1 / 35

Understanding Three-Input Scenarios in Calculus: Perspectives on Change and Volume

In this lesson, we explore the intricacies of three-input scenarios in calculus by changing perspectives. We’ll delve into concepts such as derivatives, volume changes, and integrating functions using geometric interpretations. Through relatable examples like slicing a cake among friends, we’ll highlight how changes in variables affect outcomes. Additionally, we will work through calculations involving integrals and derivatives to solidify your understanding. Join us as we break down these complex ideas into manageable parts, enhancing your calculus skills.

africa
Télécharger la présentation

Understanding Three-Input Scenarios in Calculus: Perspectives on Change and Volume

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Calculus Lesson 7

  2. Three inputs, 3 changing perspectives to include

  3. George Frank

  4. dg f • dg df• dg g g •df df f

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

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

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

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

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

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

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

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

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

  14. derivative of derivative of 5 6 6 = a = a (x2 + 3x + 1)6 da/dx = 2x + 3 2dx (x2 + 3x + 1)

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

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

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

  18. Three inputs, 3 changing perspectives to include

  19. Scenario With 2 Parts Fuzzy Viewpoint A’s changes B’s changes + A + B …

  20. x2 x2 x2

  21. x2 x2 x2

  22. dg f * dg g g * df df f

  23. Calculus Week 8

  24. 2 * pi * r dr (from 0 to r) r dr r

  25. 2πr dr r Work backwards to the integral. If that means

  26. πy2 dx r y x

  27. πy2 dx r y x x

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

  29. &= 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

  30. dV dr

  31. volume change /thickness change

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

More Related