Smoothing Function in Math

Introduction for smoothing function in math:

In smoothing function in math, let f(x) be a real-valued function of a real variable x and let a be some fixed point in the domain of f. The number L is the limit of f (x) as x approaches a if, given any positive number `epsi`, we can find a positive number `delta` such that f (x) is in the range

|f(x) – L| `<` `epsi`

Whenever x is in the domain

0 `<` |x – a| `<` `delta`

We write

`^lim_(x -> a)` f(x) `=` L

More about for Smoothing Function in Math:

In smoothing function in math, the notation

`^lim_(x -> a)` f(x) `=` L

is read “the limit of F(x) as x approaches a is L.

f(x) `->` L  as x `->`a;

This is read as “f(x) approaches L as x approaches a” or “f(x) goes to L as x goes to a”. I like to explain limits by writing

f(x) `~~` L when x `~~`a (but x `!=` a).

The notation A `~~` B means A is approximately equal to B. In ? We will study a limit

f’(a) `=` `^lim_(x -> a)` `(f(x)-f(a))/(x-a)`

called the derivative. The derivative is a limit, but the concept of limit is more general. Note that in the definition of limit we consider values of x close to a but never plug in x = a. In the interesting examples (like the derivative) setting x = a in f(x) leads to something undefined like `0/0` for smoothing function in math. Is this topic 8th grade math practice problems hard for you? Watch out for my coming posts.

Example for Smoothing Function in Math:

In smoothing function in math,

`^lim_(x -> 1)` 4x `-` 2 `=` 2

`^lim_(x -> 1)` `x^2` `+` 3 `=` 12

Some limits are more difficult to find if they contain a denominator that goes to zero as x `->` a. These limits must be treated carefully using, for example, L Hopitals rule.

Proof that a limit exists requires manipulation of inequalities to determine the relationship between `epsi` and `delte`. It is also necessary to verify that the inequalities |f(x) `-` L| `<` `epsi` and 0 `<` |x `-` a| `<` `delta` are satisfied as required by the definition of a limit.

Suppose we have the limits

`^lim_(x -> a)` f(x) `=` L

And

`^lim_(x -> a)` g(x) `=` M

Then

`^lim_(x -> a)` [ f(x) `+` g(x)] `=` L + M

That is, the limit of the sum of two functions is the sum of the limits.

`^lim_(x -> 1)` [`x^2` `+` 3 `+` 4x `-` 2 ] `=` 2 `+` 12

Answer is     `^lim_(x -> 1)` [`x^2` `+` 4x `+` 1 ] `=` 14

Statistics of Small Numbers

Introduction to Statistics of Small Numbers:
Statistics means that the formal science that are generating learn with the effective use of arithmetical terms. We are also help with the concepts like mode, median, mean and range. They are help with interpretation and also analysis of the terms. In this article, we are going to see examples with using of small numbers in statistics. Having problem with Mode of Numbers keep reading my upcoming posts, i will try to help you.

Mean and Median in Statistics of Small Numbers

Let see the examples of mean and median that are having the small numbers.

Mean:

Mean of the given numbers could be equivalent to the adding up of all the terms and total numbers of terms in the given list. The formula is,

Mean = Average value of the given numbers

Example problem:

Find the mean of the numbers, 3,7,1,4,9,12,6.

Solution:

Given numbers are, 3,7,1,4,9,12,6.

The formula for mean is,

Mean = Average value of the given numbers

Here, the adding up of all the numbers = 3+7+1+4+9+12+6

= 42

Total number of given numbers = 7

Therefore, Mean = `(42)/(7)`

= 6

Answer: Mean = 6

Median:

For measuring the median value for collection of numbers, we must arrange those numbers in an order.

i) When the total number of terms = Even number

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ii) When the total number of terms = Odd number

Median = Middle value of the ordered list.

Example problem:

Find the median of the numbers, 5,2,9,13,4,7.

Solution:

Given numbers are, 5,2,9,13,4,7.

The ascending order of the given group of numbers is,

2,4,5,7,9,13

Here, total numbers = 6

This 6 should be an even number.

Therefore, Median = `(5+7)/(2)`

= `(12 )/(2)`

= 6

Answer: Median = 6

These are the examples of mean and median that is having the small numbers.

Mode and Range in Statistics of Small Numbers

Let see the examples of mode and range that are having the small numbers.

Mode:

The mode value of the collection of numbers could be a number that should occur most frequently in the given list.

Example problem:

Find the mode of the numbers, 4,1,3,9,5,1,2,1.

