Learn Exponential functions

Definition of Exponential functions: Exponential functions  are those function in which independent variable appear as an exponent .
exponential functions examples : y = 3x , f(x) = ex , y = 32x+5  are an example of exponential function.
Exponential function
Y = ax, where a is a positive real number and a ?0  is the most popular exponential function . It is an increasing function (as the value of x increasing the value of y increases ) when a>1, It is a decreasing function (as the value of x decreases the value of y decreases ) when a<1.
Derivative of exponential function :
We will use the following property to derivate  exponential function such as
  (ex) = ex

Least common multiple and Least common denominator

Least common multiple or LCM of two numbers is the smallest number that is multiple of both.
How to find the least common multiple?
Methods to find the LCM:
1. Prime factorization method – Following steps are followed to find LCM by prime factorization method:
• Find the factors of numbers whose LCM has to be obtained.
• Find the common factors of all the numbers.
• Find the factors that are not common.
• LCM is the product of common factors and the factors that are not common
LCM = (Product of common factors) X (Product of uncommon factors)
2. Common division method – Following steps are followed to find LCM by common division method: -
• Arrange numbers together separated by commas.
• Start with smallest prime number and keep on dividing till none of the numbers can further be divided.
• Multiply all the factors together to find the LCM.
The common division method is useful for finding the LCM of more than two numbers.
Least common denominator
Least common denominator is calculated for adding and subtracting the fractions. We cannot add or subtract any fraction before finding the least common denominator.
 How to find the least common denominator? (Finding the least common denominator)
Least common denominator finder
Suppose if we have two fractions such as 7/4 and 5/6.
To add these fractions, we need to find the least common denominator and o find the least common denominator of these two fractions, we find the least common multiple of their denominators i.e. LCM of 4 and 6.
4 = 2 X 2
6 = 2 X 3
2 is the only common factor and 2, 3 are the factors that are not common.
Product of common factors = 2
Product of factors that are not common = 2 X 3 = 6
LCM = (Product of common factors) X (Product of factors that are not common)
= (2) X (2 X 3) = 2 X 6 = 12
Hence the LCM of 4 and 6 is 12.
The least common denominator of 7/4 and 5/6 is 12
7/4 + 5/6 = 7*3/4*3 + 5*2/6*2 = 21/12 + 10/12 = 31/12

Properties of Real Numbers

What is a Real Number?
You can define real number as any valid number, be it whole number or rational number or irrational number. For example: 1, 1.234, 1/8, π, √7 are real numbers.

Properties of Real Numbers
There are certain properties that can be applied to all the real numbers. The different properties of real numbers are:

Cumulative Property
• Cumulative Property of Addition: This property of real numbers states that if there are two numbers, they can be added in any order. For example, 10 + 5 = 5 + 10

• Cumulative Property of Multiplication: This property states that the numbers can be multiplied in any order. For example, 10 X 5 = 5 X 10, both return the same value.

Associative Property
• Associative Property of Addition: If more numbers have to be added together, then you can associate any of them together in any way. For example, 10 + (5 + 2) = (10 + 5) + 2.

• Associative Property of Multiplication: If more numbers have to be multiplied together, then they can be associated in any way. For example, 10 X (5 X 2) = (10 X 5) X2.

Identity Property
• Identity Property of Addition: Any number added to zero will result in the number itself.  For example, 10 + 0 = 10.

• Identity Property of Multiplication: Any number added to one will result in the number itself. For example, 10 X 1 = 10.

Inverse Property
• Inverse Property of Addition: A positive number when added to its inverse results in zero. For example, 10 + (-10) = 0.

• Inverse Property of Multiplication: A number when multiplied by (1/same number) will result in 1. For example, 10 X (1 / 10) = 1.

Zero Property
Any number multiplied with zero results in zero. For example, 10 X 0 = 0

Density Property
 As per density property, it is always feasible to find a number existing between two real numbers. For example, between 10.1 and 10.2 you have a lot of numbers like 10.11, 10.12, and 10.13 and so on.

