A word problem on sets

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

Here is an example from algebra 1 word problems


Question:-

In a school there are 20 teachers who teach mathematics or physics,of these ,12 teach mathematics and 4 teach both physics and mathematics .How many teach physics

Answer:-

The total number of students =20 (this is we call what is Universal Set U)

Let M be the teachers who teach mathematics

and P be the teachers who teach physics.

the word 'or' gives a clue of union and the word 'and' gives a clue of intersection.

n(M U P)=20

n(M)=12

n(M Ω P)=4

you can see a example of Venn diagram

using the formula

n(M U P)=n(M)+n(P)-n(M Ω P)

20 = 12+n(P)-4

20 = 8+n(P)

subtract 8 on both sides

12 = n(P)

So 12 teachers teach physics

How to find critical points for an equation

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical is a critical value of the function. These definitions admit generalizations to functions of several variables, differentiable maps between R^m and Rⁿ, and differentiable maps between differentiable manifolds.

Lets see a problem from calculus help with probability problems,which explains us
more about this .

Question:-

f(x)=3x⁴-4x³-12x²+6

Differentiating with respect to x

Solution:-

f'(x)=12x³-12x²-24x

we know f'(x)=0 ,so

12x³-12x²-24x=0

12x(x²-x-2)=0

12x(x²-2x+x-2)=0

12x(x-2)(x+1) = 0

x(x-2)(x+1) = 0

Now we use the zero product property ,which states that
if ab=0 ,then a=0 and b=0,this property is from geometry terms and definitions .

so x=0 (x-2)=0 (x+1)=0

x=0 x=2 x=-1

Therefore,x=0,-1,2 are the critical points for y=f(x)

Simple Algebraic Equation Problem

Topic: Algebraic Equation

An equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in
2 + 3 = 5
9 - 2 = 7
The equations above are examples of an equality: a proposition which states that two constants are equal. Equalities may be true or false
How to solve Equations ?
Its simple.Here is a example of 7th grade math equations :

Solving a integral problem by using substitution method

Topic:- Integration

Integration is an important concept together with differentiation, forms one of the main operations in calculus help
Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

Let's work out a simple example on this.
Question:-

solve ∫ ( x / √x²-9 )

Answer:-

We do it by substitution method.
 Let u = x2-9
    
    du
    ---- = 2x
    dx
 
   du
or ---- = xdx
   2
substituting these values ,the integral becomes

∫ (du/2) / √u

= 1/2 ∫ (u)-1/2 du

        (u)-1/2 + 1
= 1/2 ---------------
         -1/2 + 1
by equalizing the denominators
   
    -1+2
   ------- = 1/2
      2
So the integral becomes
    
        (u)1/2
= 1/2 ---------------   + c
         1/2

= (x2-9)1/2+C (as u =x2-9)

= √( x2-9 ) +c  is the Answer

For more help on this ,you can reply me .

A problem on Integrating Factor

Topic:Integrating Factor

In simple term integrating factor in mathematics, is an function that is chosen to facilitate
the solving of a given equation involving differentials. It is generally used to solve ordinary differential equations, but is also used within multivitamin calculus,in this case often multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field).
In simple term integrating factor is denoted as I.F.

Question:
   
           dy
solve     ____ + = ex
           dx

Answer:

   Standard form: dy
                 ____ + p(x) * y = q(x)    Here p(x) = 1
                  dx                            q(x) = ex


 Now Inergrating factor = e√pdx = e√idx = ex

 Now the equation is:

                     y = i
                        ___
                        I.F  √I.F * Q(x) dx + c

Plugging in all the values:

                     y = 1
                        ___
                         ex √(ex * ex) dx + c

                     y = 1
                        ___
                         ex √(ex)2 dx + c

                     y = 1 * e2x
                        ___ ___ + c
                        (ex) 2

                     y = ex
                        ___ + c
                         2        

For help you can contact us

a problem on quadrilaterals

Topic:-quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad and lateral . Quad means four and lateral means sides. The interior angles of a quadrilateral add up to 360 degrees of arc.

Question:-

In a quadrilateral, the third angle is 2 times the fourth and the second angle is 3 times the fourth. The first angle is given as 120. Find the fourth angle.

Solution:-

Let the fourth angle be 'x'.

Here, the third angle = 2 times the fourth angle = 2x.

The second angle = 3 times the fourth angle = 3x.

We know the sum of the angles of a quadrilateral = 360°.

Therefore x + 2x + 3x + 120 = 360.

ie 6x + 120 = 360.

ie 6x = 360 - 120 = 240.

ie x = 240 / 6 = 40.

Hence the fourth angle = 40°.

For more help on this ,you can reply me

Solving Linear Equations

Topic:- Linear Equation


A mathematical expression that has an equal sign and linear expressions is called a Linear equation.

A number that you don't know,often represented by "x" or "y" is called a variable.

We solving linear equations by clicking and dragging a number to the "other" side of the equal sign. Remember that you are "isolating" the unknown "X" to solve the problem.
(Example is provided below.)

Question:-

Solve 2x-18=15

Answer:-

This math help is a part of linear equations.

2x-18=15

Add 18 to both sides

2x-18+18 = 15 + 18

2x = 15+18

2x = 33


divide by 2 on both sides


2x      33
----- = -------
2        2


  x = 16.5

16.5 is the Answer.