Solution:

Given numbers are,

4,1,3,9,5,1,2,1

The ascending order of these numbers = 1,1,1,2,3,4,5,9.

Here, the number ‘1’ is coming three times in the group of given numbers.

Therefore, the mode of those collection of numbers = 1

Answer: Mode = 1

Range:

The differentiation among the values like largest value and smallest value is said to be range.

Example problem:

Find the range of the numbers, 4,8,6,2,1,9,5.

Solution:

Given group of numbers are,

4,8,6,2,1,9,5

The formula for range is,

Range = Biggest value – Smallest value

Here, Biggest value = 9

Smallest value = 1

Therefore, Range = 9 – 1

= 8

Answer: Range = 8

These are the examples of mode and range that is having the small numbers.

That’s all about the statistics of small numbers.

Deference in Math

Introduction to difference in math:

In mathematics, difference or subtraction is one of the four basic arithmetic operations; it is the inverse of addition operation, meaning that if we start with any number and add any number and then subtract the equal number we added, we return to the number and we started with. Subtraction or difference is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula,

a − b = c,

are minuend (a) − subtrahend (b) = difference (c). Instead we say that a and −b are terms, and treat subtraction as addition of the opposite. The answer is called difference.

(Source from Wikipedia)

Example Problems for Difference in Math:

Example 1:

Find the difference value of the given numbers, using subtraction operation, 100 and 80.

Solution:

Given numbers using subtraction operation for, 100 - 80

We are going to subtract the two numbers,

100 - 80 = 20

The difference of given number is 20

Example 2:

Find the difference value of the given numbers, using subtraction operation, 550 and 200.

Solution:

Given numbers using subtraction operation for, 550 - 200

We are going to subtract the two numbers,

550 - 200 = 350

Finally we get the difference answer for given numbers are 350.

Example 3:

Find the difference value of the given numbers, using subtraction operation, 235 and 647.

Solution:

Given numbers using subtraction operation for, 235 - 647

We are going to subtract the two numbers,

235 - 647 = -412

The difference of given number is  -412.

Example Decimal Problems for Difference in Math:

Example 4:

Find the difference value of the given decimal numbers, using subtraction operation, 6.5 and 2.6.

Solution:

Given decimal numbers using subtraction operation for, 6.5 – 2.6

We are going to subtract the two numbers,

6.5 – 2.6 = 3.9

The difference of given decimal number is 3.9

Example 5:

Find the difference value of the given decimal numbers, using subtraction operation, 20.865 and 15.358.

Solution:

Given decimal numbers using subtraction operation for, 20.865 – 15.358

We are going to subtract the two numbers,

20.865 – 15.358 = 5.507

The difference of given decimal number is 5.507.

Example 6:

Find the difference value of the given decimal numbers, using subtraction operation, 3.4 and 9.7.

Solution:

Given decimal numbers using subtraction operation for, 3.4 – 9.7

We are going to subtract the two numbers,

3.4 – 9.7 = -6.3

The difference of given decimal number is -6.3.

Regression Slope Coefficient

Introduction to regression slope coefficient:
In mathematics, regression is one of the interesting topics in statistics. Linear regression is defined as the process of determining the relationship between two variables. It is a statistical analysis method which can be used to assessing the association between the two different variables. Let us see some example problems using regression slope coefficient.

Regression Slope Coefficient - Formula for Regression:

Formula for Regression:

Regression Equation (y) = a + bx
Regression Slope coefficient  (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
Intercept(a) =`(sumY - b(sumX)) / N `

Where
x and y are the variables.
b = the slope of the regression line (also called as regression slope coefficient)
a = the intercept point of the regression line and in the y-axis.
N = Number of values or elements
X = First Score
Y = Second Score
`sumXY` = Sum for the product of the first and Second Scores
`sumX` = Sum of First Scores
`sumY` = Sum of Second Scores
`sumX^2` = Sum of square First Scores. I like to share this Adding and Subtracting Significant Figures with you all through my article.

Regression Slope Coefficient - Example Problem:

Example 1:

Find the regression equation by using the regression slope coefficient value.

