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Scientists: Which scientific advance has had the most impact on people’s everyday lives?

Scientists: Which scientific advance has had the most impact on people’s everyday lives?. #3: Darwin’s theory of evolution. Scientists: Which scientific advance has had the most impact on people’s everyday lives?. #3: Darwin’s theory of evolution. #2: Einstein’s relativity (cold war).

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Scientists: Which scientific advance has had the most impact on people’s everyday lives?

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  1. Scientists: Which scientific advance has had the most impact on people’s everyday lives? #3: Darwin’s theory of evolution

  2. Scientists: Which scientific advance has had the most impact on people’s everyday lives? #3: Darwin’s theory of evolution #2: Einstein’s relativity (cold war)

  3. Scientists: Which scientific advance has had the most impact on people’s everyday lives? #3: Darwin’s theory of evolution #2: Einstein’s relativity (cold war) #1: Newton’s Calculus

  4. Calculus: the Science of Change No single thing abides,but all things flow - Heraclitus

  5. Example: Climate change • Global mean temperature • Rate of change consistent with natural causes? • OR is human activity implicated? • What else changes due to global warming? • Sea ice extent • …? ? ?

  6. Think of a quantity you might measure

  7. Think of a quantity you might measure • How fast is it changing: • over a decade? • a year? • a month? • a day? • a second? • right now?

  8. Calculus: the Science of Change Monday Sept. 13Univariate Calculus 1 Derivative: the RATE OF CHANGE Taylor series approximations Differentiating data

  9. Functions as modelsof changing quantities

  10. Univariate function and slope

  11. Univariate function and slope

  12. “Slope” of a function: the tangent line

  13. The derivative as a limit

  14. The derivative as a limit

  15. The derivative as a limit

  16. The derivative as a limit

  17. Alternative symbols

  18. Examples

  19. Rules of differentiation

  20. The Chain Rule:Latitude 1o = 110km

  21. Unit conversion

  22. Unit conversion The CHAIN RULE

  23. The Chain Rule EXAMPLES:

  24. Higher-order derivatives EXAMPLES

  25. Extrema maximum minimum inflection point

  26. Taylor series power of h=x-x0 constant derivative at x0

  27. Taylor series example

  28. Taylor series example

  29. Taylor series example

  30. Negating the argument EXAMPLE:

  31. Negating the argument EXAMPLE:

  32. Approximating the derivative Observational data analysis Numerical modeling Forwarddifference approximation Error ~ O(h)

  33. Differentiating data:Forward difference example

  34. Approximating the derivative Observational data analysis Numerical modeling Forwarddifference approximation Error ~ O(h) Backwarddifference

  35. Forward and backward differences FD actual BD

  36. Approximating the derivative Forwarddifference Backwarddifference SUM

  37. Approximating the derivative Forwarddifference Backwarddifference SUM /2

  38. Approximating the derivative Forwarddifference Backwarddifference SUM /2 Centereddifference Error ~ O(h2)

  39. Centered difference example

  40. Approximating the 2nd derivative Forwarddifference Backwarddifference SUBTRACT /h Centereddifference Error ~ O(h2)

  41. Application: error analysis Floodwaters in the Kalama Gap V?

  42. Curvature

  43. Curvature Gentle turn Sharp turn

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