The number "e" tess

Hi I’m Professor Richards and today I will be teaching you about the number “e”! the start off the number “e” = 2.71 and is between 2<e<3. The reason why “e” is equal to 2.71 was figured out by Leonard Euler proof. For (obituary) M we see the pattern of the expression  (1+1/m)^m.

M	(1+1/m)m
1	2
10	2.59
100	2.70
1000	2.716
10000	2.718
	“e”

 ^for (1+1/m)^m as m= (1+1/m)^m =e converges to “e”

this is because (1+1/m)^m =e any positive base “b” can be written as a power of “e” (ie, e^k=b)

·      So any expentical function f(9)=ab^t can be rewritten in terms of “e” as f(t)=ab^t=a(e^k)^t=ae^kt

·      Now “k” is called the continuous growth rate   

F(t)= ae^kt is “b” when “k” is the continuous growth rate

·      If b>1 when “k” is positive

·      If 0<b<1 then “k” is negative

An example is

Let e^k=2 and b>1

then in e^k=in2

K=in2

K²=0.69 K is the positive when b>1

An example

Let e^k=1/2 and 0<b<1

Ine^k=in1/2

K²=in1/2

K²=-0.69

K in negitice 0<b<1

Here is an example from an everyday situation

An amount of $2,340.00 is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.

The Solution is:

Use the continuous compound interest formula, A = Pe rt, with P = 2340, r = 3.1/100 = 0.031, t = 3. Recall that e stands for the Napier's number (base of the natural logarithm) which is approximately 2.7183. However, one does not have to plug this value in the formula, as the calculator has a built-in key for e. Therefore, So, the balance after 3 years is approximately $2,568.06.