# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

54 Pages

### Chiang_Ch9

Course: ECON 101, Spring 2011
School: Cambrian College
Rating:

Word Count: 3017

#### Document Preview

Optimization:ASpecial VarietyofEquilibriumAnalysis Ch.9 9.1 OptimumValuesandExtremeValues 9.2 RelativeMaximumandMinimum: FirstDerivativeTest 9.3 SecondandHigherDerivatives 9.4 SecondDerivativeTest 9.5 DigressiononMaclaurinandTaylor Series 9.6NthDerivativeTestforRelative ExtremumofaFunctionofOneVariable 1 9.1 OptimumValuesandExtremeValues Goalvs.nongoalequilibrium Intheoptimizationprocess,weneedto...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Canada >> Cambrian College >> ECON 101

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Optimization:ASpecial VarietyofEquilibriumAnalysis Ch.9 9.1 OptimumValuesandExtremeValues 9.2 RelativeMaximumandMinimum: FirstDerivativeTest 9.3 SecondandHigherDerivatives 9.4 SecondDerivativeTest 9.5 DigressiononMaclaurinandTaylor Series 9.6NthDerivativeTestforRelative ExtremumofaFunctionofOneVariable 1 9.1 OptimumValuesandExtremeValues Goalvs.nongoalequilibrium Intheoptimizationprocess,weneedto identifytheobjectivefunctiontooptimize. Intheobjectivefunctionthedependent variablerepresentstheobjectof maximizationorminimization =PQC(Q) 2 9.2 RelativeMaximumandMinimum: FirstDerivativeTest 9.21Relativeversusabsoluteextremum 9.22Firstderivativetest 3 9.2 Critical & stationary values Thecriticalvalueofxisthevaluex0if f(x0)=0 Astationaryvalueofyisf(x0) Astationarypointisthepointwith coordinatesx0andf(x0) Astationarypointiscoordinateofthe extremum 4 9.22Firstderivativetest Thefirstorderconditionornecessarycondition forextremaisthatf'(x*)=0andthevalueof f(x*)is: Arelativemaximumifthederivativef'(x) changesitssignfrompositivetonegativefromthe immediateleftofthepointx*toitsimmediate right.(firstderivativetestforamax.) y A x* f '(x*) = 0 5 9.22Firstderivativetest Thefirstorderconditionornecessarycondition forextremaisthatf'(x*)=0andthevalueof f(x*)is: Arelativeminimumiff'(x*)changesitssign fromnegativetopositivefromtheimmediateleft ofx0toitsimmediateright.(firstderivativetestof y min.) B x* f '(x*)=0 x 6 9.22Firstderivativetest Thefirstorderconditionornecessaryconditionfor extremaisthatf'(x*)=0andthevalueoff(x*)is: Neitherarelativemaximanorarelativeminimaif f'(x)hasthesamesignonboththeimmediateleftand rightofpointx0.(firstderivativetestforpointof inflection) y D x* f '(x*) = 0 x 7 9.2 Example 1 p. 225 y = f ( x) = x 3 12 x 2 + 36 x + 8 primative function f ' ( x) = 3 x 2 24 x + 36 = 0 1st derivative ( ) f ' ( x) = 3 x 2 8 x + 12 = 0 ( x 2) ( x 6) = 0 left f (1.5) = 38.375 f ' (1.5) = 6.75 f (5.5) = 9.375 f ' (5.5) = 5.25 * x1 = 2; () y * = f xi* f (2) = 40 f ' ( 2) = 0 f (6) = 8 f ' (6) = 0 * x2 = 6 extrema right f 2.5 = 38.625 f ' (2.5) = 5.25 f (6.5) = 9.625 f ' (6.5) = 6.75 max ( +,1) min ( ,+ ) 8 primitive function and 1st & 2nd derivatives 1) f ( x) = x 3 12 x 2 + 36 x + 8 2) f ( x) = 3 x 2 24 x + 36 f ( x) = x 2 8 x + 12 = 0 ( x = 2) ( x = 6) 3) f ( x) = 6 x 24 = 0 x=4 9 9.2 Example 1 (neg) p. 225 y= ( f ( x) = x 3 12 x 2 + 36 x + 8 ) f ' ( x) = 3 x 2 + 24 x 36 = 0 ( ) primative function 1st derivative f ' ( x) = 3 x 2 8 x + 12 = 0 ( x 2) ( x 6) = 0 left * x1 = 2; y = f ( xi* ) * x2 = 6 extrema right f (1.5) = 38.4 f ' (1.5) = 6.8 f (2) = 40 f ' ( 2) = 0 f ( 2.5) = 38.6 f ' (2.5) = 5.3 min ( , + ) f (5.5) = 9.4 f ' (5.5) = 5.3 f (6) = 8 f ' (6) = 0 f (6.5) = 9.6 f ' (6.5) = 6.8 max ( +,1) 10 primitive function and 1st & 2nd derivatives ( 1) f ( x ) = x 3 12 x 2 + 36 x + 8 2) f ( x ) = 3 x 2 24 x + 36 ( f ( x ) = ( x 2 ) 8 x +12 = 0 ( x 2 )( x 6 ) = 0 3) ) ) f ( x ) = ( 6 x 24 ) = 0 x =4 11 9.2 Example 2 p. 226 AC = Q 2 5Q + 8 f / ( Q ) = 2Q 5 = 0 Q* = 5 / 2 = 2.5 left f ( 2.4) = 1.76 f ' (2.4) = 0.2 primative function 1st derivative extrema () AC = f Q* f ( 2.5) = 1.75 f / ( 2.5) = 0 right f ( 2.6) = 1.76 relative min f ' (2.6) = 0.2 ( ,+ ) 12 plots: primitive convex function, as well as 1st & 2nd derivatives 2 1) AC = Q 5Q + 8 2) MC = 2Q 5 MC = 2Q 5 = 0 Q=5 2 3) MC = 2 13 9.2 Smile test 1) R = 1200Q 2Q 2 primative function dR 2) = 1200 4Q = 0 dQ 1st derivative 3) 4Q = 1200; Q * = 300 extrema d 2R 4) = 4 2 dQ 2nd derivative if f // ( x ) < 0, then relative maximum 14 Primitive