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. How fast is it changing: – over a decade? – a year? – a month? – a day? – a second? – right now?

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

9. Functions as models of changing quantities

10. Univariate function and slope

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

13. The derivative as a limit

17. Alternative symbols

18. Examples

19. Rules of differentiation

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

21. Unit conversion

22. The CHAIN RULE

23. The Chain Rule EXAMPLES:

24. Higher-order derivatives EXAMPLES

25. maximum minimum inflection point Extrema

26. Taylor series constant derivative at x 0 power of h=x-x 0

27. Taylor series example

30. Negating the argument EXAMPLE:

31. Negating the argument EXAMPLE:

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

33. Differentiating data: Forward difference example xf=x 3 f’3x 2 11(8-1)/1 = 73 28(27-8)/1 = 1912 327(64-27)/1 = 3727 464(125-64)/1 =6148 512575

34. Approximating the derivative Observational data analysis Numerical modeling Backward difference Forward difference approximation Error ~ O(h)

35. FD actual BD Forward and backward differences

36. Approximating the derivative Forward difference Backward difference SUM

37. Approximating the derivative Forward difference Backward difference SUM /2

38. Approximating the derivative Forward difference Backward difference SUM /2 Centered difference Error ~ O(h 2 )

39. Centered difference example xf=x 3 f’3x 2 11 28(27-1)/2 = 1312 327(64-8)/2 = 2827 464(125-27)/2 = 4948 5125(216-64)/2=7675 6216(343-125)/2=109108 7343

40. Approximating the 2 nd derivative Forward difference Backward difference SUBTRACT /h Centered difference Error ~ O(h 2 )

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

42. Curvature

43. Gentle turn Sharp turn