January 24, 2014
Review of Types of Numbers
Natural Numbers:
Whole Numbers:
Integers:
Rational Numbers:
Irrational Numbers:
January 24, 2014
January 24, 2014
e
What is e?
January 24, 2014
Make this table and
take out your calculators
January 24, 2014
So what is e?
(Cue: Mrs. Mayberry's "e" song.)
January 24, 2014
When do we use the number e?
Compound Growth
You put $2000 in the bank. Your money compounds
annually with an interest rate of 5%. Write an equation A(t)
that models the amount of money in the bank at time t.
January 24, 2014
Applications of e
Compound Interest
P is principle
r is the annual interest rate (decimal)
n is the number of times compounded per year
t is the time (number of years)
January 24, 2014
A man invests $10, 000 in an account that pays 8.5% interest per year,
compounded quarterly. What is the amount of money that he will have
after 3 years?
January 24, 2014
Applications of e
Continuously Compound Interest
P is principle
r is the annual interest rate (decimal)
n is the number of times compounded per year
t is the time (number of years)
A = Pe
rt
January 24, 2014
Applications of e
If I deposit $1000 in an account that pays 8% annual
interest compounded continuously, what is the balance
after one year?
A = Pe
rt
January 24, 2014
If you invest $20,000 at an annual interest rate of 1% compounded continuously,
calculate the final amount you will have in the account after 20 years.
January 24, 2014
These compound interest problems are just one place where the
constant 2.71828... arrises. Because that number plays such an
important role, we define e as 2.71828... after Leonhard Euler.
We also study the specific exponential function
January 24, 2014
New Function!
t-chart
x
Graph:
January 24, 2014
Transformations of our new function
x marks the spot: (
,
Horizontal shrink / stretch
Vertical shrink / stretch
t-chart
x
)
January 24, 2014
x marks the spot: (
,
Horizontal shrink / stretch
Vertical shrink / stretch
t-chart
x
)
January 24, 2014
Homework
Textbook p. 471 #59-64 and p. 484 #74-77