Broad Differentiation

In differentiation we use different formulas. Differentiation is the process of finding the derivative of the given function. The differentiation of the function is denoted as f'(x). The function can differentiated by different variables. Broad differentiation is the process of finding the derivative value one or more time of the given function. Broad differentiation includes third derivative and fourth derivative values. Now in this article we learn about broad differentiation and their example problems.

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Example problems for broad differentiation

Broad differentiation example problem 1:
        Find the third derivative value of the given function f(x) = 5x⁴ - 7x² + 61x - 9
Solution:
  Given function is f(x) = 5x⁴ - 7x² + 61x - 9
Differentiate the given function with respect to x, we get
                                f'(x) = 20x³ - 14x + 61
Again differentiate the given function for finding the second derivative, we get
                               f''(x) = 60x² - 14
For finding the third derivative, again differentiate the given function
                                f'''(x) = 120x
Answer:
      The final answer is 120x
Broad differentiation example problem 2:
        Find the third derivative value of the given function f(x) = 52x⁴ - 17x² + 33x
Solution:
  Given function is f(x) = 52x⁴ - 17x² + 33x
Differentiate the given function with respect to x, we get
                           f'(x) = 208x³ - 34x + 33
Again differentiate the given function for finding the second derivative, we get
                          f''(x) = 624x² - 34
For finding the third derivative, again differentiate the given function
                          f'''(x) = 1248x
Answer:
The final answer is 1248x

Broad differentiation example problem 3:
Find the fourth derivative value of the given function f(x) = 10x⁴ + 4x³ + 14x² - x
Solution:
  Given function is f(x) = 10x⁴ + 4x³ + 14x² - x
Differentiate the given function with respect to x, we get
                                f'(x) = 40x³ + 12x² + 28x - 1
Again differentiate the given function for finding the second derivative, we get
                                 f''(x) = 120x² + 24x + 28
For finding the third derivative, again differentiate the given function
                                 f'''(x) = 240x + 24
For fourth derivative, we get
                                f''''(x) = 240
Answer:
      The final answer is 240

Mean Frequency Table

In Statistics, mean is a basic term used frequently. A frequency table is one in which the mean and other terms can be determined. Basically, the frequency table consists of mean, median and mode. In this article, we shall discuss about the terms used in frequency table. Also we shall solve some sample problems based on mean of a frequency table.

Terms in a frequency Table:

Mean:
Mean is the method of the addition of values divided by the number of values in the given set of data.
Mean = addition of the given values / Entire number of values

Median:
Median is actually a middle number of a table. Some cases may have even values. In such cases, there will be 2 middle values. The median for such cases will be the average of the two vales.

Mode:
The most commonly repeated value in the given set of data is called as the mode for the given table..
Range:
The Variation between the greatest and the least values in the given set is said to be the range of the particular set.

Example problems for Mean Frequency Table:

Example 1:
Determine the mean, median, mode, and the range of the set of data given below.
13  17   15  16  22  25  22  27 14
Solution:
Given set of data can be arranged in the order.
13  14   15  16  17  22  22  25  27

Mean:
                Mean = ((13 + 14 + 15+ 16 + 17+ 22+ 22+ 25 + 27)/9)
                          = 171/9
                          = 19
              Median:
                  The middle number is 17. Hence the median is 17
              Mode:
                  Commonly repeated value is 22. Therefore mode is 22
              Range:
                  27 - 13 = 14.
The results are
                Mean = 19
              Median = 17
                Mode = 22
               Range = 14.

