The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. It provides examples of calculating exponential expressions using rules for positive integer, fractional, and real number exponents. Exponential functions are important in fields like finance, science, and computing. Common exponential functions include y = 10x, y = ex, and y = 2x. An example shows how to calculate compound interest monthly over several periods using the exponential function formulation.

1. The Exponential Functions
* The meaning of 10π
* Compound interest and the periodic PINA

2. The Exponential Functions
The meaning positive integral exponents such as x2 is clear.

3. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
The Exponential Functions
K
N
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

4. b0 = 1 b–K =
b = ( b ) b =
K
N
The Exponential Functions
K
N
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

5. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 =
8 =
The Exponential Functions
K
N
3
2
3
2
8 –2 =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

6. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 =
The Exponential Functions
K
N
3
2
3
2
8 –2 =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

7. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 =
82
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

8. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

9. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 = ( ) = 1/4
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

10. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
8 –2 = =
8 = ( ) = 1/4
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

11. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

12. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 =
3
2
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

13. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

14. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 = 10
61
50
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

15. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
 b
1
Example A.
80 = 1
8 = (  8 ) = 4
3 2
82
1
3
2
 8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 = 10 = ( 10 )  16.59586….
61
50
50 61
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:

16. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159..
10 
Example C.
The Exponential Functions

17. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10 
Example C.

The Exponential Functions

18. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10
Example C.
31
10

The Exponential Functions
≈1258.9..

19. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10
Example C.
31
10
314
100

The Exponential Functions
≈1258.9.. ≈1380.3..

20. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..

21. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..

22. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b  1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..

23. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b  1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.

24. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b  1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.
The most used exponential functions are
y = 10x, y = ex and y = 2x.

25. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
  3.14159.. 3.1 3.14 3.141 3.1415
10  10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b  1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.
The most used exponential functions are
y = 10x, y = ex and y = 2x.
Let’s use $ growth as applications below.

26. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
Compound Interest

27. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

28. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

29. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01)
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

30. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

31. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

32. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2

33. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2

34. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3

35. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01)
= 1000(1 + 0.01)4 = $1040.60
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3

36. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01)
= 1000(1 + 0.01)4 = $1040.60
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
Continue the pattern, after N periods, we obtain the
exponential periodic-compound formula (PINA): P(1 + i)N = A.

37. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
The PINA Formula (Periodic Interest)

38. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)

39. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
0 1 2 3 Nth period
N–1

40. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward

41. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i)

42. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2

43. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3

44. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1

45. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N

46. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?

47. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01,

48. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months

49. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months
so by PINA, there will be 1000(1 + 0.01) 720

50. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth period
N–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months
so by PINA, there will be 1000(1 + 0.01) 720 = $1,292,376.71
after 60 years.

51. Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

52. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

53. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

54. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

55. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

56. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

57. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,000
0.09
12
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

58. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,000
0.09
12 or
(1 + ) 480
P = 250,000
0.09
12
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

59. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,000
0.09
12 or
(1 + ) 480
P = 250,000
0.09
12
P = $6,923.31
by calculator
Hence the initial deposit is $6,923.31.
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f

60. x -4 -3 -2 -1 0 1 2 3 4
y=2x 1/16 1/8 1/4 1/2 1 2 4 8 16
Graphs of the Exponential Functions
Here is a table of y = 2x for plotting its graph.

61. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)
(-2,1/4)
y=2x
Graph of y = 2x
x -4 -3 -2 -1 0 1 2 3 4
y=2x 1/16 1/8 1/4 1/2 1 2 4 8 16
Graphs of the Exponential Functions
Here is a table of y = 2x for plotting its graph.

62. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)
(-2,1/4)
y=2x
Graph of y = 2x
x -4 -3 -2 -1 0 1 2 3 4
y=2x 1/16 1/8 1/4 1/2 1 2 4 8 16
Graphs of the Exponential Functions
Graph of y = bx where b>1
Here is a table of y = 2x for plotting its graph.
This is the shape of the graphs of y = bx for b > 1.

