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Chem 302 - Math 252

Chem 302 - Math 252. Chapter 4 Differentiation & Integration. 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

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Chem 302 - Math 252

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  1. Chem 302 - Math 252 Chapter 4Differentiation & Integration

  2. 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

  3. Linear Differentiation

  4. Linear Differentiation Smaller spacing not necessarily better

  5. 3 point Differentiation • Linear differentiation ignores actual point • Make exact for

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

  7. Multi-point Differentiation

  8. Multi-point Differentiation

  9. Multi-point Differentiation

  10. Example

  11. Numerical Integration

  12. Midpoint Formula • Uses value of function and slope at midpoint of interval • Determine w1 & w2

  13. Composite Midpoint Formula • n subintervals (equal spacing)

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

  15. Trapezoidal Integration • Alternate derivation • Linear combination of endpoints that give best estimate of integral • Determine w1 & w2

  16. Composite Trapezoidal Integration • n subintervals (equal spacing)

  17. Simpson’s Rule • Combines Trapezoidal and Midpoint • Also referred to as 3 - point • Determine w1w2 & w3

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

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

  20. Newton-Cotes Formula

  21. Gaussian Quadratures • So far evaluated function at fixed points & optimized coefficients • Optimize locations also • Optimize wi & zi

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

  23. Gaussian Quadratures • 2-point • Need four equations • Make exact for f(z) = 1, f(z) = z, f(z) = z2, f(z) = z3 • Does not give unique solution • Make symmetric about 0

  24. Gaussian Quadratures

  25. Gaussian Quadratures

  26. Gaussian Quadratures

  27. Gaussian Quadratures

  28. Gaussian Quadratures • Other forms

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

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