Example 2:
Consider the frequency distribution shown in scores of 20 students in a science test.
The mean of these marks = total marks /number of students
Frequency table:

Marks(x)	Frequency(f)
40	1
50	2
60	4
70	3
80	5
90	2
100	3
Total	20

                             
Then,  x = (40*1)+ (50*2)+ (60*4)+ (70*3)+ (80*5)+ (90*2)+ (100*3)
             =  40  + 100 + 240 + 210 + 400 + 180 + 300
             =  1470 / 20
             = 551 / 7
              = 73.5 marks

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    It will be very easy to discover the products of the scores and the frequencies by adding an extra column to the frequency table and the work as below:

Marks(x)	Frequency(f)	fx
40	1	40
50	2	50
60	4	240
70	3	210
80	5	400
90	2	180
100	3	300
Total	20	1470

                                                                
   Mean = sum of the xf column/ sum of the f column  
            = 1470/20

   Mean = 73.5 marks

Saxon Math Calculus

• Saxon math calculus deals with solving basic calculus problems for test preparation. Most of the Saxon’s math book deals with solving test problems for exam preparation. Saxon’s Calculus mainly deals with the differentiation of functions with one or more variables. Calculus can be divided into two parts namely differential calculus and integral calculus. Mostly Saxon’s calculus test involves in solving differential problems. The following are the test problems in math calculus solved by Saxon.

  
  

  Example problems on saxon math calculus:

  The following are the Saxon’s test problems in calculus with detailed solution.
  

  Ex:1 Solve the given function by differentiation f(u) = u ³ – 48u + 10

  Sol: The given function f(u) is the set of all real numbers. The first derivative f' (u)is given as
  f '(u) = 3 u ² - 48
  f '(u) is defined for all real numbers. Let us now solve f '(u) = 0
  3 u ² - 48 = 0
  Add 48 on both sides,
   3 u ² – 48 + 48 = 48
  3 u ² = 48
  u ² = 16
  u = 4 or u = -4
  Since u = 4 and u = -4 are in the domain of f so these both are  critical numbers.
  
  Ex:2 Solve the given function by differentiation. f(u) = | u - 8 |
  Sol:
  The domain of f(u) is the set of all real numbers. Let us use the fact sqrt (s ²) = | s | to modify function f as follows
  f(u) = sqrt (s ²) , with s = u - 8
  Using the chain rule, f '(u) is given by
  f '(u) = (1/2) 2 s s'(u) / | s |
  Since s '(u) = 1, f '(u) simplifies to
  f '(u) = (u - 8) / | u - 8 |
  f ' is undefined at u = 8 and 8 is in the domain of f. u = 8 is a critical number of function f given above.
  
  Q:3 Determine the critical number(s) of the absolute value function f given by f(x) = | x - 8 |
  Sol: The domain of f is the set of all real numbers. Let us use the fact sqrt (u ²) = | u | to modify function f as follows
  f(x) = sqrt (u ²) , with u = x - 8
  Using the chain rule, f '(x) is given by
  f '(x) = (1/2) 2 u u'(x) / | u |
  Since u '(x) = 1, f '(x) simplifies to
  f '(x) = (x - 8) / | x - 8 |
  f ' is undefined at x = 8 and 8 is in the domain of f. x = 8 is a critical number of function f given above.
  
  Practice problems on saxon math calculus:
  
  The Saxon math test problems in calculus are given below for practice.
  Q:1 Solve the given function by differentiation. f(x) = u ⁴ - 108u + 100
  Ansr:  u = 3 or u = -3
  
  Q:2 Solve the given function by differentiation  f(u) = | u - 7 |
  Ans:  f ' is undefined at u = 7 and 7 is in the domain of f. u = 7 is a critical number of function f given above. 

Math Names of Polygons

In this article we see about some names of each polygon in math, three sided polygon is triangle, some 4 sided polygons are square, rectangle, rhombus, parallelogram, trapezoid and quadrilateral. Each polygons have several properties. In math a closed geometric figure with three or more sides are also called polygon. Polygons have regular and irregular polygons shapes.

Math names of polygons:

Three sided polygons in math:
In 3-sided polygon the total angle of a triangle is 180 degrees.   
                                   
The triangle consists of many names in math:
• Right triangle
• Acute triangle
• Obtuse triangle
• Equilateral triangle
• Isosceles triangle
• Scalene triangle
Names of special polygons in math with 4 sides:
                                          
• In math 4-sided polygons, the total angle of a quadrilateral is 360 degrees.
• In math 4-sided polygon having all right angles. The total angle of a rectangle is 360 degrees.
• In math 4-sided polygon having equal length for all sides meeting at right angles. The total angle of a square is 360 degrees.

• In math 4-sided polygon have two pairs of parallel sides. The total angle of a parallelogram is 360 degrees.
• In math 4-sided polygon having all four sides of equal length. The total angle of a rhombus is 360 degrees.
• In math 4-sided polygon having accurately one pair of parallel sides. The two sides that are parallel in trapezoid are called the bases. The total angle of a trapezoid is 360 degrees.
Some 3d polygons in math are
Cube, pyramid, cone etc
                                              

Math names of other polygons with many sides:

Regular and Irregular polygons with many sides:
• Regular polygon – all angles are equal and all sides are the same length. Regular polygons have both equiangular and equilateral.
• Equiangular – all the angles are equal
• Equilateral – all the sides are same length
• Irregular polygon –A polygon whose all sides are not the same length or whose interior angles do not all have the same measure.
• Convex polygon – you can draw a straight line through a convex polygon crosses at most two sides. Every interior angle is less than 180 degree.
• Concave polygon –You can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angles is more than 180 degree.


Other polygons:
If a figure contains 5 sides, then the name of the polygon is Pentagon
If a figure contains 6 sides, then the name of the polygon is Hexagon
If a figure contains 7 sides, then the name of the polygon is Heptagon
If a figure contains 8 sides, then the name of the polygon is Octagon
If a figure contains 9 sides, then the name of the polygon is Nonagon
If a figure contains 10 sides, then the name of the polygon is Decagon
If a figure contains 12 sides, then the name of the polygon is Dodecagon

Quadrilaterals 4 Types

Quadrilaterals are four sided polygons. They are classified by their sides and angles. an important distinction between quadrilaterals is whether or not one or more pairs of sides are parallel. One of the more familiar quadrilaterals is a parallelogram. You will see that a square, a rectangle, and a rhombus are all different types of a parallelogram. The quadrilaterals 4 types basically but there are some other types that satisfy the properties of quadrilaterals. Let us see the 4 types of quadrilaterals.

quadrilaterals 4 types:

Trapezoid:
A trapezoid is a quadrilateral that has one pair of parallel sides.

                                      
Parallelogram:
A parallelogram is a quadrilateral of  two pairs of parallel sides.

                                     
Additional properties:

•  Opposite sides parallel

•  Opposite sides equal in measure

•  Opposite angles equal in measure

Rectangle:
A rectangle is a parallelogram with four right angles.

                                       
Additional properties:

•  Opposite sides parallel

• Opposite sides equal in measure

•  All angles measure 90°

•  Diagonals equal in length

Square:
A square is a rectangle with all sides equal.

                                
Additional properties:

•  Opposite sides parallel

• All sides equal in measure

•  All angles measure 90°

•  Diagonals equal in length

Rhombus:
A rhombus is a parallelogram with all sides equal.

                                        
Additional properties:

•  Opposite sides parallel

•  All sides equal in measure

•  Opposite angles equal in measure

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Isosceles Trapezoid:
An isosceles trapezoid is a quadrilateral.
                             
Additional properties:

• One pair of parallel sides

• Nonparallel sides are equal in length

quadrilaterals 4 types:

Example 1:
          Find the base of a parallelogram if its area is 512 cm² and altitude is 14 cm.
Solution:
                   Area = base × height.
                      512 = base × 14.
                         b = 512 / 14
                            = 512 cm.
                  Base = 36.5 cm.
Example 2:
Find the perimeter of square whose sides are 11 cm.
Solution:
                given the side if square is 11cm
                Perimeter of the square, P = 4a
                                                                 = 4 × 11 cm
                                                                 = 44 cm
               Hence the perimeter of square is 44 cm.

Meaning of the Triangle

Geometric figures are congruent if they have the same shape and the same size. We shall learn some properties of geometric figures that are of the same shape but not necessarily of the same size. Such figures are said to be similar. It is obvious that the congruent figures are similar but the converse is not necessarily true.
         
