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# 2.2 exponential function and compound interest

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### 2.2 exponential function and compound interest

1. 1. Exponents
2. 2. Exponents A quantity A multiplied to itself N times is written as AN. N times A x A x A ….x A = AN
3. 3. Exponents A is the base. N is the exponent. A quantity A multiplied to itself N times is written as AN. N times A x A x A ….x A = AN
4. 4. Multiply–Add Rule: Divide–Subtract Rule: Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
5. 5. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
6. 6. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
7. 7. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
8. 8. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. (multiply–add) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
9. 9. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
10. 10. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. =55 52 Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
11. 11. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
12. 12. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) (divide–subtract) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
13. 13. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) (divide–subtract) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
14. 14. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk c. (22*34)3 = Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) (divide–subtract) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
15. 15. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk c. (22*34)3 = 26*312 Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) (divide–subtract) (power–multiply) Exponent–Rules A is the base. N is the exponent. N times A x A x A ….x A = AN
16. 16. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56 An Ak b. = 55–2 = 5355 52 Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk c. (22*34)3 = 26*312 Exponents A quantity A multiplied to itself N times is written as AN. = An – k (multiply–add) (divide–subtract) (power–multiply) Exponent–Rules ! Note that (22 ± 34)3 = 26 ± 38 A is the base. N is the exponent. N times A x A x A ….x A = AN
17. 17. 0-power Rule: A0 = 1 (A≠0) Special Exponents
18. 18. 0-power Rule: A0 = 1 (A=0) Special Exponents because 1 = A1 A1
19. 19. 0-power Rule: A0 = 1 (A=0) Special Exponents because 1 = = A1–1 = A0A1 A1 (divide–subtract)
20. 20. 0-power Rule: A0 = 1 (A=0) 1 Ak Special Exponents because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = (divide–subtract)
21. 21. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because (divide–subtract)
22. 22. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k (divide–subtract) (divide–subtract)
23. 23. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k (divide–subtract) (divide–subtract)
24. 24. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k (divide–subtract) (divide–subtract)
25. 25. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k (divide–subtract) (divide–subtract)
26. 26. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n (divide–subtract) (divide–subtract)
27. 27. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A Example B. because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n c. 641/3 = b. 81/3 = a. 641/2 = (divide–subtract) (divide–subtract)
28. 28. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A Example B. because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n c. 641/3 = b. 81/3 = a. 641/2 = 64 = 8 (divide–subtract) (divide–subtract)
29. 29. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A Example B. because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n c. 641/3 = b. 81/3 = 8 = 2 3 a. 641/2 = 64 = 8 (divide–subtract) (divide–subtract)
30. 30. 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak Special Exponents ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A Example B. because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n c. 641/3 = 64 = 4 3 b. 81/3 = 8 = 2 3 a. 641/2 = 64 = 8 (divide–subtract) (divide–subtract)
31. 31. Special Exponents By the power–multiply rule, the fractional exponent A k n±
32. 32. Special Exponents By the power–multiply rule, the fractional exponent A k n± (A )n 1 is take the nth root of A
33. 33. Special Exponents By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power
34. 34. Special Exponents To calculate a fractional power: extract the root first, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power
35. 35. Special Exponents a. 9 –3/2 = To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
36. 36. Special Exponents a. 9 –3/2 = (9 ½ * –3) To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
37. 37. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
38. 38. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
39. 39. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
40. 40. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 =
41. 41. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 = (271/3)-2 = (27)-23
42. 42. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 3
43. 43. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9 3
44. 44. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = (161/4)-3 = (16)-34 b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9 3
45. 45. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-34 b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9 3
46. 46. Special Exponents a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 To calculate a fractional power: extract the root first, then raise the root to the numerator–power. Example C. Find the root, then raise the root to the numerator–power. By the power–multiply rule, the fractional exponent A k n± (A ) kn ±1 is take the nth root of A then raise the root to ±k power c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-3 = 1/23 = 1/84 b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9 3
47. 47. 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
48. 48. 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
49. 49. 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..
