1. Review Slides
From week#2 discussion on exponential functions

2. Why are exponential relationships important? Where do we encounter them?
Populations tend to growth exponentially not linearly
When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the ambient temperature (the surrounding temperature)
Radioactive substances decay exponentially
Bacteria populations grow exponentially
Money in a savings account with at a fixed rate of interest increases exponentially
Viruses and even rumors tend to spread exponentially through a population (at first)
Anything that doubles, triples, halves over a certain amount of time
Anything that increases or decreases by a percent

3. If a quantity changes by a fixed percentage, it grows or decays exponentially.

4. Solving Exponential Equations
Remember that exponential equations are in the form:
y = P(1+r)x
P is the initial (reference, old) value
r is the rate, a.k.a. percent change (and it can be either positive or negative)
x is time (years, minutes, hours, seconds decades etc…)
Y is the new value

5. Savings Accounts
Applying exponential formula to saving account applications…

6. Earning Interest in a Savings Account
Putting your money into a savings account is like loaning the bank your money
Buying savings bonds you actually loan money to the government
In return the bank/government pays you interest…
And gets to use your savings to generate more money
Through investments, loans, etc…

7. Annual Percentage Rate (APR)
The amount of interest you are paid for loaning your money
Formula for calculating APR is
A=P*(1+r/n)^(nY)
P = beginning balance
r = annual interest rate
n = compounding frequency (1=annually, 4 = quarterly, 12 = monthly)
Y = number of years

8. Example
You deposit $800 into a savings account that has and annual percentage rate of 2.1% compounded quarterly.
What is your balance after the first year?
A=P*(1+r/n)^nY)
A=800*(1+.021/4)^(4*1)
A=$816.93
What is your balance after 5 years?
How long would it take your money to double?
Hint: Use logs

9. Using Logs Use the percentage increase/decrease formula
In this case Y=P*(1+r)^x
The equation?
1600 = 800*(1+.021/4)^4x
Divide by 800
2 = ( )^4x
Take log of both side
Log 2 = log( )^4x
Follow rule #2
Log 2= 4x* log( )
Divide by log( )
33.35 years to double your money

10. Annual Percentage Yield (APY)
Percentage rate reflecting the total interest to be earned based on:
the interest rate
an institution’s compounding method
assuming funds remain in account for a 365-day year.
Formula
Use Percentage change formula for 2 consecutive years
=(new-old)/old
Change value to a %, show 2 decimal places

11. More on Calculating Interest
Check out this link
ABC's of Figuring Interest