1. Calculus Lesson 7

2. SystemDerivative A + B + C[ ] + [ ] + [ ] A * B * C[ ] + [ ] + [ ] A^(B^C)[ ] + [ ] + [ ] A^B / C[ ] + [ ] + [ ] Three inputs, 3 changing perspectives to include

3. George Frank

4. g g f f f dg g df df dg dg df

5. Slicing A Cake Among Friends A A B B C C A A B B C C New person? Cut a slice from everyone New person? Cut a slice from everyone Cake D

6. Slicing A Cake Among Friends Cake A A A A B B C C New person? Cut a slice from everyone New person? Cut a slice from everyone D B B A A B B C

7. A’s changes B’s changes C’s changes ++ Scenario With 3 PartsChange Simplifies To A A B B C C ++ A A B B C C ** A A B B C C ^^ A A B B C C */ …

8. F’s changes G’s changes + Simplifies to Scenario With 2 Parts F F G G + F F G G * F F G G ^ F F G G / …

9. Scenario With 3 Parts A A B B * C C * A’s changes B’s changes + C’s changes +

10. Simplifies to Scenario With 2 Parts F F G G + F F G G * F F G G ^ F F G G / … F’s changes G’s changes + Convert df to dx X’s changes + Convert dg to dx

11. F’s changes G’s changes + X’s changes + Convert dg to dx Convert df to dx

12. Hours G’s changes X’s changes + Convert dg to dx Seconds Seconds/Hour [ f(g(x)) ]’ = f’(g(x)) * g’(x)[ f(A) ]’ = f’(A) * A’ If you stop analyzing at A… then A’ = 1 dA/dA = 1 df/dx = df/dg * dg/dx dollars/yen = dollars/euro * euro/yen

13. f f g g * derivative of =+ f f dg * df g g *= dg/dx = 2 df/dx = 1 (x + 3) (2x + 7) 2dx 1dx (x + 3) (2x + 7) *

14. a a 6 derivative of = a a = da/dx = 2x + 3 (x 2 + 3x + 1) 2dx (x 2 + 3x + 1) 6 5 6

15. x’s changes F’s changes df/dx df’s changes df/dx dx’s changes dg’s changes dg ----- dx dx’s changes dx’s changes Paint $

16. Wood $ Paint $ + Wood ¥ Paint ¥ + Convert Wood $ to ¥ Convert Paint $ to ¥

17. F’s Changes G’s Changes + X’s changes + Convert df to dx Convert dg to dx

18. SystemDerivative A + B + C[ ] + [ ] + [ ] A * B * C[ ] + [ ] + [ ] A^(B^C)[ ] + [ ] + [ ] Three inputs, 3 changing perspectives to include

19. SystemDerivativeFuzzy Derivative A * B * C[ ] + [ ] + [ ] A^(B^C)[ ] + [ ] + [ ] Scenario With 2 PartsFuzzy Viewpoint A A B B + … A’s changes B’s changes +

20. x2x2 x2x2 x2x2

21. x2x2 x2x2 x2x2

23. g g f f f * dg g * df dg df

24. Calculus Week 8

25. InteractionOverall Change Addition Multiplication Powers Inverse Division

26. InteractionOverall ChangeAnalogy AdditionTrack changes from each part MultiplicationGrow a rectangle Powers N viewpoints of “my change times the others” InverseSharing cake, new guy walks in DivisionImagine f * (1/g)

27. X-Ray StrategyVisualizationStep-by-Step LayoutStep Zoom In Ring-by-ring r dr SymbolicSolutionStep Zoom In r dr (from 0 to r) 2 * pi * r

28. StrategyVisualizationStep-by-Step LayoutSingle Step Zoom Ring-by-ring Timelapse r dr 2πr2πr Symbolic DescriptionSolutionNotes Work backwards to the integral. that meansIf

29. StrategyVisualizationHeight of PlateSingle Step Zoom Plate-by-plate Timelapse dx π y2π y2 x y r

30. StrategyVisualizationHeight of PlateSingle Step Zoom Plate-by-plate Timelapse x dx π y2π y2 x y r

31. SymbolicSolutionNotes Write height in terms of x Work backwards to find integrals Find volume at full radius (x=r)

32. &= 2 \int_0^r \pi y^2 \ dx \\ &= 2 \int_0^r \pi (\sqrt{r^2 - x^2})^2 \ dx \\ &= 2 \pi \int_0^r r^2 - x^2 \ dx \\ &= 2 \pi \left( (r^2)x - \frac{1}{3}x^3 \right) \\ &= 2 \pi \left( (r^2)r - \frac{1}{3}r^3 \right) \\ &= 2 \pi \left( \frac{2}{3}r^3 \right) \\ &= \frac{4}{3} \pi r^3

33. StrategyVisualizationShell Analysis Shell-by-shell X-Ray StrategyVisualizationShell Analysis Shell-by-shell X-Ray dr dV

34. StrategyVisualization Shell-by-shell X-Ray volume change / thickness change

35. SymbolicSolutionNotes Express height (y) in terms of x Work backwards to the integral Get volume for full radius (x=r)