1 / 28

Review 6

Review 6. Infinite integrals – rules U-substitution Slope Fields Separation of Variables. Change tanu to: And do a u-substitution for cosu. Change cotu to: And do a u-substitution for sinu:. Change. Change. U-Substitution.

cayla
Download Presentation

Review 6

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. Review 6 • Infinite integrals – rules • U-substitution • Slope Fields • Separation of Variables

  2. Change tanu to: • And do a u-substitution for cosu

  3. Change cotu to: • And do a u-substitution for sinu:

  4. Change

  5. Change

  6. U-Substitution • You replace the part that cannot be integrated – it’s a composition function. The inside gets replaced. • Take the derivative of the u with respect to x. This needs to match what is left in the original integral. You can only differ by a constant! • If there are limits, then you need to adjust the limits in terms of the u, then integrate and evaluate.

  7. examples Missing constant of 3, so divide the 3 to the other side to du

  8. Examples The quantity to the 5th power is u Matches exactly

  9. Examples Tan is the issue since it is raised to a power and its derivative is sec squared. No 4 present so divide it to the du

  10. Limits with u-sub examples Change the limits to terms of u Evaluate Stop here unless a mc question

  11. examples Replace the limits with respect to u Done unless a mc question

  12. Slope fields • Slope fields are mapping every possible tangent line for a function that is offset by a different constant c.

  13. Create the slope field for dy/dx=xy Show calculations for the slope fields on a free response question

  14. Create the slope field for dy/dx = 1/(x+1) Show calculations for the slope fields on a free response question

  15. Separation of variables. • Relocate the y to the side with dy, the dx to the side with dx. • Integrate both sides, remembering to include the constant c on the x side. • Find c now! It’s in an easier format. • Write the solution in terms of y = • you might have to take into consideration the initial condition (to find c) where it either lies or what the slope at that point is to determine the sign associated with the final answer. (only if you have to remove an absolute value or if you have square rooted the equation).

  16. Solve dy/dx=xy for the initial condition of (2,1) Find c first, then e both sides Since the solution is at (2, 1) and the slope is positive, you can drop the absolute value.

  17. Solve the differential equation of dy/dx = 1/(x+1) with the initial condition of (0, 2) Technically, you should do a u-sub for x+1 Find c

More Related