advertisement

“I feel like I’m diagonally parked in a parallel universe” Math Review Monday June 7 2003 A) Introduction a. Symbols b. Operations c. Central Tendencies B) Linear Algebra C) Correlation/Regression Analysis D) Applied Calculus Basic Math Review B) System of equations a) 3x - y = -7 5y + 5 = -5x b) 3x + 4y = 2 2y = 4 - 3/2x Basic Math Review D) Applied Calculus Rate of change (slope): Dy/Dx or (y2-y1)/(x2-x1) Here Dy/Dx is constant regardless of the “limit” 160 140 Street Number 120 100 80 60 40 20 0 0 1000 2000 Broadway Address 3000 4000 Basic Math Review D) Applied Calculus How long does it take to fill one beaker (1L)? dV t 1000ml / dt V(t) dV 1000ml/t dt t Basic Math Review D) Differential equations A differential equation is an equation in which one or more unknowns depend on its/their rate of change (or that of other variables included in the equations) F ma where dv(t ) d 2 x(t ) a 2 dt dt Newton’s principia Basic Math Review D) Definition of a derivative f ( x0 ) lim Dx 0 f ( x Dx) f ( x) Dx The derivative of a function (x) at a point “a” is the slope of the straight line tangent to (x) at “a” instantaneous rate of change! One is pushing to limit to “0”: the slope is close to real as (x) approaches 0 Basic Math Review D) Definition of a derivative f ( x0 ) lim Dx 0 f ( x Dx) f ( x) Dx The derivative of a function (x) = f’(x) Important derivatives: f(x) = C f’(x) = 0 f(x) = xn f’(x) = nxn-1 f(x) = ex f’(x) = ex f(x) = lnx f’(x) = 1/x Basic Math Review D) Maxima, Minima One of the great applications of calculus (particularly in economics) is to determine the “maxima” and “minima” of functions. The derivatives of the maxima and minima = 0 The function neither increase nor decreases! f(x) = x3 – 3x2 – 24x + 5 f’(x) = 3x2 – 6x – 24 3 (x2 – 2x – 8) 3 (x + 2)(x –4) f’(x) vanishes (reaches a critical point) only when f’(x) = 0 D) Maxima, Minima f”(x) < 0 maximum f”(x) > 0 minimum 250 200 150 100 50 0 -8 -6 -4 -2 0 -50 -100 -150 -200 2 4 6 8 10 D) Application: Derivatives Ji = - DS ([Ci ]/z) DS = D0 2 J Cz=0 = -3 D0 (8 oC) ([Ci ]/z)z=0 D) Application: Derivatives Ji = - DS ([Ci ]/z) DS = D0 2 J Cz=0 = -3 D0 (8 oC) ([Ci ]/z)z=0 “A River (used to) Run Through It”: Part I Reservoirs and soil erosion Fo (%) 0% 2% 4% 6% 8% 10% 0 12% 1200 5 10 Depth (cm) 15 20 25 30 35 40 Fo 45 C/N 50 12 14 16 (C/N)a 18 20 D) Application: Integral 1.2 J-Co (gC/m2.yr) 0.0 1.0 0.2 0.4 0.6 0.8 1.0 0 1200 5 10 15 0.6 Depth (cm) Jo (gC/m2.yr) 0.8 0.4 0.2 20 25 30 35 40 0.0 0 45 10 50 20 30 40 50 Age (years) 60 70 80 90 D) Application: Integral (x) = Polynomial (6th degree) ∫ (xn) = [n(n+1)/(n+1)] + C 1.2 1.0 Jo (gC/m2.yr) 0.8 0.6 0.4 0.2 y = -2E-10x6 + 4E-08x5 - 3E-06x4 + 0.0001x3 - 0.0013x2 + 0.0136x + 0.3082 R2 = 0.9795 0.0 0 10 20 30 40 50 60 70 80 90 Age (years) Integral between two limits (0 and 85 yrs) ∫ (xn)100 - ∫ (xn)0 C-C