Approximation Calculus

In approximation calculus, we use both the differentiation and integration process. Approximation calculus gives the approximate values. It does not give the complete solution of the problem. The resultant value of the function is not a exact solution of the problem. Approximation calculus mainly uses the differentiation process. Linear function variables are also used in approximation calculus.

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Example problems for approximation calculus

Approximation calculus problem 1:

Find the approximate value of (3.9)4. The given function f(x) = x4. The equation of the tangent line to f(x) at x = 4 can be given as y = mx + b.

Solution:

Given function f(x) = x4

Differentiate the above function with respect to x, we get

f'(x) = 4x3

Find the slope of tangent line at x = 4 is given as,

f'(4) = 4 * (4)3

= 256

In point slope form, the line passes through the points (4, 44) and has the slope 256 is given as,

y - 256 = 256 * (x - 4)

y = 256x - 768

Therefore,

x4 = 256x - 768 at x = 4

Finally,

(3.9)4 = 256(3.9) - 768 = 230.4

Answer:

The final answer is 230.4

Approximation calculus problem 2:

Find the approximate value of (4.5)3. The given function f(x) = x3. The equation of the tangent line to f(x) at x = 5 can be given as y = mx + b.

Solution:

Given function f(x) = x3

Differentiate the above function with respect to x, we get

f'(x) = 3x2

Find the slope of tangent line at x = 5 is given as,

f'(4) = 3 * (5)2

= 75

In point slope form, the line passes through the points (5, 53) and has the slope 75 is given as,

y - 125 = 75 * (x - 5)

y = 75x - 250

Therefore,

x3 = 75x - 250 at x =5

Finally,

(4.5)3 = 75(4.5) - 250 = 87.5

Answer:

The final answer is 87.5

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Practice problems for approximation calculus

Approximation calculus problem 1:

Find the approximate value of (3.7)4. The given function f(x) = x5. The equation of the tangent line to f(x) at x = 2 can be given as y = mx + b.

Answer:

The final answer is 140.8

Approximation calculus problem 2:

Find the approximate value of (5.9)2. The given function f(x) = x2. The equation of the tangent line to f(x) at x = 6  can be given as y = mx + b.

Answer:

The final answer is 34.8

what is the median in math

In statistics theory, the median is the middle value of the set of data after the arrangement of ascending or descending order. If the total of elements in a set is even then take the two middle elements (n and n+1) after the arrangement of the elements and find the average (n + (n+1))/2 which is the required median.

Median in Math – Example Problems

Example 1: In a class nine students’ marks as follows 28, 37, 45, 98, 76, 71, 65, 49, and 58. What is the median?

Solution:

Arrange the elements of a data set in ascending order {28, 37, 45, 49, 58, 65, 71, 76, 98}

Median:

{28, 37, 45, 49, 58, 65, 71, 76, 98}

Total number of elements in a data set is 9 which is an odd number. So find the middle number which is the median.

Here, 58 is the middle value.

Therefore 58 is the median.

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Example 2: In a class ten students’ weights as follows 36, 42, 54, 68, 72, 67, 49, 57, 62, and 74}. What is the median?

Solution:

Arrange the elements of a data set in ascending order {36, 42, 49, 54, 57, 62, 67, 68, 72, 74}

Median:

{36, 42, 49, 54, 57, 62, 67, 68, 72, 74}

Total number of elements in a data set is 10 which is an even number.

Divide 10 by 2

10 / 2 = 5 = n

So take the 5th (n) and 6th (n+1) elements and find the average.

(57 + 62) / 2 = 59.5

Therefore 59.5 is the median.

Example 3: Ten students’ height in a class as follows 157, 157, 178, 160, 120, 146, 165, 135, 182, and 184. What is the median?

Solution:

Arrange the elements of a data set in ascending order {120, 135, 146, 157, 157, 160, 165, 178, 182, 184}.

Median:

{120, 135, 146, 157, 157, 160, 165, 178, 182, 184}

Total number of elements in a data set is 10 which is an even number.

Divide 10 by 2

10/12 = 5 = n

So take the 5th (n) and 6th (n+1) elements and find the average.

(157 + 160) / 2 = 158.5

Therefore 158.5 is the median.

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Median in Math – Practice Problems

Problem 1: Ten students’ weight in a class as follows 56, 42, 64, 72, 89, 79, 56, 61, 65, and 56. What is the median?

Answer: Median: 62.5

Problem 2: Nine students’ height in a class as follows 165, 146, 170, 140, 162, 137, 193, 179, and 160. What is the median?

Answer: Median: 162

Ordering Fractions Calculator

Fractions :

A certain part of the whole is called as fractions. The fractions can be denoted as `a/b` , Where a, b are integers. We can multiply two or more fractions.

Calculator :

The device that used to get an output by giving proper input is known as calculator.

ordering fraction calculator

By using the Ordering Fractions calculator we can order the given fraction form least to greater or greater to least.  Let us see some problems on ordering fractions calculator.

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Steps involved in ordering fractions:

The following steps are used to order the fractions.

Step 1 :   Rewrite the given input as the fraction if its needed.

Step 2:   Find the least common denominator for the following fractions.

Step 3:  Then rewrite the given fraction with the least common denominator.

Step 4:  Order the fractions by using the numerator.

By using the above steps we can get the order of the given fractions.

Problems on ordering fractions calculator :

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Problem 1 :

Ordering the following fractions `3/4 , 89 / 67 , 65 , 90 / 156 , 23 / 45`

Solution :

Given , `3/4 , 89 / 67 , 65 , 90 / 156 , 23 / 45`

We need to ordering the given fractions,

Step 1:  Rewrite the given input as the fraction if its needed.