Distributive Property
Distributive property is applied when an expression includes addition and also multiplication. If a number is multiplied with a result of addition, then the multiplication has to be distributed over all the numbers participating in addition. For example, 2 X (5 + 10) = (2 X 5) + (2 X 10)

If you understand these properties clearly, then you can easily solve the algebra problems that include even complex expressions.

Derivatives of cotangent function

Derivative of cot:
Based on the above discussion we see that cot function exists (or is defined) only if sin(x) at that point is not equal to zero. Therefore for the derivative of cot function to exist, the following condition has to be met:
 Interval (a,b) should not contain n(pi) for any n Є Z, (Z = set of all integers). That is because, for x = n(pi), sin (x) = sin (n(pi)) = 0. And we already established that sin function cannot be equal to zero for cot function to exist at that point.

Derivative of cot x:
If (a,b) does not contain 

Assuming that sin and cos are differentiable and x ≠ n(pi), then sin x ≠ 0.


Proof for derivative of cotx:

The above derivation was using the quotient rule. We can also obtain the above form using the limit definition of derivative as follows:
Derivative of cot x (using limit definition):

Derivative of cot(-x):

(using chain rule, therefore multiplying by (d/dx) (-x) = -1)

= csc^2(x)

Improper Fractions

What is an improper fraction?
An improper fraction is a fraction in which the numerator, (the top number) is greater than the denominator (the bottom number).  Like in a fraction a/b, it is improper when a>b and b not zero.

For example, 5/3 is an improper fraction (5>3)
Improper fractions can be written as a mixed number, mixed number is a fraction in which we have a whole part and fraction part. 3 ½ is a mixed number, with 3 as the whole part and ½ , the fraction part
Let us convert a mixed numbers to improper fractions.

To convert a mixed number to improper fraction, we first find the product of the whole part and the denominator and then add the numerator to this product. This gives us the numerator of the improper fraction, the denominator remains the same

Convert  the mixed number, 5 ¾ to an improper fraction
(5=whole part, 3=numerator, 4=denominator)

Improper Fraction example

we find the product of the whole part and the denominator
Product = (whole part x denominator) = (5 x 4) = 20
then, we add the numerator to the product
      = 20 +3 = 23 (numerator)
While converting, the denominator remains the same
So, the improper fraction will be 23/4

Converting improper fraction to mixed number
Example: Convert the improper fraction, 8/3 to a mixed number
To convert, we need to divide the improper fraction using the long division method. The quotient will be the whole part, the remainder the numerator and the denominator remains the same
When we divide 8 with 3, we get
Quotient = 2 and remainder = 2
So, the required improper fraction would be 2 ⅔      
(2 is the whole part, ⅔ is the fraction part)

Improper fraction to mixed number
To understand the converting of an improper fraction to a mixed number, let us take a look at the example given below:
Convert the improper fraction 24/5 to mixed number
Divide 24 with 5 using the long division method
24 ÷ 5 = 4 (quotient) Remainder = 4
The mixed number is written as 4 4/5

Fraction and its types

Let's learn all about fractions in today's post.

What is a fraction? A fraction is a rational number having one numerator and a denominator.

What does fraction represents? Fractions represents ratio, decimals and percentages.

Types of fractions: There are three types of fractions: proper fraction, improper fraction and mixed fraction.

Proper fraction: A fraction where numerator is smaller than the denominator.
Improper fraction: A fraction where denominator is smaller than the numerator.
Mixed fraction: A fraction formed of a whole number and a fraction.

Fraction operations: There are different operations involved with fractions: ordering fractions, comparing fractions, reducing fractions, adding, subtracting multiplying and dividing fractions and so on.

For more help connect to an online tutor and get your help. Not just fractions help but you can get help with every concept such as geometry help from geometry tutors.

Do post your comments.

how to do math problems

Let us learn how to do math problems?

Math problems can be fun when you know what you're doing.

Read the question and visualize problem.

Underline what you need to find & underline where you finding difficulty

Circle the important numbers

Choose a strategy

Make a table or chart

Draw a picture

Use logical reasoning

Refer books & double check

In our next blog we shall learn about zinc nitrate formula I hope the above explanation was useful.Keep reading and leave your comments.