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- 1. Exponents
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 0-power Rule: A0 = 1 (A≠0) Special Exponents
- 18. 0-power Rule: A0 = 1 (A=0) Special Exponents because 1 = A1 A1
- 19. 0-power Rule: A0 = 1 (A=0) Special Exponents because 1 = = A1–1 = A0A1 A1 (divide–subtract)
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Special Exponents By the power–multiply rule, the fractional exponent A k n±
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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