from week#2 discussion on exponential functions. populations tend to growth exponentially not...

Review SlidesFrom week#2 discussion on exponential functions

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

2

If a quantity changes by a fixed percentage,

it grows or decays exponentially.

3

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

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Savings AccountsApplying exponential formula to saving account applications…

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…

Annual Percentage Rate (APR)

The amount of interest you are paid for loaning your money

Formula for using APR is A=P*(1+r/n)^(nY)

P = beginning balance r = annual interest rate (APR) n = compounding frequency (1=annually, 4 =

quarterly, 12 = monthly) Y = number of years

Example

You deposit $800 into a savings account that has an 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

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 = (1.00525)^4x

Take log of both side Log 2 = log(1.00525)^4x

Follow rule #2 Log 2= 4x* log(1.00525)

Divide by log(1.00525) 33.35 years to double your money

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

More on Calculating Interest

Check out this link ABC's of Figuring Interest