Transcript Chapter 4
• College Algebra K/DC Monday, 31 April 2014
OBJECTIVE
solve equations involving compound and continuous interest.
TSW use properties of exponents to
•
ASSIGNMENT DUE
– Sec. 4.1: p. 413 (55-75 odd, 77-82 all) wire basket • RELOCATION TOMORROW (5
th
– Room 2266
period only)
1
4.2
Exponential Functions
Compound Interest Compounding ▪ ▪ The Number e and Continuous Exponential Models and Curve Fitting 4-2
The Number e
e
is a real number, not a variable. To nine decimal places,
e
≈ 2.718281828.
e
is irrational.
4-3
Simple Interest and Compound Interest
The
simple interest
formula is
I
=
Prt
, where
P
is the principal (amount deposited)
r
is the annual rate of interest (expressed as a decimal)
t
is the time in years.
Compound interest
is interest paid on both principal
and
interest.
4-4
Compound Interest
(Don’t copy this yet.)
Suppose
t
= 1 yr. Some amount
P
is invested. At the end of one year, the total amount is
P
P r t P r
P r
P
1
r
, the original principal plus interest.
If this new balance earns interest at the same rate for another year, the balance at the end of that 2 nd year will be
P
1
r
P
1
r
r
P
1
r
1
r
P
1
r
2
Previous balance Interest on previous balance
4-5
Compound Interest
(Don’t copy this yet.)
After the third year, this will grow to
P
1
r
2
P
1
r
2
r
P
1
r
1
r
P
1
r
3 .
This leads to the formula for interest compounded annually:
A
P
1
r
t
And this last formula gives us the
compound interest formula
.
4-6
Compound Interest Formula
If
P
dollars are deposited in an account paying an annual rate of interest
r
compounded (paid)
n
times per year, then after
t
years the account will contain
A
dollars, where
A
P
1
r n
t n
Memorize this formula!!!
Memorize this formula!!!
Memorize this formula!!!
Memorize this formula!!!
4-7
Using the Compound Interest Formula
Suppose $2500 is deposited in an account paying 6% per year compounded semiannually (twice per year). Find the amount in the account after 10 years with no withdrawals.
Compound interest formula
P
= 2500,
r
= .06,
n
= 2,
t
= 10
Application problem; answer in context.
Round to the nearest hundredth.
There is $4515.28 in the account after 10 years.
4-8
Using the Compound Interest Formula
How much interest is earned over the 10-year period?
The interest earned over the 10 years is $4515.28 – $2500 = $2015.28
4-9
Finding Present Value
Leah must pay a lump sum of $15,000 in 8 years. What amount deposited today at 4.8% compounded annually will give $15,000 in 8 years?
Compound interest formula
A
= 15,000,
r
= .048,
n t
= 8 = 1, Simplify, then solve for
P
.
Round to the nearest hundredth.
If Leah deposits $10,308.63 now, she will have $15,000 when she needs it.
4-10
Finding Present Value
If only $10,000 is available to deposit now, what annual interest rate is necessary for the money to increase to $15,000 in 8 years?
Compound interest formula
A
= 15,000,
P
= 10,000,
n
= 1,
t
= 8 Simplify, then solve for
r
.
Use a calculator.
An interest rate of about 5.20% will produce enough interest to increase the $10,000 to $15,000 by the end of 8 years.
4-11
How e Is Determined
Suppose $1 is invested at 100% interest per year, compounded
n
times per year.
A
1 1
n n
What happens if the compounding (
n
) increases?
Look what happens: n A What is A getting close to as n gets bigger and bigger?
e!!!!!!
4-12
Continuous Compounding
If
P
dollars are deposited at a rate of interest
r
compounded continuously for
t
years, the compound amount in dollars on deposit is
A
Pe r t
Memorize this formula!!!
Memorize this formula!!!
Memorize this formula!!!
Memorize this formula!!!
4-13
Solving a Continuous Compounding Problem
Suppose $8000 is deposited in an account paying 5% interest compounded continuously for 6 years. Find the total amount on deposit at the end of 6 years.
Continuous compounding formula
P
= 8000,
r
= .05,
t
= 6 Round to the nearest hundredth.
Answer in context.
There will be about $10,798.87 in the account at the end of 6 years.
4-14
Comparing Interest Earned as Compounding is More Frequent Suppose $2500 is invested at 6% in an account for 10 years. Find the amounts in the account at the end of 10 years if the interest is compounded quarterly, monthly, daily, and continuously.
4-15
Using Data to Model Exponential Growth
If current trends of burning fossil fuels and deforestation continue, then future amounts of atmospheric carbon dioxide in parts per million (ppm) will increase as shown in the table. The data can be modeled by the function What will be the atmospheric carbon dioxide level in 2015?
Sec. 4.2: p. 429 (72-80 even, 82 omit d, 83, 84)
Friday, 04 April 2014.
Complete-sentence answers.
Due on
72) Find the future value and interest earned if $56,780 is invested at 5.3% compounded a) quarterly for 23 quarters b) continuously for 15 yr.
74) Find the present value of $45,000 if interest is 3.6% compounded monthly for 1 yr.
76) Find the required annual interest rate to the nearest tenth of a percent for $65,000 to grow to $65,325 if interest is compounded monthly for 6 months.
78) Find the required annual interest rate to the nearest tenth of a percent for $5000 to grow to $8400 if interest is compounded quarterly for 8 yr.
80) Suppose $10,000 is invested at an annual rate of 5% for 10 yr. Find the future value if interest is compounded as follows.
a) annually b) quarterly c) monthly d) daily (365 days)