function and 1st & 2nd derivatives 1) R = 1200Q 2Q 2 2) MR = 1200 4Q 1200 4Q = 0 Q = 300 3) MR = 4 15 9.3 SecondandHigherDerivatives 9.31Derivativeofaderivative 9.32Interpretationofthesecondderivative 9.33Anapplication 16 9.31Derivativeofaderivative Giveny=f(x) Thefirstderivativef'(x)ordy/dxisitselfa functionofx,itshouldbedifferentiablewith respecttox,providedthatitiscontinuousand smooth. Theresultofthisdifferentiationisknownasthe secondderivativeofthefunctionfandisdenoted asf''(x)ord2y/dx2. Thesecondderivativecanbedifferentiatedwith respecttoxagaintoproduceathirdderivative, f'''(x)andsoontof(n)(x)ordny/dxn 17 9.3 Example 1) 2) 3) 4) R = f ( Q ) = 1200 2Q 2 f (Q) = 1200 4Q f ( Q ) = 4 f ( Q ) = 0 primative function 1st derivative 2nd derivative 3rd derivative 18 Primitive function and 1st & 2nd derivatives 1) R = 1200Q 2Q 2 2) MR = 1200 4Q 1200 4Q = 0 Q = 300 3) MR = 4 19 Example 1, p. 228 1) y = f ( x) = 4 x 4 x 3 + 17 x 2 + 3 x 1 primative function 2) f ' ( x ) = 16 x 3 3 x 2 + 34 x + 3 1st derivative 3) f " ( x) = 48 x 2 6 x + 34 2nd derivative 4) f (3) ( x ) = 96 x 6 3rd derivative 5) 6) f ( 4) ( x ) = 96 f ( 5) ( x ) 0 4th derivative 5th derivative 20 9.32Interpretationofthesecondderivative f'(x)measurestherateofchangeofa function e.g.,whethertheslopeisincreasingor decreasing f''(x)measurestherateofchangeintherate ofchangeofafunction e.g.,whethertheslopeisincreasingor decreasingatanincreasingordecreasingrate howthecurvetendstobenditself(p.230) 21 9.33Thesmiletest () x* < 0, If f * then x is a relative maximum () If f x* > 0, then x* is a relative minimum 22 9.3 Example 1) TR = 1200Q 2Q 2 primative function 2) MR = 1200 4Q = 0 1st derivative 3) Q* = 300 MR = 4 extrema 4) 5) MR < 0 2nd derivative maximum 23 9.4 SecondDerivativeTest 9.41Necessaryandsufficientconditions 9.42Conditionsforprofitmaximization 9.43Coefficientsofacubictotalcost function 9.44Upwardslopingmarginalrevenue curve 24 9.41Necessaryandsufficientconditions Thezeroslopeconditionisanecessary conditionandsinceitisfoundwiththefirst derivative,werefertoitasa1storder condition. Thesignofthesecondderivativeis sufficienttoestablishthestationaryvaluein questionasarelativeminimumiff"(x0)>0, the2ndorderconditionorrelativemaximum 25 iff"(x0)<0. 9.42Profitfunction:Example3,p.238 1) TR = 1200Q 2Q 2 revenue function 2) TC = Q 3 61.25Q 2 + 1528.5Q + 2000 cost function 3) = TR-TC = 1200Q 2Q 2 Q 3 61.25Q 2 + 1528.5Q + 2000 profit function ( = Q 3 + 59.25Q 2 328.5Q 2000 ) simplified 5) = 3Q 2 + 118.5Q 328.5 = 0 1st derivative 6) * Q2 = 36.5 quadratic eq. extrema 7) * Q1 = 3 = 6Q + 118.5 8) (3) = 100.5 (36.5) = 100.5 solving 2nd der. w/ extrema 9) * Q1 > 0 min * Q2 < 0 max applying the smile test () 2nd derivative () 26 9.4 Quadratic equation Given a quadratic function in the form ax + bx + c = 0 the roots of the quadratic function are 2 ( ( b b 4ac x ,x = 2a * 1 * 2 ) 2 ) 1 2 27 9.42Profitfunction:example3,p.238 revenue and cost functions 1) TR = 1200Q 2Q 2 2) TC = Q 3 61.25Q 2 + 1528.5Q + 2000 profit function 3) = TR-TC = Q 3 + 59.25Q 2 328.5Q 2000 1st derivative of profit function 4) = 3Q 2 + 118.5Q 328.5 = 0 5) * * Q1 = 3 Q2 = 36.5 2nd derivative of profit function 6) = 6Q + 118.5 7) (3) = 100.5 (36.5) = 100.5 applying the smile test 8) () () * * Q1 > 0 min Q2 < 0 max 28 9.4 Imperfect Competition, Example 4, p. 240 Total revenue function 1) ( ) TR AR * Q = 8000 23Q + 1.1Q 2 0.018Q 3 Q 29 Example 4, p. 240 TR and 1st , 2nd & 3rd derivatives Average and total revenue functions 1) AR = f ( Q ) = 8000 23Q + 1.1Q 2 0.018Q 3 4) MR = 8000 23Q + 1.1Q 2 0.018Q 3 (1) 2) TR = f ( Q ) Q Marginal revenue and the product rule 3) MR = f (Q )Q + Qf (Q ) ( ( +Q 23 + 2.2Q .054Q 2 ) ) = 8000 46Q + 3.3Q 2 0.0.72Q 3 2nd derivative 5) MR = f ( MR ) 6) 7) MR = 46 + 6.6Q 0.216Q 2 = 0 * Q1 = 10.76 * Q2 = 19.79 3rd derivitive 8) MR = f ( MR ) 9) MR (Q) = 6.6 0.432Q 10) MR (10.76) = 1.95 MR" (19.79) = 1.94 smile test 11) * MR (Q1 ) > 0 min. * MR" (Q2 ) < 0 max. 30 9.4 Strict concavity Strictlyconcave:ifwepickanypairofpointsM andNonitscurveandjointthembyastraight line,thelinesegmentMNmustlieentirelybelow thecurve,exceptatpointsMN. Astrictlyconcavecurvecannevercontainalinear segmentanywhere Test:iff"(x)isnegativeforallx,thenstrictly concave. 