Continuosly compound interest and a comparison of exponential growth phenomena

R e v i e w !• Interest: I= Prt • Simple interest: A = P + Prt = P(1 + rt)• Compound Interest: A = P(1 + r)t

• Other compounding periods: semiannually(2), quarterly(4), monthly(12), weekly(52), daily(365)…

mt

m

rPA

1

You deposit $10000 in an account that pays 12%

annual interest. Find the balance after I year if the interest is compounded with the given frequency.a.Annuallyb.Quarterlyc.Monthlyd.Weeklye.Daily

a) annually b) quarterlyc)monthly

d) weekly e.) daily

A=10000(1+ .12/1)1x1

= 10000(1.12)1

≈ $11200

A=10,000(1+.12/4)4x1

=10000(1.03)4

≈ $11225.09

A=10,000(1+.12/365)365x1

≈10,000(1.000329)36

5

≈ $11,274.75

A=10,000(1+.12/12)12x1

=10000(1.01)12

≈ $11268.25

A=10,000(1+.12/52)52x1

=10000(1.00231)52

≈ $11273.41

A=P(1+r/m)mt

How Frequent?

Compounded annually, quarterly, monthly, weekly or daily… ?

C O N T I N U O U S

C O M P O U N D INTEREST

A=P(1+r/m)mt

How many periods?

Construct a new formula

mt

m

rPA

1

A Little Math Trick

11

rtk

Pk

As m gets large...

Call it “e”

Continuous Compound Interest

Note that here the exponent is “ rt ”, NOT “ mt ” as in the earlier formula.

Compare

How oftencompounded Computation

yearly

semi-annually

quarterly

monthly

weekly

daily

hourly

every minute

every second

e = 2.718 281 828 459 …

• Just like π, e is an irrational number which can not be represented exactly by any finite decimal fraction. • However, it can be approximated by

for a sufficiently large x

e

e

ex

x

11

A = P e rt

Example

Another Example

1. If $ 8000 is invested in an account that pays 4% interest compounded continuously, how much is in the account at the end of 10 years.

2. How long will it take an investment of $10000 to grow to $15000 if it is invested at 9% compounded continuously?

1. If interest is compounded continuously at 4.5% for 7 years, how much will a $2,000 investment be worth at the end of 7 years.

2. How long will it take money to triple if it is invested at 5.5% compounded continuously?

If $ 8000 is invested in an account that pays 4% interest

compounded continuously, how much is in the account at the end

of 10 years.

Formula: A =P ert A= $ 8000 e .04(10)

A= $ 11,934.60

How long will it take an investment of $10000

to grow to $15000 if it is invested at 9% compounded continuously?

Formula: A =P ert 15000 = 10000 e .09t

1.5 = e .09t

Ln (1.5) = ln (e .09t) Ln (1.5) = .09 t So t = ln(1.5) / .09 t = 4.51

It will take about 4.51 years

If interest is compounded continuously at 4.5% for 7

years, how much will a $2,000 investment be worth at the end

of 7 years.

Formula: A =P ert A= $2,000 e .045(7)

A= $ 2,740.52

How long will it take money to triple if it is

invested at 5.5% compounded continuously?

Formula: A =P ert 3P = P e .055t

3 = e .055t

Ln 3 = ln (e .055t) Ln 3 = .055t So t = ln3 / .055 t = 19.97

It will take about 19.97 years

Which function eventually

exceeds the other as x

approaches infinity?

y= 100x30

y= 3.5x

C O M P A R I S O N OF EXPONENTIAL

GROWTH PHENOMENA

y=x3

y=2x

X x3 2x

1 1 2

2 8 4

3 27 8

4 64 16

5 125 32

6 216 64

7 343 128

8 512 256

9 729 512

10 1000 1024

In the long run, exponential growth will always end up

ahead of polynomial growth.

Which function eventually

exceeds the other as x

approaches infinity?

y= 100x30

y= 3.5x

₱50 and increases by ₱50 each week

₱5 and doubles each week

Or

W 0 1 2 3 4 5 6 7 8

1 5 10 20 40 80 160 340 680 1,360

2 50 100 150 200 250 300 350 400 450

₱5 and doubles each week

Or₱50 and increases by ₱50 each week

y= 5(2)w

y= 50 + 50w

Option A: ₱ 1000 would be deposited on Dec. 31st in a bank account bearing your name and each day an additional ₱1,000 would be deposited ( until January 31st)Option B: One penny (.01 ) would be deposited on Dec. 31st in a bank account bearing your name. Each day, the amount would be doubled ( until January 31st )

B(t)= 0.01(2)t

t= time in # of days since

Dec. 31

A(t) = ₱ in

account after t days

t= time in # of days since

Dec. 31

A(t) = ₱ in

account after t days

0 1000 0 .01

1 2000 1 .02

2 3000 2 .04

10 11000 10 10.24

21 22000 21 20,971.52

31 32000 31 21,474,836.48

A(t)=1000t + 1000

Linear function grows by addition and exponential

function grows by multiplication

I. Solve the ff.1. An amount of $2,340.00 is

deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.

2. How long will it take $4000 to triple if it is invested at 5% compounded continuously?

II. Compare the ff.a. polynomial and exponential

growth.b. Linear and exponential growth.

God bless!

T H A N K S F O R

L I S T E N I N G ! ! !

Ma’am DianN:)