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Chiang Ch. 10 Exponential and Logarithmic Functions

2. 10.1(a) Types of functions (Chiang, p.26). 3. 4. 7.3.2 Inverse-function rule. This property of one to one mapping is unique to the class of functions known as monotonic functions:Definition of a function (p. 17)Function one y for each x andMonotonic function one x for each yOne x for each y, akainverse function.

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Chiang Ch. 10 Exponential and Logarithmic Functions

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    1. 1 Chiang Ch. 10 Exponential and Logarithmic Functions 10.1 The Nature of Exponential Functions 10.2 Natural Exponential Functions and the Problem of Growth 10.3 Logarithms 10.4 Logarithmic Functions 10.5 Derivatives of Exponential and Logarithmic Functions 10.6 Optimal Timing 10.7 Further Applications of Exponential and Logarithmic Derivatives

    2. 2

    3. 3

    4. 4 7.3.2 Inverse-function rule This property of one to one mapping is unique to the class of functions known as monotonic functions: Definition of a function (p. 17) Function one y for each x and Monotonic function one x for each y One x for each y, aka inverse function

    5. 5 7.3 Rules of Differentiation Involving Functions of Different Variables

    6. 6

    7. 7 10.1 The Nature of Exponential Functions 10.1(a) Simple exponential function 10.1(b) Graphical form 10.1(c) Generalized exponential function 10.1(d) A preferred base

    8. 8 10.1(a) Simple exponential function

    9. 9 10.1(c) Generalized exponential function Where y = dependent variable b = base t = independent variable a = vertical scale factor (directly related) c = horizontal scale factor (inversely related)

    10. 10 10.1(b) Graphical Exponential Functions y=bt where b=3,e,2

    11. 11 10.1(d) A preferred base (e)

    12. y1=e and y2=(1+1/m)m as m?? 12

    13. 10.5 Derivative of the inverse of y=et, i.e. y=ln t (p. 277) 13

    14. 10.5 Derivative of y=et using the Inverse function rule 14

    15. 15 10.5(d) The case of base b

    16. 16 10.1(d) A preferred base (e) (e) = 2.71828, irrational number w/ beautifully simple characteristics

    17. 17 10.1(d) A preferred base (e) (e) = 2.71828, irrational number w/ beautifully simple characteristics)

    18. 18 10.1(b) Graphic for f(x)=ex

    19. 19 10.1(d) A preferred base (e) (e) = 2.71828, irrational number w/ beautifully simple characteristics)

    20. 20 10.1(d) A preferred base (e) (e) = 2.71828, an irrational number w/ beautifully simple characteristics)

    21. 21 10.2 Natural Exponential Functions and the Problem of Growth 10.2(a) The number e 10.2(b) An economic interpretation of e 10.2(c) Interest compounding and the function Aert 10.2(d) Instantaneous rate of growth 10.2(e) Continuous vs. discrete growth 10.2(f) Discounting and negative growth

    22. 22 10.2(a) The number e

    23. 23 10.2(b&c) Interest compounding and the function Aert y = Aert is the value of an $A investment at nominal interest rate r / compounding period compounded t times over the investment period (# days, months, or years) (growth in an investment) As m ? ?, e ? $2.71828 $1 = initial investment $1 = 100% return (if m = 1 at the end of the year, then e = 2 if m > 1, then r = 100% is a nominal rate and e > 2) $.71828 = interest on the interest received during the year as m ? ?, r = 171.8% (effective rate)

    24. 24 10.2(e) Continuous vs. discrete growth

    25. 25 10.2(e) Continuous vs. discrete growth

    26. Relation between effective (i) and nominal (r) rates of interest 26

    27. 27 10.2(d) Instantaneous rate of growth of a stock at two points in time (Chiang, pp. 278-289)

    28. 28 10.2(d) Instantaneous rate of growth of a stock at two points in time (Chiang, pp. 278-289)

    29. 29

    30. 30 10.2(f) Discounting as negative growth or rate of decay future V= present A= f(compounding the present A) f(discounting the future V)

    31. 31 10.3 Logarithms 10.3(a) The meaning of logarithm 10.3(b) Common log and natural log 10.3(c) Rules of logarithms 10.3(d) An application

    32. 32 10.3(a) The meaning of logarithm Exponents Common logs (solve for y given t) (solve for t given y)

    33. 33 10.3(b) Common log and natural log Exponent Natural log

    34. 34 10.3(c) Rules of logarithms in the land of exponents Product Quotient Power Base inversion Base conversion

    35. 35 10.4 Logarithmic Functions 10.4(a) Log functions and exponential functions 10.4(b) The graphical form 10.4(c) Base conversion

    36. 36 10.4(a) Log functions and exponential functions Exponential function Log function Dependent variable on left Monotonically increasing functions If ln y1 = ln y2, then y1 = y2

    37. 37 10.4(b) The graphical form y=et (blue), y=2t (red-top), y=ln(t) (red), 45o (green)

    38. 38 10.4(c) Base conversion Let er = bc Then ln er = ln bc r = ln bc = c ln b Therefore er = e c ln b And y = Abct = Ae(c ln b)t =Aert

    39. 39

    40. 40

    41. 41 10.5 Derivatives of Exponential and Logarithmic Functions 10.5(a) Log-function rule 10.5(b) Exponential-function rule 10.5(c) The rules generalized 10.5(d) The case of base b 10.5(e) Higher derivatives

    42. 10.5 Derivative of y=ln t 42

    43. 43 10.5(a) Log-function rule Derivative of a log function with base e Simple General

    44. 44 10.5(b) Exponential-function rule Derivative of an exponential function with base e Simple General

    45. 45 10.3(d) An application The common use of the logarithmic transformation in economic research is when estimating production functions and other multiplication nonlinear theoretical specifications Transforming a multiplicative production function into a logarithmic one suitable for estimation by linear regression. Let Q = output and L & K are labor and capital inputs respectively

    46. 46 Total differential of Q using logs

    47. Exponent example 47

    48. 48

    49. 49 10.5(d) The case of base b

    50. 50 10.5(e) Higher derivatives

    51. 51 10.6 Optimal Timing 10.6(a) A problem of wine storage 10.6(b) Maximization conditions 10.6(c) A problem of timber cutting

    52. 10.6(a) A problem of wine storage

    53. 53 10.6(a) A problem of wine storage

    54. 54 10.6(b) Maximization conditions

    55. 55 10.6(a) A problem of wine storage plot of .5(t)-.5=r, r=.10, t=25

    56. 56

    57. 57

    58. 58 10.6(c) Timber cutting problem plot of .5(t)-.5ln(2)=r, r=.05, t=48

    59. 59 10.7 Further Applications of Exponential and Logarithmic Derivatives 10.7(a) Finding the rate of growth 10.7(b) Rate of growth of a combination of functions 10.7(c) Finding the point elasticity

    60. 60 10.7(a) Finding the rate of growth

    61. 61 10.7(b) Rate of growth of a combination of functions; Example 3 consumption & pop.

    62. 62 10.7(b) Rate of growth of a combination of functions; Example 3 consumption & pop.

    63. 63 10.7(c) Finding the point elasticity

    64. 64

    65. 65

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