50. 50. 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..
51. 51. 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..
52. 52. 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..
53. 53. 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..
54. 54. 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 often used exponential functions are y = 10x, y = ex and y = 2x.
55. 55. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? The Exponential Functions
56. 56. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems The Exponential Functions
57. 57. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 The Exponential Functions
58. 58. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 The Exponential Functions
59. 59. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 The Exponential Functions
60. 60. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 The Exponential Functions
61. 61. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t The Exponential Functions
62. 62. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions
63. 63. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x).
64. 64. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),
65. 65. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x),
66. 66. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25".
67. 67. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions
68. 68. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table:
69. 69. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t
70. 70. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1
71. 71. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1 2 4 8 16
72. 72. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/2 1 2 4 8 16
73. 73. Example C. A germ splits into two germs once every day. How many germs will there be after one day? Two days? Three days? Four days? t days? No. of days 0 1 2 3 4 t No. of gems 1 2 4 8 16 2t Hence P(t) = 2t gives the number of germs after t days. The Exponential Functions The exponential function bx is also written as expb(x). For example, 52 = exp5(2),10x = exp10(x), and "4x = 25" may be expressed as "exp4(x) = 25". Graph of Exponential Functions To graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
74. 74. The Exponential Functions t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
75. 75. (0,1) The Exponential Functions Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
76. 76. (0,1) (1,2) The Exponential Functions Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
77. 77. (0,1) (1,2) (2,4) (3,8) The Exponential Functions Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
78. 78. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) y=2t The Exponential Functions Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
79. 79. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) y=2t The Exponential Functions Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
80. 80. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) Graph of y = bx where b>1 y=2t The Exponential Functions Graph of y = 2x
81. 81. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) Graph of y = bx where b>1 y=2t The Exponential Functions To graph f(t) = (½)t, we make a table. Graph of y = 2x
82. 82. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) Graph of y = bx where b>1 y=2t The Exponential Functions To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t Graph of y = 2x
83. 83. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) Graph of y = bx where b>1 y=2t The Exponential Functions To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t 1 1/2 1/4 1/8 1/16 Graph of y = 2x
84. 84. (0,1) (1,2) (2,4) (3,8) (-1,1/2)(-2,1/4) Graph of y = bx where b>1 y=2t The Exponential Functions To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t 16 8 4 2 1 1/2 1/4 1/8 1/16 Graph of y = 2x
85. 85. (0,1) (-1,2) (-2,4) (-3,8) (1,1/2) (2,1/4) y= (½)t The Exponential Functions Graph of y = (½)x
86. 86. (0,1) (-1,2) (-2,4) (-3,8) (1,1/2) (2,1/4) y= (½)t Graph of y = bx where 0<b<1 The Exponential Functions Graph of y = (½)x
87. 87. (0,1) (-1,2) (-2,4) (-3,8) (1,1/2) (2,1/4) y= (½)t Graph of y = bx where 0<b<1 The Exponential Functions Graph of y = (½)x Compound Interest When an account offer interest on top of previously accumulated interest, it is called compound interest.
88. 88. 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
89. 89. 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)
90. 90. 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)
91. 91. 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)
92. 92. 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)
93. 93. 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)
94. 94. 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
95. 95. 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
96. 96. 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
97. 97. 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
98. 98. 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.
99. 99. 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
100. 100. Example E. 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
101. 101. Example E. 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 numbers 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
102. 102. Example E. 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 numbers 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
103. 103. Example E. 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 numbers 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
104. 104. Example E. 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 numbers 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
105. 105. Example E. 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 numbers 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
106. 106. Example E. 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 numbers 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
107. 107. Example E. 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 numbers 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 in \$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
108. 108. The graphs shown here are the different returns with r = 20% with different compounding frequency, from “yearly” to “compounding continuously” – which is our next topic. Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia) Compound Interest