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. Some key points made in the document include: - The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained. - Exponential functions are defined for all real numbers x. - Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents. - Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y

1. The Exponential Functions
http://www.lahc.edu/math/precalculus/math_260a.html

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
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:

5. 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:

6. 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:

7. 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:

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
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:

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
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:

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.
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:

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 = 9 =
3
2
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 = (9 ) = 27
3
2
3
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 = 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:

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 = ( 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:

15. 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

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.. 3.1 3.14 3.141 3.1415
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  10
Example C.
31
10

The Exponential Functions
≈1258.9..

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 10
Example C.
31
10
314
100

The Exponential Functions
≈1258.9.. ≈1380.3..

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 10 10
Example C.
31
10
314
100
3141
1000
31415
10000

The Exponential Functions
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..

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 10≈1385.45..
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
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..

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..
Exponential functions show up in finance, bio science,
computer science and physical sciences.

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.
The most used exponential functions are
y = 10x, y = ex and y = 2x.

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.
Let’s use $ growth as applications below.

25. 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

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
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)

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?
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)

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.
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)

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) = 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)

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)
= 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)

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)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2

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
After 3 months: 1020.10(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

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: 1020.10(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

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: 1020.10(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030.30(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

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: 1020.10(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030.30(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.

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

37. 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)

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)
We use the following time line to see what is happening.
0 1 2 3 Nth periodN–1

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.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward

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 periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i)

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 periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2

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 periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3

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 periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1

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 periodN–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

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 periodN–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?

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 periodN–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 =

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 periodN–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

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 periodN–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

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 periodN–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.

50. 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

51. 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

52. 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

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,
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

54. 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

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. 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

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, so by PINA:
Compound Interest
P (1 + ) 480 = 250,0000.09
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

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 = 250,0000.09
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

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 = 250,0000.09
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

59. 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.

60. (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.

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
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.

62. 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

63. (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

64. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)x
Graph of y = bx where 0<b<1Graph 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.

65. 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

66. 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

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
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

68. 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 of 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.

69. 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 of 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.