Show an Algebraic Number Line

Here we are going to see the article as show an algebraic number line. Generally number lines will be drawn in horizontally, it means the lines have the number for both the negative and the positive, it have the equal width from the left to the right side. Using the number line we do the algebraic operation as addition, subtraction and division like that.

Algebraic number line:

Algebraic number line means using the number line we do the algebraic operation it will be shown in below,
Addition - show an algebraic number line:
Example 1:

7+4

Solution:
              After drawing the horizontal number line we have to number that line after that mark the first digit as 7 and then we do the operation as addition so we move to 4 places to the right it will mark the digit as 11 this is the answer for this. So the algebraic number line will be shown in below,

Subtraction –shown an algebraic number line:
Example 2:

8-3

Solution:
          After drawing the horizontal number line we have to number that line after that mark the first digit as 8 then we do the operation as subtraction so we move to 3 places to the left it will mark the number as 5 it is the answer for this  ,it will be shown below ,

Division - shown an algebraic number line:

Divide the number 15/3 using the number line
Solution:
Here we are going to draw the horizontal line .Then starting to give the numbers (positive )from 0 to18.After that we are going to do the operation as Division .The number line division diagram will be shown as bellows ,(here the denominators are 3 so we are going to divide the number line into 3 and we will be noted the numerator value here the numerator has contains the value 15 ,so the division will be taken place up to  15 after that we are counting the parts )here 5 parts are divided 15 with 3 equal intervals so 5 is the answer for this problem.

Rational Functions Calculus

In calculus, we use both the differentiation and integration methods. In rational functions calculus both the integration and differentiation process plays an important role. Rational function consists of both the numerator and denominator value. Rational function is denoted as,

f(x) = (g (x)) / (h (x))

Here, g (x) = numerator value, h (x) = denominator value.

The denominator value h (x) is not equal to zero. It consists of all real numbers.

Example problems for rational function calculus

Rational function calculus problem 1:

Differentiate the given function f (x) = (1 / (2x))

Solution:

Given function is (1 / (2x))

Rearrange the given equation, we get

f (x) = - 2x -1

Differentiate with trespect to x, we get

f' (x) = (- 2 * - 1) (x -2)

= 2 / x^2

Answer:

The final answer is 2 / x^2

Rational function calculus problem 2:

Differentiate the given function f (x) = (1 / (4x^2 + 4))

Solution:

Given function is (1 / (4x^2 + 4))

Rearrange the given equation, we get

f (x) = (- 4x -2 - 4)

Differentiate with trespect to x, we get

f' (x) = (- 4 * - 2) (x -3) + 0

= 8 / x^3

Answer:

The final answer is 8 / x^3

Rational function calculus problem 3:

Integrate the given rational function ∫ (dx / x)

Solution:

Given integration function (dx / x)

Integate the given function with respect to x, we get

∫ (dx / x) = ∫ (1 / x) dx

= log x + c

Answer:

The final answer is log x+ c

Rational function calculus problem 4:

Integrate the given rational function ∫ (dx / (x - a))

Solution:

Given function is dx / (x - a)

Integrate the given function with respect to x, we get

∫ (dx / (x - a)) = ∫ (1 / (x - a)) dx

= log (x - a) + c

Answer:

The final answer  is log (x - a) + c

Practice problems for rational functions calculus

Rational function calculus problem 1:

Integrate the given rational function ∫ (dx / (x^2 - a^2))

Answer:

The final answer is (- tan h^-1 (x / a)) / (a)

Rational function calculus problem 2:

Integrate the given rational function ∫ (dx / (x + a))

Answer:

The final answer is log (a + x) + c



Rational function calculus problem 3:

Differentiate the given function f (x) = (1 / (x^4))

Answer:

The final answer is (- 4 / x^5)

Rational function calculus problem 4:

Differentiate the given function f (x) = (1 / (5x^7))

Answer:

The final answer is (- 7 / (5x^8))