The document discusses continuous compound interest and how the accumulated value increases as the compounding period decreases. It provides an example where $1000 is deposited at 8% annual interest. When compounded 4 times per year, the accumulated value after 20 years is $4875.44. When compounded 100, 1000, and 10000 times per year, the accumulated values gradually increase to $4949.87, $4952.72, and $4953 respectively, demonstrating that more frequent compounding results in higher returns. In the limit of compounding continuously, the accumulated value approaches $4953.03.

1. Continuous Compound Interest

2. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest.

3. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N

4. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?

5. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?
P = 1000, yearly rate is 0.08,

6. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?
0.08
P = 1000, yearly rate is 0.08, so i = 4 = 0.02,

7. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?
0.08
P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,
N = (20 years)(4 times per years) = 80 periods

8. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?
0.08
P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,
N = (20 years)(4 times per years) = 80 periods
Hence A = 1000(1 + 0.02 )80 4875.44 $

9. Continuous Compound Interest
In the last section, we gave the formula for the return of
periodic compound interest. Let
P = principal,
i = periodic rate,
N = total number of periods
A = accumulated value
then A = P(1 + i )N
Example A. We deposited $1000 in an account with annual
compound interest rate r = 8%, compounded 4 times a year.
How much will be there after 20 years?
0.08
P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,
N = (20 years)(4 times per years) = 80 periods
Hence A = 1000(1 + 0.02 )80 4875.44 $
What happens if we keep everything the same but compound
more often, that is, increase K, the number of periods?

10. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?

11. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,

12. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100

13. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000

14. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000

15. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $

16. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,

17. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000

18. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000

19. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000 4952.72 $

20. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000 4952.72 $
0.08
For 10000 times a year, i = 10000= 0.000008,

21. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000 4952.72 $
0.08
For 10000 times a year, i = 10000= 0.000008,
N = (20 years)(10000 times per years) = 200000

22. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest rate r = 8%. How much will be there after
20 years if it's compounded 100 times a year? 1000 times a
year? 10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000 4952.72 $
0.08
For 10000 times a year, i = 10000= 0.000008,
N = (20 years)(10000 times per years) = 200000
Hence A = 1000(1 + 0.000008 )200000

23. Continuous Compound Interest
Example B. We deposited $1000 in an account with annual
compound interest fvHow much will be there after 20 years if
it's compounded 100 times a year? 1000 times a year?
10000 times a year?
P = 1000, r = 0.08, T = 20,
For 100 times a year, i = 0.08 = 0.0008,
100
N = (20 years)(100 times per years) = 2000
Hence A = 1000(1 + 0.0008 )2000 4949.87 $
0.08
For 1000 times a year, i = 1000 = 0.00008,
N = (20 years)(1000 times per years) = 20000
Hence A = 1000(1 + 0.00008 )20000 4952.72 $
0.08
For 10000 times a year, i = 10000= 0.000008,
N = (20 years)(10000 times per years) = 200000
Hence A = 1000(1 + 0.000008 )200000 4953.00 $

24. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.

25. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $

26. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $

27. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $

28. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $

29. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $

30. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $
4953.03 $

31. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $
4953.03 $
We call this amount the continuously compounded return.

32. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $
4953.03 $
We call this amount the continuously compounded return.
This way of compounding is called compounded continuously.

33. Continuous Compound Interest
We list the results below as the number compounded per year
K gets larger and larger.
4 times a year 4875.44 $
100 times a year 4949.87 $
1000 times a year 4952.72 $
10000 times a year 4953.00 $
4953.03 $
We call this amount the continuously compounded return.
This way of compounding is called compounded continuously.
The reason we want to compute interest this way is because
the formula for computing continously compound return is
easy to manipulate mathematically.

34. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)

35. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…

36. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
There is no “f” because
it’s compounded continuously

37. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?

38. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20.

39. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20

40. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6

41. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$

42. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?

43. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20

44. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4

45. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18$

46. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18$
c. If r = 16%, how much will be there after 20 years?

47. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18$
c. If r = 16%, how much will be there after 20 years?
r = 16%, A = 1000*e0.16*20

48. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18$
c. If r = 16%, how much will be there after 20 years?
r = 16%, A = 1000*e0.16*20 = 1000*e 3.2

49. Continuous Compound Interest
Formula for Continuously Compounded Return (Perta)
Let P = principal
r = annual interest rate (compound continuously)
t = number of year
A = accumulated value, then
Per*t = A where e 2.71828…
Example C. We deposited $1000 in an account compounded
continuously.
a. if r = 8%, how much will be there after 20 years?
P = 1000, r = 0.08, t = 20. So the continuously compounded
return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18$
c. If r = 16%, how much will be there after 20 years?
r = 16%, A = 1000*e0.16*20 = 1000*e 3.2 24532.53$

50. Continuous Compound Interest
About the Number e

51. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics.

52. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.

53. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.
For example, the following sequence of numbers zoom–in on
the number,
( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, …
1 4 2.71828…
2 3

54. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.
For example, the following sequence of numbers zoom–in on
the number,
( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, …
1 4 2.71828…which is
2 3
the same as ( 2.71828…)

55. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.
For example, the following sequence of numbers zoom–in on
the number,
( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, …
1 4 2.71828…which is
2 3
the same as ( 2.71828…)

56. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.
For example, the following sequence of numbers zoom–in on
the number,
( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, …
1 4 2.71828…which is
2 3
the same as ( 2.71828…)
This number emerges often in the calculation of problems in
physical science, natural science, finance and in mathematics.

