in your syllabus because there queries solution requires implementation of their common properties, which are explored in detail in this article.
In the successive article of this one, we will discuss the next included algebra topic in grade XII syllabus and that is co-relation of data.
After reading this article still there is any issue or doubt remains in your mind or you want to solve related math queries of it, and then access the online math tutoring websites where students like you can ask their queries and surely will have immediate assistance by online math tutor. These math education websites provides proper interactive session to students so that they can ask their problems on the regular basis to be clear in every math concept. Students can use some available options like online chat, relevant study material, video seminars, unsolved and solved math worksheets, online math calculators for quick solution of the problem, and many more. All these options increase the ease level of the students’ study that causes their better involvement in studies.
Grade XII math syllabus is comparatively vast than any other standard and the main difficulty in it is that all the topics of it usually have the math application problems whose introductory part was studied by you guys in your previous classes. So always learn every math topic with your pure attention to execute it with ease in the final examination because math is a kind of practical subject that can enrich your aggregate percentage when potentially studied otherwise it also can ruin your overall performance in other subjects.
In upcoming posts we will discuss about Patterns and Functions in Grade XII and Standard distributions. Visit our website for information on 12th physics syllabus Maharashtra board
Let us first start with the general description of exponent after that we will move to the main topic of Exponential functions with their growth and decay property. Before coming into 12 standard students learn almost all the aspects of logarithm and in respect of those exponents as well. Whenever an exponent is described it is clear that presence of Euler constant 'e' is surely there in the function. The main reason to use exponent in any function is that the corresponding constant change in any variable in parallel with other independent variable is measured with the help of graph. For more on exponents visit here.
If any function is described as f(x) = yp then 'y' is called base and 'p' is its relative exponent. There are some pre-defined properties for exponential function as follows:
Suppose two integer variables are m and n, then
• yn = y × y × · · · × y with n factors
• y1/n means the nth root of a. That is, a1/n is that positive number which satisfies
(y1/n ) × (y1/n )) × · · · × (y1/n ) = y
Where there are n factors on the left hand side.
• ym/n = (y1/n ) × (y1/n ) × · · · × (y1/n ) (y1/n )
• y−n = 1/ yn
The above situations are used when x is an integer but if ‘x’ is a real number then:
Suppose ‘p’ is a rational number
p=m/n; here m, n are integers
Now assume ‘x’ as a real number that cannot be written in a form of rational number. Two common examples of this type or irrational numbers are x = √2 and x = π.
For other possible real numbers that can be easily converted into fraction of rational form use decimal to fraction conversion for example:
x = 3.914712334317 . . .
Then, if we are working to 3 decimal points we would write
x ≈ 3.915
and this number can certainly be expressed as a rational number:
x ≈ 3.915 = 3915/1000
So, in this case
yx = y3.914712... ≈ y3.915 = y 3915/1000
and the final term: y 3915/1000 can be determined in the usual way by using your calculator.
This is all about Exponents; now let us move towards the next part of this session that includes the description of Exponential functions.
Exponential Functions: - suppose a function is as f(x) = yx
Now for this function the value of ‘x’ clearly varies according to a fixed base value and its appropriate exponent value.
So the functions (yx) (as different values are chosen for a) are called exponential functions. Let us explain in more practical manner:
If y>z> 0 then yx > zx only if x>0
And if x<0 then yx < zx
Euler constant ’e’ is the most important and widely used exponential function whose particular base value is 2.7182818 . . . ,
e = 2.7182818 . . .
The number ‘e’ has its own importance in mathematics like number π and, like π it is also an irrational. So ‘e’ also cannot be written as the quotient of two integers. The value of ‘e’ is stored in most of the science calculators to sort out exponential function.
There are various ways of calculating the value of ‘e’. For example, ‘e’ represents the end point of sequence of numbers as follows:
(2/1)1 . (3/2)2 . (4/3)3 . ...... (18/17)17..... .. (65/64)64.....
Decimal form of these ( up to 6 decimal points) is as: 2.000000, 2.250000, 2.370370, ..., 2.637929, . . . , 2.697345, ...
This slowly converging unit can represent as single unit as ‘e’s expression as:
e= Lim m-> infinity [( m + 1) /m]m
There is one more way to calculate the value of ‘e’ and that is infinite series:
e = 1 + 1/1! + ½! + 1/3! + ¼! .........+ 1/m! + .....
Here students know that
m! = m × (m − 1) × (m − 2) × . . . (3) × (2) × (1)
According to the standard analysis it can be shown that first eleven terms of this series represents the value of ‘e’ with almost negligible error of 3 × 10−8.
In general all functions of the form of ax are called exponential functions but still to be more specific we refer to ex as
The exponential function.
Exponential growth:-
If s > 1 then it can be easily represented no matter how large K is:
sx/xk → ∞ as x → ∞
It means if K is a fixed value and can be chosen as large as wanted, still if the value of ‘x’ increases then sx will also increase and being ahead of it always. It will happen until the value of‘s’ is greater than 1. This phenomenon of sx growth as x increases is called exponential growth.
Exponential decay
As explained in previous point that the behavior of ‘ex’ as x → ∞ is known as exponential growth. Just in similar passion,
if we explain or characterize the expansion of e–x then that procedure will called as Exponential decay.
This is all about Exponential functions and their relative properties. As the definition of exponential function defines that in it we observe the relative change in any variable in respect of any other variable. This variation may occur in two different styles as Direct and Indirect.
Let us first talk about the direct variations in exponential function. If two variables are said to be direct variant of each other then it means that if one of them get doubles then other one also changes in same ratio, similarly if the second one is reduced in some amount then first one is also decreased in same ratio. For example: assume a function as y = x + 1
If value of ‘x’ grows up then at the same time value of ‘y’ also gets increased.
Now let’s see “what is Inverse variation?”, this type of variation causes opposite change in any value whenever the related value has increased or decreased. For example, if x and y are two values that are related to each other with Inverse relation then if value if x increases as it doubles then at the same time the value of y gets decreased by half of its value. This type of change gets complicated when more than two independent variables are included in the function. Let us take example of Newton’s kepler theorem:
Formula is as F = Gm/r2
So it represents that whenever mass of the body gets doubled then the strength of force also gets doubled and when distance is increased then the strength of the force gets fall in its fourth.
The thing is that both variations: direct and Inverse, are occurring together so determination of the value gets complicated.
This is all about Exponents, Exponential function with their growth and decay functionality, and types of variations in exponents; all of these grade XII Algebra topics comes as a single chapter in your syllabus because there queries solution requires implementation of their common properties, which are explored in detail in this article.
In the successive article of this one, we will discuss the next included algebra topic in grade XII syllabus and that is co-relation of data.
After reading this article still there is any issue or doubt remains in your mind or you want to solve related math queries of it, and then access the online math tutoring websites where students like you can ask their queries and surely will have immediate assistance by online math tutor. These math education websites provides proper interactive session to students so that they can ask their problems on the regular basis to be clear in every math concept. Students can use some available options like online chat, relevant study material, video seminars, unsolved and solved math worksheets, online math calculators for quick solution of the problem, and many more. All these options increase the ease level of the students’ study that causes their better involvement in studies.
Grade XII math syllabus is comparatively vast than any other standard and the main difficulty in it is that all the topics of it usually have the math application problems whose introductory part was studied by you guys in your previous classes. So always learn every math topic with your pure attention to execute it with ease in the final examination because math is a kind of practical subject that can enrich your aggregate percentage when potentially studied otherwise it also can ruin your overall performance in other subjects.
In upcoming posts we will discuss about Patterns and Functions in Grade XII and Standard distributions. Visit our website for information on 12th physics syllabus Maharashtra board
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