31 9.5 DigressiononMaclaurinandTaylor Series 9.51Maclaurinseriesofapolynomial function 9.5-2 Taylorseriesofapolynomialfunctions 9.53Expansionofanarbitraryfunction 9.54Lagrangeformoftheremainder 32 9.5-2 Taylorseriesofapolynomialfunctions Taylor series for a polynominal function, the wt. sum of its derivatives f ( x0 ) f / ( x0 ) f // ( x0 ) f (n) ( x0 ) 0 1 2 ( x x0 ) + ( x x0 ) + ( x x0 ) + + ( x x0 ) n f ( x) = 0! 1! 2! n! Taylor series for an arbitrary function, any function by the wt. sum of its derivatives f ( x0 ) f / ( x0 ) f // ( x0 ) f (n) ( x0 ) 0 1 2 ( x x0 ) = ( x x0 ) + ( x x0 ) + + ( x x0 ) n + R f ( x) 0! 1! 2! n! f (n+1 ) ( p ) ( x x0 ) n+1 Where R = ( n + 1)! If n = 0, then rise run = rise > < 0 run where p is a point on the curve between x and x 0 f ( x) f ( x0 ) = R = f / ( p ) ( x x0 ) = whose tangent is parallel to the secant from x and x 0 33 8.1 Differentials d 1) y = f ( x ) = lim y x derivative x 0 dx 2) y x f ( x ) = as x 0 then 0 3) y x = f ( x ) + 4) y = f ( x ) x + x 5) y = f ( x ) x as finite x 0 6) dy = f ( x ) dx differential (see 8.4) where f ( x ) is the factor of proportionality A first - order Taylor series approximation of y1 is : 7) f ( x1 ) = f ( x0 ) + f ( x0 ) ( x1 x0 ) 8) y = f ( x0 ) x 34 Difference Quotient, Derivative & Differential f(x) y=f(x) f(x0+ x) B x y f(x0) D A x0 x f(x) f(x0) C x x x0+ x 35 Exercise 9.5-2(a): Geometric series Find the first five terms of Maclaurin the series, f // ( x0 ) f (3) ( x0 ) f ( n ) ( x0 ) 2 3 ( x x0 ) + ( x x0 ) + + ( x x0 ) n f ( x ) = f ( x0 ) + f ( x0 )( x x0 ) + 2! 3! n! i.e., choose n equals 4 and let x 0 = 0 for / 1 1 x 2 f / ( x) = ( 1) ( 1) (1 x ) = (1 x ) 1 f ( x0 ) = (1 - 0 ) = (1 x ) 2 f / ( x0 ) = (1 0) f // ( x) = ( 2) ( 1) (1 x ) = 2(1 x ) 3 f // ( x0 ) = 2(1 0 ) = 6(1 x ) 4 f (3) ( x0 ) = 6(1 0) f(x) = 3 f (3) ( x) = (3)(2)( 1) (1 x ) 4 f ( 4) ( x) = (4)(3)(2)( 1) (1 x ) 5 = 24(1 x ) 5 1 =1 2 =1 3 =2 4 f ( 4 ) ( x0 ) = 24(1 0 ) 5 =6 = 24 2 ( x 0) 2 + 6 ( x 0) 3 + 24 ( x 0) 4 + 2 6 24 1 a 1 2 3 4 n n f ( x ) = 1 + x + x + x + x + ... = x = ax = ax n = a 1 1 x 1 x 1 x n =0 n =0 n =1 Geometric series : each term is obtained from the proceeding one by multipling f ( x ) = 1 + 1( x 0) + it by x, convergent if x < 1 Chiang & Wainwright p. 243 and Stewart, Calculus, pp 610 - 11 36 y = (1 a ) 1 where a = 0 0.9 2 3 4 y = 1+ a + a + a + a + 37 5.7LeontiefInputOutputModels Structure of an input-output model Miller & Blair, p. 102 A1 A 1 = A0 AA-1 = I A0 = I Taylor series approximation (Chiang & Wainwright, p. 250, 9.5 - 2(a)) (1 x ) 1 = x n = 1 + x + x 2 + ... + x n 1 if x < 1 scalar algebra n =0 ( I A) 1 = A n = I + A + A2 + ... + A n 1 by analogy n =0 2 1 0 0.15 0.25 0.15 0.25 1.2541 0.3300 = + 0.20 0.05 + 0.20 0.05 + ... 0.2640 1.1221 0 1 Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 38 The stock (V) - flow (v) problem : Given interest rate (r) prove that v v v v + + ... + = (1 + r ) (1 + r ) 2 (1 + r ) n r v v v V= + + ... + + ... (1 + r ) (1 + r ) 2 (1 + r ) n V= 1 1 1 = v + + ... + + ... (1 + r ) (1 + r ) 2 (1 + r ) n 1 Let x = and applying the geometric series 1+ r 1 V = v x + x2 + + xn = v 1 1 x 1 1 = v 1 = v 1 1 1+ r 1 1 1+ r 1+ r 1+ r 1+ r r 1 = v 1 = v = v r r r 1 v 1 1 V = v + + ... + + ... = (1 + r ) (1 + r ) 2 r (1 + r ) n ( ) Walter Nicholson, Microeconomic Theory, 1972, p. 381, 39 Diewerts Quadratic Lemma 1) Y j = f ( X ij , j ) 2) f ( x1 ) = f ( x0 ) + f ( x0 )( x1 x0 ) + ( f ( x0 ) / 2!)( x1 x0 ) 2 + ... + ( f n ( x0 ) / n!)(n t ) n + Rn ( x1 , x0 ) 3) Let 3) ln Y1 ln Y0 = f ( x ) = ln Y and f ( X i 4) i0 x = ln x ) ( ln X i1 ln X i 0 ) + 1 2 f ( X i 0 ) ( ln X i1 ln X i 0 ) f ( X i 0 ) = ln Y / ln X i = ( Y X i ) ( X i 0 Y0 ) = Pi 0 X i 0 Py 0Y0 = Pi 0 X i 0 ( 2 P i0 X 0 = si 0 i ) 5) Py 0 MPPxi = Pi 0 6) Y X i = Pi 0 Py 0 7) f ( X i 0 ) = f ( f ( X i 0 ) ) = si ln X i = ( si1 si 0 ) ( ln X i1 ln X i 0 ) 8) ln Y1 ln Y0 = = s i i0 i 9) s ln ( Y1 Y0 ) = Y1 Y0 = ( ln X i1 ln X i 0 ) + 1 2 ( si1 si 0 ) ( ln X i1 ln X i 0 ) ( ln X i1 ln X i 0 ) 2 i ( ln X i1 ln X i 0 ) + 1 2( si1 si 0 ) ( ln X i1 ln X i 0 ) = 1 2( si1 + si 0 ) ln( X i1 1 2 ( s i 10) i0 ( X i1 i i0 ( ) + si1 ) ln X i1 X i 0 + ln (1 + ) X i 0 ) 1 2( si 0 + si1 ) (1 + ) X i0 ) i (Cooke and Sundquist, SJAE, July 1991) i 40 Diewerts Quadratic Lemma Let i = set of inputs in the production of a commodity in the economy, where e.g., i = KLEFMA inputs of capital, labor, enery, fertilizer, materials, and planted acres j = set of geographic regions such that 0 base region and 1 is the first region Yj = the average yield per acre of commodity Y in region j Pyj = the price of output Y in region j X ij = the average quantity of input X i in region j Pij = the price of input i in region j s ij = the share of wage expenditures in sector i in region j j = the productivity index in region j relative to the base region 41 Diewerts Quadratic Lemma 1) W j = f (Wij , j ) 2) f ( x) = f (a ) + f ( a )( x a ) + ( f ( a ) / 2!)( x a ) 2 + ... + ( f n (a ) / n!)(n t ) n + Rn ( x, a ) 3) ln W1 ln W0 = f (Wi 0 ) ( ln Wi1 ln Wi 0 ) + i ( )( 1 2 f (Wi 0 ) ( ln Wi1 ln Wi 0 ) 2 ) 4) f (Wi 0 ) = ln W / ln Wi = W Wi Wi 0 W0 = N i 0Wi 0 N 0W0 = si 0 5) Wj = W N ij Nj ij i 6) W Wi = N ij N j 7) f (Wi 0 ) = f ( f (Wi 0 ) ) = si ln Wi 8) f (a )( x a ) + ( f ( a ) / 2!)( x a) 2 = = 9) s ( i i0 i i0 ( ln Wi1 ln Wi 0 ) + 1 2( si i ln Wi ) ( ln Wi1 ln Wi 0 ) 2 ( ln Wi1 ln Wi 0 ) + 1 2( si1 si 0 ) ( ln Wi1 ln Wi 0 ) = 1 2( si1 + si 0 ) ln(Wi1 i ) 1 2 ( si0 + si1 ) ln(Wi1 ln W1 W0 = i 10) s W1 W0 = (W i1 ) Wi 0 + ln (1 + ) Wi 0 ) i Wi 0 ) 1 2( si 0 + si1 ) (1 + ) i 42 Diewerts Quadratic Lemma Let i = set of standard industrial classifications for the sectors in the economy, where e.g., i = 1 and 2 for agriculture and non - agriculture sectors j = set of geographic regions of the U.S. where j = 0 to 9 such that 0 is the U.S. and 1 is the first region within the nation W j = the average wage per job in region j Wij = the average wage per job in sector i in region j N ij = the employment in sector i in region j Sij = the share of jobs in sector i in region j s ij = the share of wage expenditures in sector i in region j j = the industry mix coefficient in region j 43 9.5 Maclaurin Series of a Polynomial Function f ( x) = a0 + a1 x + a2 x 2 + a3 x 3 + ... + an x n primative function f / ( x) = a1 + 2a2 x + 3a3 x 2 + ... + nan x n 1 1st derivative f // ( x) = 2a2 + 6a3 x + ... + n(n 1)an x n 2 2 nd derivative f /// ( x) = 6a3 + ... + n(n 1)(n 2)an x n 3 3rd derivative f n ( x) = n(n 1)(n 2)(n 3) (3)(2)(1) an n th derivative Evaluating each function at x = 0, simplifying & solving for the coefficient f (0) = a0 f (0) = 0! a0 a0 = f (0) 0! f / (0) = a1 f / (0) = 1!a1 a1 = f / (0) 1! f // (0) = 2a2 f // (0) = 2!a2 a2 = f // (0) 2! f /// (0) = 6a3 f /// (0) = 3!a3 a3 = f /// (0) 3! f n (0) = n(n 1)(n 2)(n 3) (3)(2)(1)an an = f n (0) n! Substituti ng the value of the coefficients into the primative function f ( 0) 0 f / ( 0) 1 f // ( 0 ) 2 f (n) ( 0 ) n ( x) + ( x) + ( x ) + + ( x) f(x) = 0! 1! 2! n! 44 9.5 Maclaurin Series of a Polynomial Function Given f(x) = 2 + 4 x + 3 x 2 f( 0 ) = 2 f /(x) = 4 + 6 x f /( 0 ) = 4 f //(x) = 6 f //( 0 ) = 6 f ( 0) 0 f / ( 0) 1 f // ( 0 ) 2 f (n) ( 0 ) n ( x) + ( x) + ( x ) + + ( x) f(x) = 0! 1! 2! n! 2 ( x ) 0 + 4 ( x )1 + 6 ( x ) 2 = 2 + 4 x + 3x 2 f(x) = 0! 1! 2! 45 9.5 Taylor Series of a Polynomial Function Let x = x0 + where x 0 is a chosen point and is the deviation from x 0 f ( x) g ( ) = a0 + a1 ( x0 + ) + a2 ( x0 + ) 2 f / ( x) g / ( ) = a1 + 2a2 ( x0 + ) + 3a3 ( x0 + ) primitive function 2 1st derivative f // ( x) g // ( ) = 2a2 + 6a3 ( x0 + ) 2 nd derivative f /// ( x) g /// ( ) = 6a3 3rd derivative At = 0, we know from the Maclaurin series expansion that g ( 0) g / ( 0) 1 g // ( 0 ) g (n) ( 0 ) 0 2 ( ) + ( ) + ( ) + + ( ) n g ( ) = 0! 1! 2! n! Therefore x = x0 and = x-x0 = 0. Since g ( ) f(x), then g( 0 ) = f ( x) = f(x0 ) and f ( x0 ) f / ( x0 ) f // ( x0 ) 0 1 ( x x0 ) + ( x x0 ) + ( x x0 ) 2 + f ( x) = 0! 1! 2! f (n) ( x0 ) ( x x0 ) n + n! 46 9.5 Taylor Series of a Polynomial Function Expand to quadratic function below around x 0 = 1, with n = 1 f ( x) = 5 + 2 x + x 2 f (1) = 8 f / ( x) = 2 + 2 x f / (1) = 4 f // ( x) = 2 f // (1) = 2 f ( x0 ) f / ( x0 ) f // ( x0 ) 0 1 ( x x0 ) + ( x x0 ) + ( x x0 ) 2 + f ( x) = 0! 1! 2! f (n) ( x0 ) ( x x0 ) n + n! f ( x0 ) f / ( x0 ) 0 ( x x0 ) + ( x x0 )1 f ( x) = 0! 1! 8 4 0 ( x 1) + ( x 1)1 = 8 + 4( x 1) + R = 4 + 4 x + R f ( x) = 0! 1! f // ( x0 ) ( x x0 ) 2 R= 2! 47 9.5 Taylor Series of a Polynomial Function Expand to quadratic function below around x 0 = 1, with n = 1 f ( x) = 5 + 2 x + x 2 f (1) = 8 f / ( x) = 2 + 2 x f / (1) = 4 f // ( x) = 2 f // (1) = 2 f ( x0 ) f / ( x0 ) f // ( x0 ) 0 1 ( x x0 ) + ( x x0 ) + ( x x0 ) 2 + f ( x) = 0! 1! 2! f (n) ( x0 ) ( x x0 ) n + n! f ( x0 ) f / ( x0 ) f // ( x0 ) 0 1 ( x x0 ) + ( x x0 ) + ( x x0 ) 2 f ( x) = 0! 1! 2! 8 ( x 1) 0 + 4 ( x 1)1 + 2 ( x x0 ) 2 f ( x) = 0! 1! 2! = 8 + 4( x 1) + 1( x 1) 2 = 4 + 4x + x2 2x +1 = 5 + 2x + x2 48 9.54Lagrangeformoftheremainder Y = ( 7-x)4 primitive function Y = -4( 7-x) 1 deriative Y = 0 at x = 7 the critical value / / 3 st * Y (7) = 12( 7-x) = 0 2 deriviative Y (3) (7) = -24( 7-x) = 0 3rd derivative Y ( 4 ) (7) = 24 4 th derivative // 2 nd Because first nonzero derivative Y ( n ) is even (4) and Y (4) > 0 (24), critical value is a min. 49 9.54Lagrangeformoftheremainder Y = ( 7-x)4 primitive function Y ( 4) = 24 4 th derivative decision rule : n is even (4) and > 0 therefore a minimum 50 9.6 NthDerivativeTestforRelative ExtremumofaFunctionofOneVariable 9.61Taylorexpansionandrelativeextremum 9.62Somespecificcases 9.63Nthderivativetest 51 9.6Taylorexpansionandrelativeextremum A function f(x) attains a relative max (min) value at x0 if f(x) -f(x0 ) is neg. (pos.) for values of x in the immediate neighborhood of x0 (the critical value) both to its left and right Taylor series approximation for a small change in x f(x) - f(x0 ) = f / ( x0 ) ( x x )1 0 + f // ( x0 ) ( x x ) 2 0 1! 2! At the max., min., or inflection, f / (x0 ) = 0 +R and if f //(x0 ) = 0, and if f ( 3)(x0 ) = 0, then f(x) - f(x0 ) = R What is the sign of R for the first nonzero derivative? 52 9.62Somespecificcases 53 9.63Nthderivativetest If the first derivative of a function f(x) at x0 is f / ( x0 ) = 0 and if the first nonzero derivative value at x 0 encountered in successive derivation is that of the N th derivative, f (n)(x0 ) 0, then the stationary value f ( x0 ) will be : a relative max if N is even and f (n)(x0 ) < 0 a relative min if N is even and f (n)(x0 ) > 0 an inflection point if N is odd 54
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Cambrian College - ECON - 101
ChiangCh.10ExponentialandLogarithmicFunctions10.1 The Nature of Exponential Functions10.2 Natural Exponential Functions andthe Problem of Growth10.3 Logarithms10.4 Logarithmic Functions10.5 Derivatives of Exponential andLogarithmic Functions10.6
Cambrian College - ECON - 101
Ch.11TheCaseofMorethanOneChoiceVariable 11.1TheDifferentialVersionofOptimizationConditions 11.2ExtremeValuesofaFunctionofTwoVariables 11.3QuadraticFormsAnExcursion 11.4ObjectiveFunctionswithMorethanTwoVariables [11.5SecondOrderConditionsinRelatio
Cambrian College - ECON - 101
Ch.12OptimizationwithEqualityConstraints12.112.212.312.412.5Demand12.612.712.8EffectsofaConstraintFindingtheStationaryValuesSecondOrderConditionsQuasiconcavityandQuasiconvexityUtilityMaximizationandConsumerHomogeneousFunctionsLeastCostCom
Cambrian College - ECON - 101
Ch. 13Further Topics in Optimization13.1 Nonlinear Programming and KuhnTucker Conditions13.2 The Constraint Qualification13.3 Economic Applications13.4 Sufficiency Theorems in NonlinearProgramming13.5 Maximum-value Functions and theEnvelope Theore
Cambrian College - ECON - 101
Chapter 12 OptimizationWith Equality ConstraintsEcon 130 Class Notes fromAlpha Chiang, Fundamentals of MathematicalEconomics, 3rd EditionIntroduction Previously,all choice variables wereindependent of each other. However if we are to observe the
Cambrian College - ECON - 101
INTRODUCTIONChapter1AlphaChiang,FundamentalMethodsofMathematicalEconomics,ThirdEditionNatureofMathematicalEconomicsMathematicaleconomicsisnotadistinctbranchofeconomicsinthesensethatpublicfinanceorinternationaltradeis.itisanapproachtoeconomicanal
Cambrian College - ECON - 101
LINEARMODELSANDMATRIXALGEBRAChapter4AlphaChiang,FundamentalMethodsofMathematicalEconomics3rdeditionWhyMatrixAlgebraAsmoreandmorecommoditiesareincludedinmodels,solutionformulasbecomecumbersome.Matrixalgebraenablestodousmanythings:providesacompac
Cambrian College - ECON - 101
LINEAR MODELS AND MATRIXALGEBRA- Part 2Chapter 4Alpha Chiang, Fundamental Methodsof Mathematical Economics3rd editionVector Operations Multiplicationof vectorsAn m x 1 column vector u, and a 1 x n row vector v,yield a product uv of dimension m
