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
r
PA 





 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. Annually
b. Quarterly
c. Monthly
d. Weekly
e. Daily

a) annually b) quarterly c)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)365
≈ $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… ?

A=P(1+r/m)mt

How many periods?

Construct a new formula
mt
m
r
PA 





 1

A Little Math Trick
1
1
rtk
P
k
  
   
   

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 often
compounded
Computation
yearly
semi-annually
quarterly
monthly
weekly
daily
hourly
every minute
every second

• 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
x
x







1
1

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

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.

Ma’am DianN:)