aim: continuous compounding course: math literacy aim: how does the exponential model fit into our...

Aim: Continuous Compounding Course: Math Literacy

Aim: How does the exponential model fit into our lives?

Do Now:

(1 ) ntrP A

n

An insurance agent wishes to sell you a policy that will pay you $100,000 in 30 years. What is the value of this policy in today’s dollars, if we assume a 9% annual inflation rate?

A P(1 r

n)nt

1 300.09100,000(1 )

1P

30100,000(1.09)P

$7,537.11P


y = a • bx

6

4

2

-2

-5 5

r x = 1+1

x x

Where’d e Come From?

Graph

6

4

2

-2

-5 5

y 2.7183

y 2.7183 is asymptotic to f(x).

Exponential function

or e

Leonard Euler

x

f xx

11


The Power of e & Continuous Compounding

y = a • bxExponential function

Exponential growthin general terms

y = P(1 + r)t

A P(1 r

n)ntExponential growth

Compound Interest

x

f xx

11

Exponential growthContinuous compounding

A Pe rt


Exponential growthCompound Interest

Exponential Function & Compounding

y = a • bxExponential function

Exponential growthin general terms

y = P(1 + r)t

A P(1 r

n)nt

e 11

n

f nn

n

Exponential growthContinuous compoundingContinuous growth/decay

k is a constant (±)

A Pe rt

N Noekt


Application

You invest $1050 at an annual interest rate of 5.5% compounded continuously. How much money, to the nearest dollar, will you have in the account after 5 years?

A Pe rt

P - principal or starting amount - 1050

r - annual interest rate – 5.5%

t - time accruing interest – 5 years

0.055 51050A e

A - ending balance

$1382.36


Application

Find the amount in a continuously compounded account for the given conditions.

Principal: $2000Annual interest: 5.1%Time: 3 years

Principal: $400Annual interest: 7.6%Time: 1.5 years


Application

Compare the balance after 25 years of a $10,000 investment earning 6.75% interest compounded continuously to the same investment compounded semi-annually.

Exponential growthContinuous compounding

A Pe rt

Exponential growthCompound Interest

A P(1 r

n)nt

A 10,000(10.0675

2)2•25 52,575.00

A 10,000e 0.0675•25 54,059.49

One earns $1484.49 more when compounded continuously


Present Value

How much money must you deposit in an account at 8.65% compounded continuously for 8 years and 135 days.

A Pe rt

0.0865 8.375225,500 Pe

0.0865 8.375

225,500P

e = $109,276.64

( )rtP Ae

Present Value Formulas

( )

1nt

rP A

n


Application

On January 2, 2017, $4000 is placed in an Individual Retirement Account (IRA) that will pay interest of 4% per annum compounded continuously. a. What will the IRA be worth on January 1, 2057?

aim: continuous compounding course: math literacy aim: how does the exponential model fit into our...

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