Cambrian College - ECON - 101
LINEAR MODELS ANDMATRIX ALGEBRA- Part 3Chapter 4Alpha Chiang, Fundamental Methods ofMathematical Economics3rd editionIdentity Matrix1 0 I2 = 0 1 1 0 0 0 1 0 I3 = 0 0 1 Characteristics:It is like the number 1: a(1) = (1)a=aIA = AI = AIden
Cambrian College - ECON - 101
LINEAR MODELS AND MATRIXALGEBRA- ContinuedChapter 5Alpha Chiang, Fundamental Methodsof Mathematical Economics3rd editionConditions for Nonsingularity of a Matrix Whensquareness condition is alreadymet, a sufficient condition for thenonsingulari
Cambrian College - ECON - 101
LeontiefInputOutputModelsFrom Chapter 5Alpha Chiang, Fundamental Methods ofMathematical Economics, 4th EditionBackgroundProfessor Wassily Leontief, a Nobel Prizewinner,* deals with this particular question:&quot;What level of output should each of the
Cambrian College - ECON - 101
COMPARATIVE STATICS AND THECONCEPT OF DERIVATIVEChapter 6Alpha Chiang, Fundamental Methodsof Mathematical Economics3rd editionNature of Comparative Staticsconcerned with the comparison of different equilibriumstates that are associated with differ
Cambrian College - ECON - 101
Chapter 8COMPARATIVE STATIC ANALYSIS OFGENERAL FUNCTION MODELSAlpha Chiang, Fundamental Methodsof Mathematical Economics3rd editionComparative Static AnalysisSignificance: sometimes no explicit reduced form solution can beobtained. Hence, we will
Cambrian College - ECON - 101
Chapter 9:OPTIMIZATION: A Special Varietyof Equilibrium AnalysisAlpha Chiang, FundamentalMethods of MathematicalEconomics3rd editionOptimum Values and Extreme ValuesEconomics is by and large a science of choice.common criterion of choice among al
Cambrian College - ECON - 101
Chapter 10:Exponential andLogarithmic FunctionsAlpha Chiang, FundamentalMethods of MathematicalEconomics3rd editionExponential functionsy = f (t ) = b b &gt; 1Generalized:ty = aba and c are compressing or extending agentsctExponential functions
Cambrian College - ECON - 101
Chapter 11 - Optimization:More Than One Choice VariableAlpha Chiang, Fundamentals ofMathematical Economics, 3rd EditionOne variable casez = f ( x)dz = 0dz = f '( x )dxMax: d 2 z &lt; 0Min: d 2 z &gt; 0necessary but not sufficientfor either maximum or
Cambrian College - ECON - 101
Econ 509, Introduction to Mathematical Economics IProfessor Ariell ReshefUniversity of VirginiaLecture notes based on Chiang and Wainwright, Fundamental Methods of Mathematical Economics.1Mathematical economicsWhy describe the world with mathematica
City Colleges of Chicago - ECON - Econ
Chapter 1Ten Principles of EconomicsTRUE/FALSE1.Scarcity means that there is less of a good or resource available than people wish to have.ANS: TDIF: 1REF: 1-0NAT: AnalyticLOC: Scarcity, tradeoffs, and opportunity costTOP: ScarcityMSC: Definiti
City Colleges of Chicago - ECON - Econ
Chapter 2Thinking Like An EconomistTRUE/FALSE1.Economists try to address their subject with a scientists objectivity.ANS: TDIF: 1REF: 2-1NAT: AnalyticLOC: The study of economics and definitions of economicsTOP: EconomistsMSC: Definitional2.Ec
City Colleges of Chicago - ECON - Econ
Chapter 4The Market Forces of Supply and DemandTRUE/FALSE1.Prices allocate a market economys scarce resources.ANS: TDIF: 1REF: 4-0NAT: AnalyticLOC: Markets, market failure, and externalitiesTOP: Market economiesMSC: Definitional2.In a market
City Colleges of Chicago - ECON - Econ
Chapter 5Elasticity and Its ApplicationTRUE/FALSE1.Elasticity measures how responsive quantity is to changes in price.ANS: TDIF: 1REF: 5-0NAT: AnalyticLOC: ElasticityTOP: Price elasticity of demandMSC: Definitional2.Measures of elasticity enh
City Colleges of Chicago - ECON - Econ
Chapter 6Supply, Demand, and Government PoliciesTRUE/FALSE1.Economic policies often have effects that their architects did not intend or anticipate.ANS: TDIF: 1REF: 6-0NAT: AnalyticLOC: The study of economics and definitions of economicsTOP: Pub
City Colleges of Chicago - ECON - Econ
Chapter 7Consumers, Producers, and the Efficiency of MarketsTRUE/FALSE1.Welfare economics is the study of the welfare system.ANS: FDIF: 1REF: 7-1LOC: Supply and demandTOP: WelfareNAT: AnalyticMSC: Definitional2.The willingness to pay is the m
