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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. #2: Einstein’s relativity (cold war). #1: Newton’s Calculus. Calculus: the Science of Change. Sunday Sept. 13 Univariate Calculus 1. Derivative: the RATE OF CHANGE

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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: Einstein’s relativity (cold war) #1: Newton’s Calculus

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

  3. 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 • …? ? ?

  4. 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?

  5. Functions as modelsof changing quantities

  6. Univariate function and slope

  7. Univariate function and slope

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

  9. The derivative as a limit

  10. The derivative as a limit

  11. The derivative as a limit

  12. The derivative as a limit

  13. Alternative symbols

  14. Examples

  15. Rules of differentiation

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

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

  18. Unit conversion The CHAIN RULE

  19. The Chain Rule EXAMPLES:

  20. Higher-order derivatives EXAMPLES

  21. Curvature

  22. Curvature Gentle turn Sharp turn

  23. Extrema maximum minimum inflection point

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

  25. Taylor series example

  26. Taylor series example

  27. Taylor series example

  28. Negating the argument EXAMPLE:

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

  30. Differentiating data:Forward difference example

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

  32. Forward and backward differences FD actual BD

  33. Approximating the derivative Forwarddifference Backwarddifference SUM

  34. Approximating the derivative Forwarddifference Backwarddifference SUM /2

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

  36. Centered difference example

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

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