The document discusses compound interest and continuous compound interest formulas. It provides examples of calculating balances with different compounding periods (annually, quarterly, monthly, etc.). It then introduces the continuous compounding formula A=Pe^rt and provides examples of using it to calculate balances over time at given interest rates. It compares exponential growth to polynomial and linear growth, showing that exponential functions eventually exceed other functions as x approaches infinity due to the power of compounding.

2. 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

3. 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

4. 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

5. How Frequent?
Compounded annually, quarterly, monthly, weekly or daily… ?

7. A=P(1+r/m)mt

8. How many periods?

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

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

11. As m gets large...

12. Call it “e”

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

15. Compare

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

18. • 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

19. A = P e rt

20. Example

21. Another Example

22. 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?

23. 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

24. 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

25. 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

26. 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

27. Which function
eventually exceeds
the other as x
approaches infinity?
y= 100x30 y=
3.5x

29. y=x3
y=2x

30. 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

31. In the long run,
exponential growth will
always end up ahead of
polynomial growth.

32. Which function
eventually exceeds
the other as x
approaches infinity?
y= 100x30 y=
3.5x

33. ₱50 and increases by
₱50 each week
₱5 and doubles each
week
Or

34. 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

35. 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 )

36. 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

37. Linear function grows by
addition and exponential
function grows by
multiplication

38. 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.

39. Ma’am DianN:)