57. Continuous Compound Interest
About the Number e
Just as the number π, the number e 2.71828… occupies a
special place in mathematics. Where as π 3.14156… is a
geometric constant–the ratio of the circumference to the
diameter of a circle, e is derived from calculations.
For example, the following sequence of numbers zoom–in on
the number,
( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, …
1 4 2.71828…which is
2 3
the same as ( 2.71828…)
This number emerges often in the calculation of problems in
physical science, natural science, finance and in mathematics.
Because of its importance, the irrational number 2.71828…
is named as “e” and it’s called the “natural” base number.
http://www.ndt-ed.org/EducationResources/Math/Math-e.htm
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

58. Continuous Compound Interest
With a fixed interest rate r, utilizing the Prffta–formula,
we conclude that the more often we compound, the higher the
return would be.

59. Continuous Compound Interest
With a fixed interest rate r, utilizing the Prffta–formula,
we conclude that the more often we compound, the higher the
return would be. However the continuously compounded return
sets the “ceiling”
or the “limit” as how
much the returns
could be regardless
how often we
compound, as shown
here.
Compounded return on $1,000 with
annual interest rate r = 20% (Wikipedia)

60. Continuous Compound Interest
With a fixed interest rate r, utilizing the Prffta–formula,
we conclude that the more often we compound, the higher the
return would be. However the continuously compounded return
sets the “ceiling”
or the “limit” as how
much the returns
could be regardless
how often we
compound, as shown
here. We may think of
the continuous –
compound as
compounding with
infinite frequency
hence yielding more
return than all other Compounded return on $1,000 with
annual interest rate r = 20% (Wikipedia)
methods.

61. Continuous Compound Interest
Growth and Decay

62. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.

63. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.

64. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.
For example,
y = e1x has the growth rate of
r = 1 or 100%.

65. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.
For example, y=ex
An Exponential Growth
y = e1x has the growth rate of
r = 1 or 100%. Exponential
growths are rapid expansions
compared to other expansion–
processes as shown here.
y = 100x y = x3

66. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.
For example, y=ex
An Exponential Growth
y = e1x has the growth rate of
r = 1 or 100%. Exponential
growths are rapid expansions
compared to other expansion–
processes as shown here.
The world population may be y = 100x y=x 3
modeled with an exponential
growth with r ≈ 1.1 % or 0.011
as of 2011.

67. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.
For example, y=ex
An Exponential Growth
y = e1x has the growth rate of
r = 1 or 100%. Exponential
growths are rapid expansions
compared to other expansion–
processes as shown here.
The world population may be y = 100x y=x 3
modeled with an exponential
growth with r ≈ 1.1 % or 0.011
as of 2011. However, this rate is dropping but it’s unclear how
fast this growth rate is shrinking.

68. Continuous Compound Interest
Growth and Decay
In all the interest examples we have the interest rate r is positive,
and the return A = Perx grows larger as time x gets larger.
We call an expansion that may be modeled by A = Perx
with r > 0 as “ an exponential growths with growth rate r”.
For example, x
y=e
An Exponential Growth
y = e1x has the growth rate of
r = 1 or 100%. Exponential
growths are rapid expansions
compared to other expansion–
processes as shown here.
The world population may be y = 100x 3
y=x
modeled with an exponential
growth with r ≈ 1.1 % or 0.011
as of 2011. However, this rate is dropping but it’s unclear how
fast this growth rate is shrinking. For more information:
(http://en.wikipedia.org/wiki/World_population)

69. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.

70. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.

71. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.
For example, An Exponential Decay
–x
y = e–1x has the decay or y=e
contraction rate of r = 1 or 100%.

72. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.
For example, An Exponential Decay
–x
y = e–1x has the decay or y=e
contraction rate of r = 1 or 100%.
In finance, shrinking values is
called “depreciation” or
”devaluation”.

73. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.
For example, An Exponential Decay
–x
y = e–1x has the decay or y=e
contraction rate of r = 1 or 100%.
In finance, shrinking values is
called “depreciation” or
”devaluation”. For example,
a currency that is depreciating
at a rate of 4% annually may be
modeled by A = Pe –0.04x
where x is the number of years elapsed.

74. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.
For example, An Exponential Decay
–x
y = e–1x has the decay or y=e
contraction rate of r = 1 or 100%.
In finance, shrinking values is
called “depreciation” or
”devaluation”. For example,
a currency that is depreciating
at a rate of 4% annually may be
modeled by A = Pe –0.04x
where x is the number of years elapsed.
Hence if P = $1, after 5 years, its purchasing power is
1*e–0.04(5) = $0.82 or 82 cents.

75. Continuous Compound Interest
If the rate r is negative, or that r < 0 then the return A = Perx
grows smaller as time x gets larger.
We call a contraction that may be modeled A = Perx
with r < 0 as “an exponential decay at the rate | r |”.
For example, An Exponential Decay
–x
y = e–1x has the decay or y=e
contraction rate of r = 1 or 100%.
In finance, shrinking values is
called “depreciation” or
”devaluation”. For example,
a currency that is depreciating
at a rate of 4% annually may be
modeled by A = Pe –0.04x
where x is the number of years elapsed.
Hence if P = $1, after 5 years, its purchasing power is
1*e–0.04(5) = $0.82 or 82 cents. For more information:
http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm