Core 3 Eulers Interest

1
Core 3 Eulers Interest

• Learning Objectives
• Understand what is meant by the number e.
• Understand what a natural logarithm is.

2
Investigating interest

• 1 for I year at 100.
• 1 1 interest 2
• 1 for 1 year 50 compound interest every 6
  months.
• 1 50p 75p 2.25
• 1 for 1 year 33 1/3 compound interest every 4
  months. Ie four times in a year.
• 1 x (11/3) x (1 1/3) x (1 1/3) 2.37

3
Carrying on

• 1 for 1 year at 25 compound interest four times
  a year.
• 1 x(11/4)x(1 1/4)x(1 1/4 )x(11/4)
• 1 for 1 year at 20 compound interest five times
  a year.
• 1x(11/5)5
• Carry on as far as possible..

4
Algebraic approach

• The nth time
• n times in one year at a rate of 100/n
• 1x(1 1/n)n (1 1/n)n
• What about when n100?
• What happens as n gets larger?

5
Graphical approach

• Plot the graph of y (1 1/x)x for xgt0
• There is a limit for (1 1/x)x as x-gt infinity
• Find this limit correct to three decimal places.

6
Graphical approach
7
Graphical approach
8
Graphical approach
9
Graphical approach

• This limit is e.
• e is approximately
• 2.718281828459045235

10

Another function for e

• f(n)1/0! 1/1! 1/2! 1/3! . 1/n!
• For example f(3) 1/0! 1/2! 1/3!
• f(3)11/21/6 1 2/3
• Find f(4), f(5), f(6), . . . f(15)
• What do you notice as n gets larger?
• e again!
• Try Exercise A Core 3 page 73

11

Natural Logarithms

• Use trial and improvement to solve
• ex 7
• x is the log to base e of 7
• Use trial and improvement to solve
• ex 2
• x is the log to base e of 2

12

Natural Logarithms

• Any log to the base e is known as a
• natural logarithm.
• In French this is a
• logarithme naturel
• Which is where ln comes from.

13

Homework

• Complete the idea development B questions on page
  74 and 75.
• B2 to B13
• Note any Key Points