For the given data set of data, solve the regression slope coefficient.
Solution:

Let us count the number of values.
N = 6

Determine the values for XY, X2

Determine the following values `sumX` , `sumY` , `sumXY` , `sumX^2` .
`sumX` = 444
`sumY` = 21.4
`sumXY` = 1589.1
`sumX^2` = 32890

Substitute values in the slope formula
Regression Slope coefficient (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
= `((6)*(1589.1)-(444)*(21.4))/((6)*(32890)-(444)^2)`
= `(9534.6 - 9506.1)/ (197340 - 197136)`
=`28.5/204`
= 0.14
Substitute the values in the intercept formula given.
Intercept (a) = `(sumY - b (sumX)) / N `
=` (21.4 - 0.14(444))/6`
= `(21.4 - 62.16)/6`
= -`40.76/6`

= -6.79

Substitute the Regression coefficient and intercept values in the regression equation
Regression Equation(y) = a + bx
= -6.79 + 0.14x.

Solution:

Regression Slope coefficient (b) = 0.14

Intercept (y) = - 6.79

Regression equation y = - 6.79 + 0.14x.

Determine the approximate value for y:

When x = 75

Substitute the x value into the regression equation

Regression Equation(y) = a + bx
= - 6.79 + 0.14x.

= - 6.79 + 0.14(75)

= - 6.79 + 10.5
y = 3.71

Solution:

y = 3.71

Solve Math Equation Solutions

Introduction to solve math equation solutions:
An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that to be equal on opposite sides of an equal sign. (Source: From Wikipedia).

The following rules are used to solve the equation and the equation does not change: Add or subtract any variable or number to the both sides of the equation. Multiply or divide any variable or number to the both sides of the equation. I like to share this Solving Equations Using Addition and Subtraction with you all through my article.

Now, we are going to see some of the problems on how to solve math equations solutions.

Problems to Solve Math Equations Solutions:
Example problem 1:

How to solve an equation for the variable a: a + 23 = 26

Solution:

a + 23 = 26

Subtract 23 on both sides of the equation

a + 23 – 23 = 26 – 23

a = 3

So, the answer is a = 3.

Example problem 2:

How to solve an equation for the variable t:` t / 8 ` = 10

Solution:

`t / 8` = 10

Multiply 8 on both sides of the equation

`(t / 8)` * 8 = 10 * 8

t = 80

So, the answer is t = 80. Is this topic Mathematical Induction hard for you? Watch out for my coming posts.

Few more Problems to Solve Math Equations Solutions:

Example problem 3:

How to solve an equation for the variable m:  5m - 7 = -8

Solution:

5m – 7 = -8

Add 7 on both sides of the equation

5m – 7 + 7 = -8 + 7

5m = -1

Divide by 5 on both sides of the equation

`(5m) / 5 = -1 / 5`

m =` -1 / 5`

So, the answer is m = `-1 / 5` .

Example problem 4:

How to do solve an equation for the variable c:  -2c - 86 = -136

Solution:

-2c - 86 = -136

Add 86 on both sides of the equation

-2c - 86 + 86 = -136 + 86

-2c = -50

Divide by -2 on both sides of the equation

`(-2c) / -2 = (-50) / -2`

c = 25

So, the answer is c = 25.

Practice Problems to Solve Math Equations Solutions:

1)      Solve the equation for the variable x: 3x + 41 = 59 (Answer: x=6).

2)      Solve the equation for the variable b: b + 35 = 26 (Answer: b=-9)

3)      Solve the equation for the variable n: n - 13 = 3 (Answer: n=16)

Rounding Numbers Practice

Introduction to rounding numbers practice:

In this article we discuss to find the rounding numbers practice solving problems. The rounding numbers or digits are only approximate. An exact solutions are generally cannot be obtain using rounding numbers or digits. If the number 0, 1, 2, 3, 4, do not change the rounding numbers. All the numbers are right side of the requested rounding number will become 0.The number 5, 6, 7, 8, 9 rounding number round up by one number. Let us see to rounding numbers practice solving problems are given below.I like to share this Rounding Decimals with you all through my article.

Rounding Numbers Practice-example Problems:

Example 1:

Round off 3247 to the nearest ten.

Solution:

We note that 3247 lies between 3240 and 3250

3247 is closer to 3250 than to 3240

(Since 3250 – 3247 = 3, while 3247 – 3240 = 7)

Hence, 3247 when rounded off to the nearest ten becomes 3250.

Example 2:

Round off 3247 to the nearest hundred

Solution:

We further note that 3247 lies between 3200 and 3300

3247 is closer to 3200 than to 3300

(Since 3300 – 3247 = 53, while 3247 – 3200 = 47)

Hence, 3247 when rounded off to the nearest hundred becomes 3200.

Example 3:

Round off 3247 to the nearest thousand

Solution:

Finally, we note that 3247 lies between 3000 and 4000

3247 is closer to 3000 than to 4000

(Since 3247 – 3000 = 245, while 4000 – 3247 = 755)

Hence, 3247 when rounded off to the nearest thousand becomes 3000.