63. x -4 -3 -2 -1 0 1 2 3 4
y=(½)x 16 8 4 2 1 1/2 1/4 1/8 1/16
Here is a table of y = (½)x for plotting its graph.
Graphs of the Exponential Functions

64. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)x
Graph of y = (½)x
x -4 -3 -2 -1 0 1 2 3 4
y=(½)x 16 8 4 2 1 1/2 1/4 1/8 1/16
Here is a table of y = (½)x for plotting its graph.
Graphs of the Exponential Functions

65. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)x
Graph of y = bx where 0<b<1
Graph of y = (½)x
x -4 -3 -2 -1 0 1 2 3 4
y=(½)x 16 8 4 2 1 1/2 1/4 1/8 1/16
Here is a table of y = (½)x for plotting its graph.
Graphs of the Exponential Functions
This is the shape of the graphs of y = bx for b < 1.

66. The graphs shown here are the different returns with r = 20%
with different compounding frequencies.
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest

67. The graphs shown here are the different returns with r = 20%
with different compounding frequencies. We observe that
I. the more frequently we compound, the bigger the return
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest

68. The graphs shown here are the different returns with r = 20%
with different compounding frequencies. We observe that
I. the more frequently we compound, the bigger the return
II. but the returns do not go above the blue-line
the continuous compound return, which is the next topic.
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest

69. Compound Interest
B. Given the monthly compounded periodic rate i, find the
principal needed to obtain a return of $1,000 after the given
amount the time.
1. i = 1%, time = 60 months.
Exercise A. Given the monthly compounded periodic rate i and
the amount the time, find the return with a principal of $1,000.
2. i = 1%, time = 60 years.
3. i = ½ %, time = 60 years 4. i = ½ %, time = 60 months.
5. i = 1¼ %, time = 6 months. 6. i = 1¼ %, time = 5½ years.
.
7. i = 3/8%, time = 52 months. 8. i = 2/3%, time = 27 months.
1. i = 1%, time = 60 months. 2. i = 1%, time = 60 years.
3. i = ½ %, time = 60 years 4. i = ½ %, time = 60 months.
5. i = 1¼ %, time = 60 months. 6. i = 1¼ %, time = 60 years.
7. i = 3/8%, time = 60 years 8. i = 2/3%, time = 60 months.

70. Compound Interest
D. Given the annual rate r, convert it into the monthly
compounded periodic rate i and find the principal needed to
obtain $1,000 after the given amount the time.
1. r = 1%, time = 60 months.
C. Given the annual rate r, convert it into the monthly
compounded periodic rate i and find the return with a principal
of $1,000 after the given amount the time.
2. r = 1%, time = 60 years.
3. r = 3 %, time = 60 years 4. r = 3½ %, time = 60 months.
1. r = 1%, time = 60 months. 2. r = 1%, time = 60 years.
3. r = 3 %, time = 60 years 4. r = 3½ %, time = 60 months.
5. r = 1¼ %, time = 6 months. 6. r = 1¼ %, time = 5½ years.
.
7. r = 3/8%, time = 52 months. 8. r = 2/3%, time = 27 months.
5. r = 1¼ %, time = 6 months. 6. r = 1¼ %, time = 5½ years.
.
7. r = 3/8%, time = 52 months. 8. r = 2/3%, time = 27 months.

71. Exercise B.
1. 𝐴 ≈ 1816.7
(Answers to the odd problems) Exercise A.
3. 𝐴 ≈ 36271.41 5. 𝐴 ≈ 1077.39
7. 𝐴 ≈ 1214.87
1. P ≈ 550.45 3. P ≈ 27.57 5. P ≈ 474.57
7. P ≈ 67.55
1. 𝐴 ≈ 1051.25
Exercise C.
3. 𝐴 ≈ 6036.07 5. 𝐴 ≈ 1006.27
7. 𝐴 ≈ 1016.39
Exercise D.
1. 𝑃 ≈ 951.25 3. 𝑃 ≈ 165.67 5. 𝑃 ≈ 993.78
7. 𝑃 ≈ 983.88
Compound Interest