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

My forthcoming post is on Solving Quadratic Equations by Finding Square Roots, 10th model question paper 2012 samacheer kalvi will give you more understanding about Algebra

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


Between, if you have problem on these topics Solving first Order Linear Differential Equations, please browse expert math related websites for more help on 10th samacheer kalvi science book.

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.

Standard Form of Ellipse

Standard form of Ellipse:
             Ellipse is the two dimensional closed geometric figure formed by the intersection of the circular cone and the plane cutting through the circular cone completely. The distance between any point on the ellipse and the foci is always constant one. The addition of distance between any two fixed points is constant

standard formula

The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n

Model problems for standard form of ellipse

1.The equation of the ellipse is
                         4x²+9y²=36
a.Find the x intercept and y intercept.
b.Find the points of the foci
c.Find the length of the major and minor axis
d.Draw the graph
Solution:
          Here the standard equation is
                   The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n
Rewrite the equation in the standard form so, divide the entire equation by 36
              ` ((4x^2)/36)` + `((9y^2)/36)` = `(36/36)`
                        `(x^2/9)` +`(y^2/4)` = 1
                    Here m²= 9, m=3
                            n² =4,n=2             m>n>0
^{a.Find the x intercept and y intercept:}
   To find the x intercept put y=0 in the given  equation
                                 ` (x^2/9)` +0=1
                                         x²=1*9
                                         x²= 9
                                            x= + or – 3
   To find the y intercept put x=0 in the given equation
                                0 +`(y^2/4)` =1
                                            y²=1*4
                                             y²=4
                                               y= + or – 2


b.Find the point of foci:
                     We need to find the p
                                           p²= 9-4
                                           p²=5
                                              p = + or – 2.23
       The foci points are (2.23, 0) and (-2.23, 0)
c.The length of the major axis and minor axis
           The length of the major axis is 2m = 2*3= 6
           The length of the minor axis is 2n = 2*2=4
d.We need to draw the graph

                                                  

Glossary of Algebraic Symbols

Algebra is a branch of mathematics. Algebra plays an important role in our day to day life. The glossary of algebraic symbols involved in the four basic operations such as addition, subtraction, multiplication and division. The most important terms with the glossary of algebraic symbols are variables, constant, coefficients, exponents, terms and expressions. In Algebra, besides numerals we use symbols and alphabets in place of unknown numbers to make a statement. Hence, glossary of algebraic symbols may be regarded as an extension of Arithmetic.  

Glossary of algebraic symbols:

Most important terms for glossary of algebraic symbols:

Expressions

          An algebraic Expression is the combination of variables, constant, coefficients, exponents, terms which are combined by the following arithmetic operations Addition, subtraction, multiplication and division. The example of an algebraic expression is given below

                           2y + 5
Term

          Terms of the algebraic expression is concatenated to form the algebraic expression by the arithmetic operations such as addition, subtraction, multiplication and division. In the following example 3n² + 2n the terms 3n², 2n are combined to form the algebraic expression 3n² + 2n by the addition operation ( + )

Coefficient

          The coefficient of an algebraic expression is the value is present just before the terms. From the following example, 3n² + 2n the coefficient of 3n² is 3 and 2n is 2

Equations

          An algebraic equation equals the numbers or expressions. Most probably algebraic equation is used for the value of the variable. The example of the equation is given below

                          2y + 5

Examples for glossary of algebraic symbols:

Example 1:

8x - 3 = 2x

Solution:

8x - 3 = 2x

8x – 3 + 3 =2x + 3 (Add 3 on both sides)

8x =2x +3

8x – 2x =2x -2x + 3 (Add -2x on both sides)

6x = 3

6x /6 = 3 / 6  (both sides by divided 6)

X = 1/2

Example 2:

Solve the equation |-3x + 3| -8 = -6

Solution:

|-3x + 3| -8 = -6

|-3x + 3| -8 + 8 = -6 + 8 (Add 8 on both sides)

|-3x + 3| = 2

Case (i)

+ (-3x+3) = 2

-3x + 3 = 2 

Subtract 3 on both sides,

-3x + 3 -3 = 2-3

-3x=-1

x =1/3

Case (ii)

- (-3x+3) = 2

3x-3 =2

Add 3 on both sides

3x-3+3=2+3

3x=5

x=5/3