City Colleges of Chicago - ECON - Econ
Chapter 8Application: the Costs of TaxationTRUE/FALSE1.Total surplus is always equal to the sum of consumer surplus and producer surplus.ANS: FDIF: 2REF: 8-1NAT: AnalyticLOC: Supply and demandTOP: Total surplusMSC: Interpretive2.Total surplus
City Colleges of Chicago - ECON - Econ
Chapter 9Application: International TradeTRUE/FALSE1.Trade decisions are based on the principle of absolute advantage.ANS: FDIF: 1REF: 9-1NAT: AnalyticLOC: Gains from trade, specialization and tradeTOP: Absolute advantageMSC: Interpretive2.Th
City Colleges of Chicago - ECON - Econ
Chapter 10ExternalitiesTRUE/FALSE1.Markets sometimes fail to allocate resources efficiently.ANS: TDIF: 2REF: 10-0NAT: AnalyticLOC: Markets, market failure, and externalitiesTOP: Market failureMSC: Interpretive2.When a transaction between a bu
City Colleges of Chicago - ECON - Econ
Chapter 11Public Goods and Common ResourcesTRUE/FALSE1.When goods are available free of charge, the market forces that normally allocate resources in our economyare absent.ANS: TDIF: 2REF: 11-0NAT: AnalyticLOC: Markets, market failure, and exter
City Colleges of Chicago - ECON - Econ
Chapter 12The Design of the Tax SystemTRUE/FALSE1.The average American pays a higher percent of his income in taxes today than he would have in the late 18thcentury.ANS: TDIF: 1REF: 12-0NAT: AnalyticLOC: The role of government TOP:Tax burdenMS
City Colleges of Chicago - ECON - Econ
Chapter 13The Costs of ProductionTRUE/FALSE1.The economic field of industrial organization examines how firms decisions about prices and quantitiesdepend on the market conditions they face.ANS: TDIF: 2REF: 13-0NAT: AnalyticLOC: Costs of producti
City Colleges of Chicago - ECON - Econ
Chapter 14Firms in Competitive MarketsTRUE/FALSE1.For a firm operating in a perfectly competitive industry, total revenue, marginal revenue, and average revenueare all equal.ANS: FDIF: 2REF: 14-1NAT: AnalyticLOC: Perfect competitionTOP: Average
City Colleges of Chicago - ECON - Econ
Chapter 15MonopolyTRUE/FALSE1.Monopolists can achieve any level of profit they desire because they have unlimited market power.ANS: FDIF: 2REF: 15-0NAT: AnalyticLOC: MonopolyTOP: MonopolyMSC: Interpretive2.Even with market power, monopolists
City Colleges of Chicago - ECON - Econ
Chapter 16Monopolistic CompetitionTRUE/FALSE1.The &quot;competition&quot; in monopolistically competitive markets is most likely a result of having many sellers in themarket.ANS: TDIF: 1REF: 16-1NAT: AnalyticLOC: Monopolistic competitionTOP: Monopolistic
City Colleges of Chicago - ECON - Econ
Chapter 17OligopolyTRUE/FALSE1.The essence of an oligopolistic market is that there are only a few sellers.ANS: TDIF: 1REF: 17-0NAT: AnalyticLOC: OligopolyTOP: OligopolyMSC: Definitional2.Game theory is just as necessary for understanding com
City Colleges of Chicago - ECON - Econ
Chapter 18The Markets For the Factors of ProductionTRUE/FALSE1.If the marginal productivity of the sixth worker hired is less than the marginal productivity of the fifth workerhired, then the addition of the sixth worker causes total output to declin
City Colleges of Chicago - ECON - Econ
Chapter 19Earnings and DiscriminationTRUE/FALSE1.A compensating differential refers to a difference in wages that arises from nonmonetary characteristics.ANS: TDIF: 2REF: 19-1NAT: AnalyticLOC: Labor marketsTOP: Compensating differentialsMSC: De
City Colleges of Chicago - ECON - Econ
Chapter 20Income Inequality and PovertyTRUE/FALSE1.The poverty line is set by the government so that 10 percent of all families fall below that line and are therebyclassified as poor.ANS: FDIF: 1REF: 20-1NAT: AnalyticLOC: The study of economics,
City Colleges of Chicago - ECON - Econ
Chapter 21The Theory of Consumer ChoiceTRUE/FALSE1.The theory of consumer choice illustrates that people face tradeoffs, which is one of the Ten Principles ofEconomics.ANS: TDIF: 1REF: 21-0NAT: AnalyticLOC: Utility and consumer choiceTOP: Consu
City Colleges of Chicago - ECON - Econ
Chapter 22Frontiers of MicroeconomicsTRUE/FALSE1.The science of economics is a finished jewel, perfect and unchanging.ANS: FDIF: 1REF: 22-0NAT: AnalyticLOC: The Study of economics, and definitions in economicsTOP: economicsMSC: Definitional2.