A triangle is a three-sided polygon. In fact, it is the polygon with the least number of sides. We write D ABC instead of writing “Triangle ABC”.

Meaning of the triangle - Properties:

We know already two important properties of a triangle,

            (i) The sum of the angles of a triangle is 180 degree
            (ii) The sum of any two sides of a triangle is greater than the third side.
Observe that one of these statements is about the angles of a triangle, while the other is about the sides of a triangle.

1. Classify the following triangles on the basis of the sides into,
Scalene triangle:
A triangle in which all the sides are of different lengths and no two sides are equal, the triangle is called a scalene triangle.


Isosceles triangle:
A triangle in which two sides are of equal lengths is called an isosceles triangle.



 Equilateral triangle
A triangle in which all the three sides are of equal lengths is called an equilateral triangle.



2. Classify the following triangle on the basis of the angles into,

Acute angled triangle
A triangle whose all angles are acute is called an acute angled triangle or simply an acute triangle
    
 Right angled triangle:
A triangle whose one of the angles is a right angle is called a right angled triangle, or simply a right triangle.
    
Obtuse angled triangle:
A triangle one of whose angles is obtuse is called an obtuse angled triangle or simply an obtuse triangle.

Meaning of the triangle - Example problems:

Meaning of the triangle problem 1:
Two angles of a triangle measure 55 degree and 85 find the measure of the third angle.



Solution:
            Let the measure of the third angle be x degree.
            We know that the sum of the angles of a triangle is 180 degree
                        55 + 85 + x = 180
                        140 + x = 180
                                    X = 180 – 140
                                        = 40


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Meaning of the triangle problem 2:
            The angles of a triangle are in the ratio 4: 2:3, Find the angles of a triangle.
Solution:
            Let the angles of the given triangle be 4x, (2x), (3x).
            The sum of the angles of a triangle is 180 degree
                        4x + 2x + 3x = 180
                                    9x = 180
                                    X = 180 / 9
                                    X = 20
            4x = 4 * 20 = 80
            2x = 2 * 20 = 40
And     3x = 3 * 20 = 60

The angles of the triangle are 80, 60, 40

Parallelogram Activity

Parallelogram is a special case of a quadrilateral where the opposite sides are parallel and equal. The sum of the angles in the quadrilateral is equal to 360 degree.We will learn about parallelograms in this lesson

Brief Explanation about the activities of Parallelogram

Meaning of the word Parallelogram:

There are 4 sides in the Parallelogram. Here the opposite sides are parallel. It means there is no intersection point between these two sides they are not met. In this Parallelogram we have two pair of sides so it is named us Parallelogram.

They are

Activity 1: The sum of angles in the Parallelogram is equal to 360 degree.

Activity 2: The opposite angles are equal in measure.

Activity 3: The sum of the adjacent angles is equal to the measure of 180 degree.

Activity 4: The opposite sides are equal in measure.

Activity 5: The height of the Parallelogram is the distance between the parallel sides.

Activity 6: The base of the Parallelogram is b and then the height of the Parallelogram is defined by h then the area of the Parallelogram is b x h square units.

Activity 7: The perimeter of the Parallelogram is the sum of the four sides of the Parallelogram. That is 2(b + h) units.

Activity 8: The two diagonals of the Parallelogram bisect at an angle 90 degree.

Example Problems

Example 1:

The angles p, 3p, p, 3p are the angles of the Parallelogram. Find the measure of p and identify the adjacent and opposite angles of the Parallelogram

Solution:

The sum of angles in the Parallelogram is 360 degree

That is p + 3p + p + 3p = 360

Then 8p = 360

Dividing 8 on both sides we have to get,

p=45 degree

Therefore 3p = 3(45) = 135 degree

Then the opposite angles are 45 degree and then the adjacent angles are 135 degree.

Example 2:

The base of the Parallelogram is 6 cm and the height of the Parallelogram is 16 cm. Calculate the value of the area of the Parallelogram.

Solution:

The area of the Parallelogram = base x height

That is 6 x 16

= 96 square cm.