Simple Interest

ABC Bank

Offers grand Scheme

Double your money in 5 years

Contact: Branch manager

This is an advertisement requesting the investors to deposit the money in ABC bank in order to double their money. It may be puzzling to understand the logic of the advertisement. In commercial sense, yes it is possible.
So What is Simple Interest?
Logic: The bank will be using depositor’s money for five years, for their financial activities such as lending, capital investments in stocks etc. During that time, it will be earning from depositor’s money. It is logical on the part of bank to share a part of its earning. The extra amount that a depositor receives is called interest.
Simple Interest word Problems:
Find the simple interest for an amount of $5000 at the rate percent of 10% for three years.
Simple Interest Vs Compound interest
Simple interest:
 A person can receive interest at regular intervals of time and collect the principal amount after a mutually agreed time.
Compound interest:
 Alternatively, he can ask the bank to add the interest amount to the principal at regular intervals and collect the total amount after the expiry of term along with the principal amount.
As the principal amount increases regularly at certain intervals of time, the amount interest received will also, be more. We call the interest calculations of the first investment pattern as simple interest and the second one as compound interest.
Formula for simple interest calculations:
Interest is calculated for the period during which the investor, keeps his money with the bank. Longer the period, greater will be amount. Hence, the interest is directly proportional to the time (T).
If a person invests more amount of money, he will be earning more interest. Thus, the interest is directly proportional to the amount deposited. We call this amount as principal (P).
Interest is certain part of principal amount. Hence,  it can be represented as a fraction. Since the amount varies, we standardize the fraction with respect to 100. Thus, the interest is always expressed in terms of percentage (R).

Simple Interest Formula:
Simple interest = Principal x Percentage of rate of interest x Time period
SI                  =          P    x                     R%                   x         T

Problem solving strategy:
Step 1: Identify the Principal amount, rate of interest and time period
Step 2: Convert rate percentage of rate of interest into a decimal
[To convert percentage into decimal, divide by 100]
Step 3: Plug in the values and calculate the simple interest
Solved Example on Simple Interest : 
Calculate the simple interest for the principal amount $5000, at the rate of 10% for three years.
Principal = 5000, rate of interest = 10%, time period = 3 years
10% = 10/100 = 0.1
Simple interest = 5000 x 0.1 x 3 = 1500

Learn Partial Derivatives

Let z be a function of x and y such that z = f(x,y). z is therefore a function of two variables x and y.

If we keep y as constant and vary z alone, then a is a function of x only. The derivative of z with respect to x, treating y as constant is called the partial derivative of z with respect to x and is denoted by one of the partial derivative symbols listed below:

,provided this limit exists.

Similarly the derivative of z with respect to y, keeping x as constant, is called the partial derivative of z with respect to y and is denoted by one of the following partial derivative symbol:

Similarly if z is a function of three or more variables, x_1, x_2, …… x_n, then partial derivative of z with respect to x_1, is obtained by differentiating z with respect to x_1, keeping all other variables constant and is denoted by dz/dx_1.

Partial derivative examples:

Rules for Derivatives in Calculus

Derivative is the measure of the rate of change at any given point on a curve. The rate of change of a function is the slope of a line.
The derivative of f(x) when the limit extends from h to 0 is given by the formula:

Derivative Rules:  

Derivative Rule of a given function in calculus can be categorized as follows:

• General Derivative Rules, 

• Logarithm Functions Derivatives, 

• Exponential Functions Derivatives,

• Trignometric Functions Derivatives and

• Derivative Rules of Inverse Hyperbolic Functions

Derivative of a constant term:
1.       Derivative of a constant term:

   d/dx (c)=0

2.       Power Rule of Derivative:

 d/dx  x^n=  n x^(n-1)

3.       Derivative of a variable with respect to itself:  

            d/dx(x)=1

4.       Power Rule for Function:

      d/dx  [f(x)]^n=n [f(x)]^(n-1)  d/dx  f(x)

5.       Derivative of a square root:

      d/dx vx=1/(2vx)

6.       Derivative of a root of a function:

      d/dx  v(f(x) )=1/(2v(f(x) ))  d/dx  f(x)

                        =1/(2v(f(x) ))  f^' (x) 

7.       Derivative of a constant times a function:

      d/dx  c.f(x)=c  d/dx  f(x)= c .f^' (x) 

8.       Derivative of sum or difference of two functions:

      d/dx  [f(x)±g(x)]=d/dx  f(x)±d/dx  g(x)= f^' (x)± g^' (x)

9.       Chain Rule of derivatives:

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

10.   Derivative Product Rule:

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

             = f(x) g’(x) + g(x) f’ (x)

11.   Derivative Quotient Rule:

d/dx  [f(x)/g(x) ]=[g(x)  d/dx  f(x)- f(x)  d/dx  g(x)]/[g(x)]²

            =  [ g(x) f^' (x)- f(x) g^' (x)]/ [g(x)]²

Derivative Rules of Logarithm Functions:

Derivative Rules of Exponential Functions:

Derivative Rules of Trignometic Functions:

Derivative of Inverse Hyperbolic Functions:

About Square Roots

To understand square roots we first need to know what is a square. If a number is multiplied by itself, the product so obtained is called the square of that number.
For example:1:  3 x 3 = 9. So we say that 9 is square of 3.
Example: 2: 16 x 16 = 256. Then we say that 256 is the square of 16.

Square root definition:

The number when multiplied by itself gives its square is called square root of the other number. In other words, if a x a = b, then we say that a is the square root of b.
For example: 1: 3 is the square root of 9, because 3 x 3 = 9 or 3^2 = 9
Example:2: 11 is the square root of 121, again because 11^2 = 121

How to find square roots (or how to do square roots):

There are primarily two methods for finding square root:

Method 1: By prime factorization
Step 1: Resolve the given number into prime factors.
Step 2: Form pairs of like factors
Step 3: From each pair, pick out one prime factor
Step 4: Multiply the factors so picked.

Example:1: Find square root of 7225
Step 1: Resolving into prime factors and forming pairs of like factors
7225 = 5 x 5 x 17 x 17
Step 2: From each pair pick out one prime factor, i.e.,
From first pair we pick 5 and from second pair we pick 17.
Step 3: Multiply the factors so picked. So 5 x 17 = 85.
Answer: square root of 7225 = 85 or √7225 = 85

Method 2: By long division
Example 1: Calculate square root of 17424
Step 1: Mark off the digits in pairs from right to left
Step 2: 1^2=1
Step 3: Twice 1 = 2
Step 4: 2 goes into 7 three times. Put 3 on top and in the divisor as shown 23 x 3 = 69
Step 5: Double 13 = 26. First digit of 26 is 2, goes into the first digit of 524 which is 5, 2 times. Place 2 on top and in divisor as shown. 2 x 262 = 524
Step 6: Subtract. Remainder is zero and square root is 132.
Answer: √17424 = 132