Example 4:

Round off 5257 to the nearest thousand

Solution:

Finally, we note that 5257 lies between 5000 and 6000

5257 is closer to 5000 than to 6000

(Since 5257 – 5000 = 257, while 6000 – 5257 = 743)

Hence, 5257 when rounded off to the nearest thousand becomes 5000.I have recently faced lot of problem while learning translation in math, But thank to online resources of math which helped me to learn myself easily on net.

Rounding Numbers Practice Problems:

Problem 1: Round off 4247 to the nearest ten

Problem 2: Round off 3247 to the nearest hundred

Problem 3: Round off 3247 to the nearest thousand

Rounding numbers practice problems-answer key:

Problem 1: 4250

Problem 2: 4200

Problem 3: 400.

Solve Math Median

Introduction to median in math:-

Median is another measure of central tendency. It is the middle value of a series of numbers.  It divides the series into two equal parts. One half contains values less than the median and the other half contains values more than the median.  So we arrange the series in ascending or descending order to find the median.  If the series contains 'n' numbers and if 'n' is odd, then the median is found by the formula `(n+1)/2` . If n is even then we must find the mean value of the `(n/2)` th and `(n + 1)/2` th  numbers to find the median.

If we have many frequencies  of the given numbers in a grouped data, then we use the formula
Median =` l +((N/2-m)/f)xxc`                                                                                                                                 f
where l is the lower limit of the median class,
m is cumulative frequency up to l,
f is the frequency of the median class,
N is total frequency and
c is the length of the class interval.

To Find Median when Frequency is not Given.
Let us find the median of the following numbers  25,29, 32, 46, 73, 25, 28
Solution:

Step 1: We must arrange the numbers in order.  So let us arrange them in ascending order 25,25,28,29,32,46,73

Step 2: Here n=7 so it is odd

Step 3: So median is `(7+1)/2` th number = 4th number = 29

Step 4: Let us count the numbers before 29.  There are 3 numbers.  25, 25, 28

Step 5: Let us count the numbers after the median.  There are 3 numbers 32,46,73

Step 6: Hence we find that 29 is the middle number.

Answer: Median is 29

Find the median of the numbers  12,29, 24, 36, 32,48,59,73
Solution:

Step 1: Let us arrange them in ascending order .12, 24, 29, 32,36,48,59, 73

Step 2 : Let us count the numbers. They are 8.  n = 8 is even

Step 3 : The formula is to find the median of `n/2` th number and `(n +1)/2` th number . That is `8/2` th number and `(8+1)/2` th number

That means we must find the median of 4th number and 5th number.

4th number is 32 and 5th number is 36

Hence median is` (32+36)/2` = `68/2` = 34

Answer: Median of the 8 given numbers is 34.Between, if you have problem on these topics adding exponents, please browse expert math related websites for more help on cbse model question papers for class 9.

To Find Median when Grouped Data is Given:

A grouped data is given.  Let us find the median using the formula for the grouped data.
The income of  the people in a factory is as follows:-
__________________________________________________________________
Income $                 80 - 100     100- 150     150- 180      180-200      200-250   250-300
__________________________________________________________________
Number of persons       16              24                 26                30                 20              7
__________________________________________________________________
Find the median.
Solution:-
Step 1 : We must find the cumulative frequency first. CI is class Interval
_____________________________________________
Income $        Number of persons Cumulative frequency
CI                            f                                  cf
_____________________________________________
80 -100                     16                            16
100 - 150                   24                            40  m
(l)       150 - 180         26 (f)                        66
180 - 200                   30                            96
200 - 250                   20                           116
250 - 300                    7                           123 -> N
Step 3 : N=123 is odd
Hence we must find `(N+1)/2` th item  = `(123 + 1)/2` th number = `124/2` th number  =  62

Step 4 : The 62nd number lies in the Class Interval 150-180 . Hence l is the lower limit 150,so  l = 150
Step 5 : The frequency of f is 150-180 is 26  f = 26
Step 6 : C is the length of the class interval = 150-180 = 30
so C = 30
Step 7 : m is the cumulative frequency up to l that is the previous Class Interval cf  m  = 40
Step 8 : Median =` l + ((N/2 - m)/f )C `
=`150 + (62-40)/26 xx 30 `
= `150 + 22 xx 30 `
= `150 + 25.38 `
= `175.38`

Answer : Median of the grouped data is 175.38

Hence we can find the median by finding the middle value of a grouped data.