UNI - CS - 121
&lt;?xml version='1.0'?&gt;&lt;prop.map version='4'&gt;&lt;prop.list&gt;&lt;prop.pair&gt;&lt;key&gt;CurrentAuthorContextID&lt;/key&gt;&lt;int type='signed' size='32'&gt;0&lt;/int&gt;&lt;/prop.pair&gt;&lt;prop.pair&gt;&lt;key&gt;FlowChartProperties&lt;/key&gt;&lt;prop.list&gt;&lt;prop.pair&gt;&lt;key&gt;NoShow&lt;/key&gt;&lt;array&gt;&lt;array.ty
Embry-Riddle FL/AZ - ACCT - 210
7/eReceivablesand Investments7Statements andthe Annual ReportPowerPoint Author: Catherine LumbattisCOPYRIGHT 2011 South-Western/Cengage LearningApple Corporation SampleAccounts ReceivableSubsidiary LedgerAcmeBaxterJonesMartinSmithGross Ac
University of the Philippines Diliman - GEOL - 11
2/24/2010Erosion, Deposition andLithificationSedimentary processesGeology 11 Principles of GeologyA. M. P. TengonciangDepartment of Physical SciencesUniversity of the Philippines, BaguioErosion The abrasion and displacement of earth materials by
University of the Philippines Diliman - GEOL - 11
2/24/2010Sedimentary EnvironmentsGeology 11 Principles of GeologyA. M. P. TengonciangDepartment of Physical SciencesUniversity of the Philippines, BaguioCONTINENTAL SEDIMENTARY ENVIRONMENTSALLUVIALFANBreccia,conglomerate,arkoseComposition Terr
University of the Philippines Diliman - GEOL - 11
2/22/2011Continental/ terrestrialSedimentary EnvironmentsTransitionalMarineGeology 11 Principles of GeologyA. M. P. Tengonciang &amp; D. N. JavierDepartment of Physical SciencesUniversity of the Philippines, BaguioContinental/terrestrialdepositional
University of the Philippines Diliman - GEOL - 11
ANGELES, GRACE ELYSSE D.Geol 11 Section: WFU201158955BS BiologyTopic: STROMBOLIAN ERUPTION
University of the Philippines Diliman - GEOL - 11
Lecture 11: Rock deformation The movements of Earth's tectonic plates creates stressesrocks subject to stress will deform (change in volume or shape of a body of rock) ways in which rocks deform are varied; which style a particular rock mass adoptsdep
University of the Philippines Diliman - GEOL - 11
GENERAL A RTICLEAlfred Wegener From Continental Drift toPlate TectonicsA J Saigeetha and Ravinder Kumar BanyalWhat is the nature of the force or mechanism that movesmassive continents thousands of miles across? What causesviolent earthquakes to dis
University of the Philippines Diliman - GEOL - 11
University of the Philippines Diliman - GEOL - 11
7/4/2011VolcanismGeology 11 Principles of GeologyA. M. P. Tengonciang &amp; D.D. N. javierDepartment of Physical SciencesUniversity of the Philippines, BaguioWhy does magma rise? Physical properties (temperature, composition,viscosity, relative buoyan
University of the Philippines Diliman - GEOL - 11
GEOLOGY 11 Principles of GeologyCourse SyllabusCourse Credit: 3.0Section: WFR, WFUClass Schedule: WF 8:30-10:00am; 10:00-11:30pmRoom: NIGS 1271st Sem AY 2011-2012Teacher: Nicole Eloise B. PellejeraConsultation Hours: TWF 1:00pm-5:00pmOffice: NIGS
University of the Philippines Diliman - GEOL - 11
Diliman Learning Resource CenterMathematics 17Second Semester AY 2010-2011Midterm Exam ReviewJanuary 17, 2011, MondayGeneral Directions. Do as indicated. Write your answers clearly and legibly. Show your complete solutions, if necessary, andbox your
University of the Philippines Diliman - MATH - 17
IIFifth LongExaml. T rueo r F alsei 2o0e. rnai ntocfw_a T heexpression q 1 2010 1 2011 b e s implified i .(b) T hel awo f s ines ndt he l awo f c osines lso oldt ruef or r ightt riangles.aah(c) T her ectangular( n,-n) c onverted t he p olarc
University of the Philippines Diliman - MATH - 17
Mathematics 17Fifth Long ExamGENERAL DIRECTION: Write the exam code of the test paper at the upper right corner of yourbluebook (front). Write your student number below the code. Show all necessary solutions and box the nalanswers. Refrain from using
University of the Philippines Diliman - MATH - 17
Math 17 TWHFW-7Ex 1 (HW 4) Part IPlease do the exercises as indicated. Write down all nal answers on aseparate piece of paper.I. Present the completely factored out form of the following:1. 2x4 20x3 2x2 y 2 z 4 + 16x2 yz 2 + 18x22. 3(x y )2 x + y 2
University of the Philippines Diliman - MATH - 17
Math 17 TWHFW-71st Sem AY 2010-11Exercise 1 Part I Answer KeySolutions to Exercise Set I. Please remember that there may be (and probably are) multiple ways to go about a given problem. As long as the techniquesare valid and the computations are caref
University of the Philippines Diliman - MATH - 17
Math 17 TWHFW-7Ex 1 (HW 4) Part IIPlease do the exercises as indicated. Write down all nal answers on aseparate piece of paper.I. Simplify the following expressions: Consider simplest assumptions on radicandsand variables and that all divisors are no
University of the Philippines Diliman - MATH - 17
Math 17 TWHFW-71st Sem AY 2010-11Exercise 1 Part II Answer KeySolutions to Set 1 Part II. Again, solutions are not absolute. There may beother ways to arrive at the same nal answer.1.15+2 5 52 5+52 5 5+2 561225+2 5 + 52 51=6152 5 5+2 520
University of the Philippines Diliman - MATH - 17
Exercise Set 2 - Part 1 (HW 8)July 18, 2011Math 17 TWHFW-71st Sem AY 2011-12Use a separate piece of paper as your answer sheet. Show as complete a solution as youare able to. Remember to box all nal answers. Submission is on Thursday, July 21.Good l
University of the Philippines Diliman - MATH - 17
Exercise Set 2 - Part 2 (HW 12)August 23, 2011Math 17 TWHFW-71st Sem AY 2011-12Use a separate piece of paper as your answer sheet. Show as complete a solution as youare able to. Remember to box all nal answers. Submission is on Friday, August 26.I.
University of the Philippines Diliman - MATH - 17
Exercise Set 4 (HW 13)September 18, 2011Math 17 TWHFW-71st Sem AY 2011-12Use a separate piece of paper as your answer sheet. Show ascomplete a solution as you are able to, unless otherwisespecied. Remember to box all nal answers whensolutions are n
University of the Philippines Diliman - MATH - 17
Exercise Set 4 KeySeptember 21, 2011Math 17 TWHFW-71st Sem AY 2011-12I.1. FALSE55is in QII, thus sin &gt; 0.443&lt; 5 &lt; 2 5 is in QIV and sin 5 &lt; 0.2xP2=fP +xP2=fx+P29.=.|3|32 = 6 P , thus tan 3x is 2 -periodic.5. TRUE The period
University of the Philippines Diliman - MATH - 17
Math 17 TWHFW-71st Sem AY 2011-12Exercise Set 5 (HW 14)October 4, 2011Use a separate piece of paper as your answer sheet. Show ascomplete a solution as you are able to, unless otherwise specied.Remember to box all nal answers when solutions are nece