Chiang_Ch10 - ChiangCh.10Exponentialand 10.1 The Nature of...

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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
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2 10.1(a) Types of functions (Chiang, p.26) functions production growth, (inverse) log x c Logarithmi g discountin & g compoundin l Exponentia exponent) w/ associated t var. independen algebraic, - (non ntal Transcende optima / ) (algebraic Rational optima Cubic equilibria and/or optima Quadratic equilibria Linear rates interest costs, fixed Constant use economic form specific name function b 3 3 2 2 1 0 2 2 1 0 1 0 0 y b y x a y x a x a x a a y x a x a a y x a a y a y x = = = + + + = + + = + = = functions)   (algebraic    ls Polynomina
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3 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 . min , 0 8 ln 4 1 ) 14 ( 4 1 2 1 1 2 1 ) 13 ( 0 8 ln 2 1 1 ) 12 ( 0 8 ln 2 1 1 ) 11 ( 8 ln 2 1 1 1 ) 10 ( 8 ln 2 1 1 ln ) 9 ( 8 ln ln ) 8 ( ) 7 ( 8 ln ) 6 ( ln 8 ln ) 5 ( 8 ) 4 ( r for solve and 8 Let 8 y (3) ) 2 ( x optimal for Solve 5946 . 2 2 1 2 2 2 8 8 8 8 8 y (1) function l exponentia Given the 255) (p. Functions ic Largarithm and l Exponentia 10. Ch 2 3 2 2 * 2 1 2 1 2 1 2 1 2 1 2 1 2 1 8 ln 2 1 2 1 x - x x - x x - x 4 1 4 1 4 4 4 3 4 1 2 1 4 1 4 1 4 1 4 1 4 1 * x - x 2 1 2 1 2 1 2 1 2 1 = = = = = - = - = - = - = - = = - = = - = = = = = = = = - - - - - - - - = - = - = - = - - x dx y d x x x x x y dx dy x dx dy y dx x y d x x y e y x x r e r x x e b e b y x x r c r c
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4 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
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5 10.1(a) Simple exponential function y = f (x) = b x where base b > 1, x is exponent, f (x) The term exponent (x) refers to the power to which a base number (b) is raised. Base exclusions: b 1 and b 0, because f (x) = 1 x = 1; f (x) = 0 x = 0, i.e., constants 0 < b < 1 excluded since they can be expressed as negative exponents b<0 excluded because many values of f (x) from the domain would be imaginary, e.g., (-b) ½ popular bases: e and 10
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6 10.1(b) Graphic for  f(x)=e x 1 0 tangent the of slope the 0,1) ( At x as axis - x : asymptote horizontal none : intercepts - x 1 : intercept - y ) 0 ( : y of range ) ( : x of domain e b where ) ( f - , , - e f(x) / x = = =
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7 10.1(b) Graphical Exponential Functions y=b t where b=3,e,2
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8 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) ct ab y =
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9 10.1(d) A preferred base (e)   ( 29 ( 29 y e e e t f dt dy t e e m b b t f t b t t b b t b t b b t b b b t b b t f dt dy t t t t m m t t t t t t t t t t t t t t t t t t = = = = = - = = + = = - = - = - - = - = - = = = + ln ) ( ) 8 1 1 lim ln ) 7 Then ... 71828 . 2 1 1 lim e 6) If 1? e ln 5) such that e base a there Is ln ) ( ) 4 Such that 1 ln ln lim 1 ln lim ln 1 lim ) 3 Let number. particular a attain to raised is base a which power to the is logarithm A 1 lim lim lim ) ( ) 2 b y 1) Given / 0 / 0 0 0 0 0 0 /
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y 1 =e and y 2 =(1+1/m) m as m →∞ 10
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Derivative of the inverse of y=e t , i.e. y=ln t (p. 277) 11
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