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“I feel like I’m diagonally parked in a parallel universe”

“I feel like I’m diagonally parked in a parallel universe”. Math Review Monday June 7 2003. Introduction Symbols Oper atio ns Central Tendencies Linear Algebra Correlation/Regression Analysis Applied Calculus. Basic Math Review. System of equations. 3 x - y = -7

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“I feel like I’m diagonally parked in a parallel universe”

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  1. “I feel like I’m diagonally parked in a parallel universe”

  2. Math ReviewMonday June 7 2003 • Introduction • Symbols • Operations • Central Tendencies • Linear Algebra • Correlation/Regression Analysis • Applied Calculus

  3. Basic Math Review System of equations • 3x - y = -7 • 5y + 5 = -5x • 3x + 4y = 2 • 2y = 4 - 3/2x

  4. Basic Math Review Applied Calculus Rate of change (slope): Dy/Dx or (y2-y1)/(x2-x1) Here Dy/Dx is constant regardless of the “limit”

  5. Basic Math Review Applied Calculus How long does it take to fill one beaker (1L)? V(t) t

  6. Basic Math Review 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) Newton’s principia

  7. Basic Math Review Definition of a derivative 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

  8. Basic Math Review Definition of a derivative 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

  9. Basic Math Review 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

  10. f”(x) < 0  maximum f”(x) > 0  minimum Maxima, Minima

  11. Application: Derivatives Ji = - DS ([Ci ]/z) DS = D02 J Cz=0 = -3 D0 (8 oC) ([Ci ]/z)z=0

  12. Application: Derivatives Ji = - DS ([Ci ]/z) DS = D02 J Cz=0 = -3 D0 (8 oC) ([Ci ]/z)z=0

  13. “A River (used to) Run Through It”: Part I

  14. Reservoirs and soil erosion

  15. Application: Integral

  16. Application: Integral (x) = Polynomial (6th degree) ∫ (xn) = [n(n+1)/(n+1)] + C Integral between two limits (0 and 85 yrs) ∫ (xn)100 - ∫ (xn)0 C - C

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