1. Exponentials Day 2
Its Thursday… 

2. Compound interest 𝐴(𝑡)=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴 𝑡 =𝑎𝑚𝑜𝑢𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝑡 𝑦𝑒𝑎𝑟𝑠
𝐴(𝑡)=𝑃 1+ 𝑟 𝑛 𝑛𝑡
𝐴 𝑡 =𝑎𝑚𝑜𝑢𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝑡 𝑦𝑒𝑎𝑟𝑠
𝑃=𝑖𝑛𝑡𝑖𝑎𝑡𝑙 𝑎𝑚𝑜𝑢𝑛𝑡/𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝑟= 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑎𝑠 𝑑𝑒𝑐𝑖𝑚𝑎𝑙
𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑠 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑒𝑑 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
𝑡=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠

3. What is compound interest
It is the calculated amount based on the previous total. Ex: 1000 dollars earns 2% interest compound annually. For first year we use 1000 dollars the principal =1000∗ 1.02 =$1020 For the year 2 we use the principal $ =$ Instead of doing it by hand each time we use the formula given already, for any year t .

4. N as different values
Compounded annually: Semi annually: n=1 n=2 Quarterly: Monthly: n=4 n=12 Weekly: Daily n=52 n=365

5. Growth vs. Decay
If we use assume its only compounding once per year( annually) 𝐴=𝑃 1+𝑟 𝑡
If r is added then “𝑏>1" so it is growth.
If r is subtracted then "0<𝑏<1“ meaning b is a fraction and it is decay.

6. Practice
Pg. 394 number 9 and 10 in homework packet

7. Bases of Exponents
Any positive number can be used as the base of an exponential function.
We will use base 2 and 10 a lot and they are useful for certain applications
However a very useful base is 𝒆

8. What is 𝒆
𝒆 is defined as 𝟏+ 𝟏 𝒏 𝒏 as n gets larger the number e becomes more precise.
𝒆 ≈𝟐.𝟕𝟏𝟖𝟐𝟖…
𝒆 is an irrational number like 𝝅
In continuous compound interest 𝑒 is a very convenient base

9. The natural exponential function
𝑓 𝑥 = 𝑒 𝑥 This is often referred to as the exponential function. There is a built in function for 𝑒 𝑥 it is 2𝑛𝑑 → ln Graph 𝑒 𝑥 on the calculator Is the Characteristic point the same as other exponential functions?

10. Evaluating Examples Evaluate: a) 𝑒 3 b) 2 𝑒 −0.53 c) 𝑒 4.8
We can use the calculator to evaluate 𝑓 𝑥 = 𝑒 𝑥 .
There is built in function for 𝑒 𝑥 it is 2𝑛𝑑 → ln
Examples
Evaluate:
a) 𝑒 3
b) 2 𝑒 −0.53
c) 𝑒 4.8

11. Compounded Continuously
Slightly different equation: 𝐴(𝑡)=𝑃 𝑒 𝑟𝑡 𝐴 𝑡 = final amount after t years P= initial amount /Principal 𝒆 is the irrational number we talked about earlier r- rate as a decimal t- time in years

12. Homework