` 3/4, 89/67, 65/1, 90/156, 23/45`

Step 2: Find the least common denominator for the following fractions.

The least common denominator (LCD) is: 156780.

Step 3:  Then rewrite the given fraction with the least common denominator

` 117585/156780, 208260/156780, 10190700/156780, 90450/156780, 80132/156780`

Step 4: Order the fractions by using the numerator.

`80132/156780`   <  `90450/156780`  <  `117585/156780`  <  `208260/156780`   <  `10190700/156780`

Answer: least to greater : `23/45`  < `90/156`   <  `3/4`   <  `89/67`   <  65

Greater to least : 65  >  `89/67`   > `3/4`   >  `90/156`   >  `23/45`

Problem 2:

Ordering the following fractions `-5/6` ,` -12/5` , `4/9` ,  `- 4/9`

Solution:

Given , ` -5/6` , `-12/5 ` , `4/9` ,  `- 4/9`

Step 1:  Rewrite the given input as the fraction if its needed.

` -5/6` , `-12/5` ,` 4/9` , ` - 4/9 `

Step 2: Find the least common denominator for the following fractions.

The least common denominator (LCD) is: 90

Step 3:  Then rewrite the given fraction with the least common denominator

` -75/90, -216/90, 40/90, -40/90`

Step 4: Order the fractions by using the numerator.

`-216/90` <  `-75/90`   < `-40/90`   <  `40/90`

Answer: least to greater : ` -12/5`   <  `-5/6`   <  `-4/9`   <  `4/9`

Greater to least : 4/9  >  -4/9  >  -5/6  >  -12/5

Reciprocal Trigonometric Functions

Trigonometric functions are the part of mathematics association through angles, triangles. Exclusively there are six trigonometry functions such as sine, cosine, and tangent which are shortening as sin, cos, and tan likewise. The supplementary trigonometric functions are secant, cosecant, and cotangent functions.  These functions are recognized as the reciprocal trigonometric functions.

Understanding Reciprocal Identities is always challenging for me but thanks to all math help websites to help me out.

Reciprocal trigonometric functions:

Within a right triangle, one angle is ninety degree and the face across as of this angle is identified as the hypotenuse. The two faces which appearance the ninety degree angle are described the legs of the right triangle. We illustrate a right triangle below. The legs are identifying as any opposite otherwise adjacent the angle A.





In the above figure we define the following trigonometric functions ratios.

sin(A) = opposite / hypotenuse

cos(A) = adjacent / hypotenuse

tan(A) = opposite / adjacent

Three additional trigonometric ratios can be identifying as the reciprocals of these essential ratios. They are cosecant, secant, and cotangent.

The ratios are specified through the subsequent equations.


cosec(A) = hypotenuse / opposite

sec(A) = hypotenuse / adjacent

cot(A) =  adjacent / opposite

Examples for Reciprocal trigonometric functions

Math is widely used in day to day activities watch out for my forthcoming posts on Define Reciprocal and online math equation solver. I am sure they will be helpful.

Example 1 for reciprocal trigonometric functions

In the following figure the angle of A is 45 degree the adjacent side of the right triangle is 15 then computes the opposite side of the triangle



Solution

The adjacent side of the triangle is 15 then to compute opposite side of the triangle using cotangent ratio.

`cot(A) = (adjacent )/ (opposite)`



`cot 45^0 = 15/x`


`x = 15 /cot 45^0`

`x = 15 / 1 `{Since the value of cot 45 degree is 1 }

x = 15

Therefore the opposite side of the triangle is 15

Example 2 for reciprocal trigonometric functions

In the triangle the angle A is 30 degree and opposite side of the right triangle is 12 then computes the hypotenuse side of the triangle





Solution

The opposite side of the triangle is 12 then to compute hypotenuse side of the triangle

Cosec (A) = `"hypotenuse" /" opposite"`

`cosec 30^0 = x/12`

`x = 12 xxcosec 30^0`

x = 12 x 2             {since the value of cosec 30 degree is 2 }

x = 24

Therefore the hypotenuse side of the triangle is 24

How do you work Fractions out

Generally fraction is defined as the ratio of the numerator to the denominator ,the general format of the fractions is a/b, here a is the numerator and b is the denominator, the denominator of the  any one of the fraction is not equal to zero ,if it is zero means the value o the fraction is infinity ,so we cannot find the exact value of the fraction. Here, we are going to see the article as how do you work fractions out.

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Example problems on How do you work out fractions:

Example 1:

How do you work out fractions of a number, such as the numbers are given below 1/4of 28, 3/7 of 35?

Solution:

If we do the operations to workout fractions of the number, you have to divide the number by the denominator, then multiply those fraction with the number to get the answer.

Here, the denominator of the fraction is 4 and the numerator of the fraction is 1 and the number is 28
So we work out the fractions by,

¼`xx` 28=28/4=7

So the answer 7 is ¼ of 28.

(ii)   Here the denominator of the fraction is 7 and the numerator of the fraction is 3 and the number is 35

So we work out the fractions by,

3/7`xx` 35=35`xx` 3/7=1

So the answer 15 is 3/7 of 35.

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Example 2:

How do you work out fractions of a number, such as the numbers are given below 1/11of 121, 5/9 of 45?

Solution:

If we do the operations, to workout fractions of the number, you have to divide the number by the denominator, then multiply those fraction with the number to get the answer .

Here the denominator of the fraction is 11 and the numerator of the fraction is 1 and the number is 121

So we work out the fractions by,

1/11*121=121/11=11

So the answer 11is 1/11 of 121.

(ii)Here the denominator of the fraction is 9 and the numerator of the fraction is 5 and the number is 45

So we work out the fractions by,

5/9*45=45*5/5=25

So the answer 25 is5/9 of 25.