Variable Selection

The variable selection represents that the process of assigning the variable to some values and functions. The differentiation process sometimes involve big functions on that time the reference variable is used to assign and then it process into the simple calculations. In this way some confusions exist for the students for which variable we can ssign and for which functions. In this article we are going to discuss about the variable selection process in detail with the applications of the differentiation and integration process.

Examples for the variable selection in integration

• Review on the variable selection with the integrable function  `int tanh^-1 2theta ` `d theta`    

Solution:

Variable Selection:

Here the confusion exist for the variable selection.

Only one function exist  `tanh^-1 2theta`        

The another function is the constant term   1 d`theta`                   

Another one confusion

For which term we can take it as u and dv

If we choose u then it should be easily differentiable.

If we choose dv then it should be easily integrable.

u = `tanh^-1 2theta` and `(du)/(d theta)` = `theta` ;   dv = 1 d`theta`

du =` 1/(1-(2theta)^2) 2 d theta = 2/(1-4theta^2) d theta ` and  v = `theta` .

Therefore,

`int tanh^-1 2theta ` = `theta`   `tanh^-1 2theta` – `int theta 2/(1-4theta^2)` d`theta`   = `theta`   `tanh^-1 2theta` – `2int theta 1/(1-4theta^2)` d`theta` .

Let     u = `1-4theta^2`  

 `(du)/(d theta)` = -8`theta` ,

`(-1/8)` du = `theta` d`theta` .

 `int tanh^-1 2theta ` = `theta`   `tanh^-1 2theta` – `2int theta 1/(1-4theta^2)` d`theta` .

 = `theta`   `tanh^-1 2theta` `-` 2`int 1/(u) (-1/8)` du

= `theta`   `tanh^-1 2theta` `+`    `(1/4) int 1/(u)` du

 =  `theta`   `tanh^-1 2theta` `+`   ` (1/4) int ` `u^(-1)du`     

=    `theta`   `tanh^-1 2theta` `+` `(1/4) (log u)` + C

 =   `theta`   `tanh^-1 2theta` `+` `1/4 log(1-4theta^2)` + C 

Problems for variable selection in differentiation

• Review on the variable selection with the differentiable function `y = e^x [sin x + cos x]` calculate   `y'' - 2y' + y `         

Solution:

Given     `y = e^x [sin x + cos x]`           

The formula is given as

    `d/dx [f(x). g(x)] = f'(x).g(x) + f(x).g'(x)`      

or      `d/dx [u.v] = u'.v + u.v'`      

Variable Selection:

Here sometime one problem exist, which variable we can take to apply in the formula.

We can take any of them.

Take v = `sinx + cosx`  u =`e^x`     

     `v'`   = `cos x - sinx`   `u'` =`e^x`    

First Differentiation:

      `y' = u'.v + u.v'`

      `y' = e^x [sin x + cos x] + e^x[cos x - sin x]`

      `y' = e^x sin x + e^x cos x + e^x cos x - e^x sin x`    

      `y' = 2 e^x cos x `    

Second  Differentiation:

      `y'' = 2e^x [-sin x] + 2e^x[cos x ]`    

      `y'' = 2e^x cos x - 2e^x sin x`      

  `y'' - 2y' + y `  = ` 2e^x cos x - 2e^x sin x``+` `[-2( 2 e^x cos x)] ` `+ e^x sin x + e^x cos x`             

  `y'' - 2y' + y `  = ` 2e^x cos x - 2e^x sin x``-` `4 e^x cos x ` `+ e^x sin x + e^x cos x`             

  `y'' - 2y' + y `  = ` [2 - 4 + 1]e^x cos x + [1-2]e^x sin x`                    

  `y'' - 2y' + y `  = ` -e^x cos x - e^x sin x`   is the required solution.   

Basic Math Facts

 Basic math facts article deals with the basic facts behind the various arithmetic operations. The method or steps need to be followed for solving the basic math problem.
                           The Basic math facts are:

• Addition Basic facts.
• Subtraction Basic facts
• Multuiplication Basic facts.
• Division Basic facts.

Basic math facts for addition and subtraction:

 Addition Basic facts:
           The basic math facts for the addition of two numbers.

1. Symbols for the both number should be same.
2. negative number  (+) negative number = negative number
3. Positive number  (+) postive number = positive number

Model problems:

1. 15+15

             Solution:
                          Here the sign for both are same. So add the two numbers
                                        = 15+15
                                        = 30 (positive number)
2. -15 -15
               Solution:
                       Here the both numbers are negative numbers, so add the two numbers.
                                       = -15-15
                                       = -30 (negative number)
Subtraction Basic facts:
           The basic math facts for the subtraction of two numbers.

1. Symbols for the numbers should be different.
2. postive number  (-) negative number = postive  number
3. Negative number  (-) postive number = negative number

Model problems:

1. 15-10

 Solution:
                          Here the sign for both are different. So subtract the two numbers
                                        = 15-10
                                        = 5 (positive number)
2. -15 +10
 Solution:
                       Here the both numbers are having different sign, so subtract the two numbers.
                                       = -15+10
                                      = -5 (negative number)

Basic math terms for multiplication and divsions:

Mulitplication Basic facts:
           The basic math facts for the multiplication of two numbers.

• Symbols for the both number may  be same or different
• negative number  (*) negative number = positive number
• Positive number  (*) postive number = positive number
• postive number (*) negative number = negative number

Model problems:
1.15*15
             Solution:
                          Here the sign for both are same. Therefore, answer is positive numbers
                                        = 15*15
                                        = 225 (positive number)
2.-15 *15
         Solution:
                       Here both numbers are  not of same sign

                                       = -15*15
                                       = -225 (negative number)
3.-15 * -15
Solution:
                      Here the sign for both are same. Therefore, answer is positive numbers
                                        = -15* -15
                                        = 225 (positive number)

Division Basic facts:
           The basic math facts for Division of two numbers.

1. Symbols for the both number may  be same or different
2. (negative number') /( negative number) = positive number
3. Positive number  / postive number = positive number
4. postive number / negative number = negative number

Model problems:
1.15/15
 Solution:
                          Here the sign for both are same. Therefore, answer is positive numbers
                                        = 15/15
                                        = 1 (positive number)
 2.-15 /15
 Solution:
                       Here both numbers are not of same sign
                                       = -15/15
                                       = - 1 (negative number)

3.-15 / -15
Solution:
                      Here the sign for both are same. Therefore, answer is positive numbers
                                        = -15/ -15
                             =  1 (positive number)