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- 1. The Exponential Functions http://www.lahc.edu/math/precalculus/math_260a.html
- 2. The Exponential Functions http://www.lahc.edu/math/precalculus/math_260a.html
- 3. The Exponential Functions The meaning positive integral exponents such as x2 is clear.
- 4. 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:
- 5. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 8 = The Exponential Functions K N 3 2 3 2 8 –2 = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 6. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = The Exponential Functions K N 3 2 3 2 8 –2 = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 7. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = 82 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 8. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 9. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = ( ) = 1/4 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 10. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. 8 –2 = = 8 = ( ) = 1/4 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 11. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 12. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = 3 2 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 13. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 14. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 10 61 50 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 15. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N b 1 Example A. 80 = 1 8 = ( 8 ) = 4 3 2 82 1 3 2 8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 10 = ( 10 ) 16.59586…. 61 50 50 61 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
- 16. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 10 Example C. The Exponential Functions
- 17. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 Example C. The Exponential Functions
- 18. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 Example C. 31 10 The Exponential Functions ≈1258.9..
- 19. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 Example C. 31 10 314 100 The Exponential Functions ≈1258.9.. ≈1380.3..
- 20. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
- 21. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
- 22. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b 1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
- 23. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b 1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences.
- 24. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b 1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences. The most used exponential functions are y = 10x, y = ex and y = 2x.
- 25. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result. 3.14159.. 3.1 3.14 3.141 3.1415 10 10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000 The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b 1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences. The most used exponential functions are y = 10x, y = ex and y = 2x. Let’s use $ growth as applications below.
- 26. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? Compound Interest
- 27. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
- 28. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
- 29. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
- 30. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
- 31. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
- 32. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
- 33. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = $1030.30 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
- 34. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = $1030.30 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
- 35. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = $1030.30 After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01) = 1000(1 + 0.01)4 = $1040.60 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
- 36. Example D. We deposit $1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = $1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = $1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = $1030.30 After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01) = 1000(1 + 0.01)4 = $1040.60 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3 Continue the pattern, after N periods, we obtain the exponential periodic-compound formula (PINA): P(1 + i)N = A.
- 37. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation The PINA Formula (Periodic Interest)
- 38. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest)
- 39. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. 0 1 2 3 Nth periodN–1
- 40. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward
- 41. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i)
- 42. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2
- 43. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3
- 44. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth 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
- 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
- 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?
- 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 =
- 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
- 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
- 50. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth 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.
- 51. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
- 52. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
- 53. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. We have r = 9% = 0.09 for one year, and f = 12 is the 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
- 54. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. We have r = 9% = 0.09 for one year, and f = 12 is the 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
- 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 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
- 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 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
- 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 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
- 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 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
- 59. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is $250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the 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
- 60. The graphs shown here are the different returns with r = 20% with different compounding frequency. Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia) Compound Interest
- 61. The graphs shown here are the different returns with r = 20% with different compounding frequency. 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
- 62. The graphs shown here are the different returns with r = 20% with different compounding frequency. 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

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