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Chem Math 252 Chapter 4 Differentiation & Integration

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Differentiation & Integration Experimental data at discrete points Need to know the rate of change of the dependent variable with respect to the independent variable Need to know area under curve Need to integrate an analytic function that is too complicated to do analytically Can do interpolation/curvefitting to get an analytic function

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Linear Differentiation

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Eqn (1)Eqn (2) xx % error Exact value × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × Linear Differentiation Smaller spacing not necessarily better

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3 point Differentiation Linear differentiation ignores actual point Make exact for

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Multi-point Differentiation Formulae only derived for equal spacing Non equal spacing solve equations numerically

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Multi-point Differentiation Coefficient Demominator-4h-3h-2h-h-h0h2h2h3h3h4h4hExact to 1 st derivative 2h2h01Quadratic 12h1-808Quartic 60h th order 840h th order

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Multi-point Differentiation Coefficient Demominator-4h-3h-2h-h-h0h2h2h3h3h4h4hExact to 2 nd derivative h2h2 121Quadratic 12h Quartic 540h th order 5040h th order

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Multi-point Differentiation Coefficient Demominator-4h-3h-2h-h-h0h2h2h3h3h4h4hExact to 3 rd derivative 2h32h Quartic 48h th order 240h th order

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Technique % error 3-point point point point Exact value Example

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Numerical Integration

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Midpoint Formula Uses value of function and slope at midpoint of interval Determine w 1 & w 2

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Composite Midpoint Formula n subintervals (equal spacing)

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Trapezoidal Integration Approximate f(x) by a linear function over interval [a,b]

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Trapezoidal Integration Alternate derivation Linear combination of endpoints that give best estimate of integral Determine w 1 & w 2

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Composite Trapezoidal Integration n subintervals (equal spacing)

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Simpson’s Rule Combines Trapezoidal and Midpoint Also referred to as 3 - point Determine w 1 w 2 & w 3

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Composite Simpson’s Rule 2n subintervals (equal spacing)

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Newton-Cotes Formula Generalization to use more than 3 points –Trapezoidal exact up to linear – (1 st order NC) –Simpson’s exact up to quadratic (by definition but turns out to be exact for up to cubic) – (2 nd order NC) –Equivalent to integration of Lagrangian interpolation functions –3 rd order NC Use 4 points and functions up to cubic –Higher orders can give larger errors

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Newton-Cotes Formula

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Gaussian Quadratures So far evaluated function at fixed points & optimized coefficients Optimize locations also Optimize w i & z i

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Gaussian Quadratures 1-point Need two equations Make exact for (z) = 1, & (z) = z

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Gaussian Quadratures 2-point Need four equations Make exact for (z) = 1, (z) = z, (z) = z 2, (z) = z 3 Does not give unique solution Make symmetric about 0

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Gaussian Quadratures

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Roots (z i )Weight Factors (w i ) Two-Point Formula ± Three-Point Formula ± Four-Point Formula ± ±

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Gaussian Quadratures Roots (z i )Weight Factors (w i ) Five-Point Formula 0 ± ± Six-Point Formula ± ± ±

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Gaussian Quadratures Roots (z i )Weight Factors (w i ) Ten-Point Formula ± ± ± ± ± Fifteen-Point Formula 0 ± ± ± ± ± ± ±

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Gaussian Quadratures Other forms

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Gaussian Quadratures - Example Simpson’s Rule –Use 100 intervals Gaussian Quadrature –3 and 15 point

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