1. A Guide To Calculus

2. Baseline Rendering

3. Baseline vs. Progressive Rendering 25% 50% 100%

4. Circle and Sphere Fun Facts

5. Dissecting a Circle ~ Disc Rings

6. Unroll the Rings Area = ½ base x height Area = ½ base x height

7. X-Ray OriginalRingsSlicesBoards

8. Ring By Ring 25%100%50%75%

9. Slice By Slice

10. Board By Board

14. StrategyVisualizationStep-by-Step LayoutSingle Step Zoom Ring-by-ring Slice-by-slice Board-by-board r p x dp dx dr

15. StrategyVisualizationStep-by-Step LayoutCalculus Lingo Ring-by-ring Slice-by-slice Board-by-board

16. CircleRingsSlicesBoards SphereShellsWedgesPlates

17. Ring-by-ring Timelapse Shell-by-Shell X-Ray Plate-by-plate Timelapse CircumferenceAreaVolumeSurface area

20. Derivative SymbolMindset Think about the resulting step- by-step pattern Think about the actual ratio of changes

21. x x 1 1 x x 1d Perimeter Change2d Area Change

22. Infinity … Image source: wikipedia

23. The Derivative 1. Choose an interval 2. Find the raw change 3. Find the rate of change 4. Make your model perfect

24. 1+ 3+ 5+ 7+ 9 5x5 =

25. 5+ 7+ 95x5 = 2x2-

26. George Frank

27. g g f f f × dg g × df df × dg dg df

28. 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

29. 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 / … Two Perspectives To Consider

30. Scenario With 3 Parts A A B B × C C × A’s changes B’s changes + C’s changes + 3 Perspectives To Consider

31. x2x2 x2x2 x2x2

32. InteractionOverall Change Addition Multiplication Powers Inverse Division

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

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

35. = + = 2 ×

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

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