1 / 28

6.1 Differential Equations and Slope Fields

6.1 Differential Equations and Slope Fields. Consider:. or. then:. Given:. find. First, a little review:. It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation:.

lily
Download Presentation

6.1 Differential Equations and Slope Fields

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. 6.1 Differential Equations and Slope Fields

  2. Consider: or then: Given: find First, a little review: It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

  3. Given: and when , find the equation for . This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation. If we have some more information we can find C.

  4. Example • Find the solution of the initial value problem:

  5. Example • Find the solution of the initial value problem:

  6. Example • Find the solution of the initial value problem:

  7. Example • Find the solution of the initial value problem:

  8. Application • A car starts from rest and accelerates at a rate of -0.6t2 + 4 m/s2 for 0 < t < 12. How long does it take for the car to travel 100m?

  9. Application • An object is thrown up from a height of 2m at a speed of 10 m/s. Find its highest point and when it hits the ground.

  10. Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

  11. When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.

  12. Indefinite Integrals • Review the list of indefinite integrals on p. 307

  13. Differential Equations: General Solution • Finding the general solution of a differential equation means to find the indefinite integral (i.e. the antiderivative)

  14. Find the general solution

  15. Separation of variables • If a differential equation has two variables it is separable if it is of the form

  16. Example

  17. Separation of variables

  18. Separation of variables

  19. Separation of variables

  20. Separation of variables

  21. Separation of variables

  22. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand. Initial value problems and differential equations can be illustrated with a slope field.

  23. Slope Field Activity • Given • Find the slope for your point • Sketch a tangent segment across your point. Now do the same for the rest of the points • Are you on an equilibrium solution? • Find your isocline. Is it vertical, horizontal, slant, etc. • Sketch a possible solution curve through your point • Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? • What is the value of d2y/dx2 at your point?

  24. Slope Field Activity • Given • Find the slope for your point • Sketch a tangent segment across your point. Now do the same for the rest of the points • Are you on an equilibrium solution? • Find your isocline. Is it vertical, horizontal, slant, etc. • Sketch a possible solution curve through your point • Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? • What is the value of d2y/dx2 at your point?

  25. Slope Field Activity • Given • Find the slope for your point • Sketch a tangent segment across your point. Now do the same for the rest of the points • Are you on an equilibrium solution? • Find your isocline. Is it vertical, horizontal, slant, etc. • Sketch a possible solution curve through your point • Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? • What is the value of d2y/dx2 at your point?

  26. Slope Field Activity • Given • Find the slope for your point • Sketch a tangent segment across your point. Now do the same for the rest of the points • Are you on an equilibrium solution? • Find your isocline. Is it vertical, horizontal, slant, etc. • Sketch a possible solution curve through your point • Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? • What is the value of d2y/dx2 at your point?

  27. Slope Field Activity • Given • Find the slope for your point • Sketch a tangent segment across your point. Now do the same for the rest of the points • Are you on an equilibrium solution? • Find your isocline. Is it vertical, horizontal, slant, etc. • Sketch a possible solution curve through your point • Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? • What is the value of d2y/dx2 at your point?

  28. Hw: p. 312/7-17odd,31-36,39-42

More Related