1. 6.1 Differential Equations and Slope Fields

2. First, a little review: Consider: then: or 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: Given:findWe 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. If we have some more information we can find C. 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.

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 -0.6t 2 + 4 m/s 2 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. Integrals such as are called indefinite integrals because we can not find a definite value for the answer. When finding indefinite integrals, we always include the “plus C”.

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

22. Initial value problems and differential equations can be illustrated with a slope field. 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.

23. Slope Field Activity Given 1.Find the slope for your point 2.Sketch a tangent segment across your point. Now do the same for the rest of the points 3.Are you on an equilibrium solution? 4.Find your isocline. Is it vertical, horizontal, slant, etc. 5.Sketch a possible solution curve through your point 6.Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? 7.What is the value of d 2 y/dx 2 at your point?

24. Slope Field Activity Given 1.Find the slope for your point 2.Sketch a tangent segment across your point. Now do the same for the rest of the points 3.Are you on an equilibrium solution? 4.Find your isocline. Is it vertical, horizontal, slant, etc. 5.Sketch a possible solution curve through your point 6.Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? 7.What is the value of d 2 y/dx 2 at your point?

25. Slope Field Activity Given 1.Find the slope for your point 2.Sketch a tangent segment across your point. Now do the same for the rest of the points 3.Are you on an equilibrium solution? 4.Find your isocline. Is it vertical, horizontal, slant, etc. 5.Sketch a possible solution curve through your point 6.Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? 7.What is the value of d 2 y/dx 2 at your point?

26. Slope Field Activity Given 1.Find the slope for your point 2.Sketch a tangent segment across your point. Now do the same for the rest of the points 3.Are you on an equilibrium solution? 4.Find your isocline. Is it vertical, horizontal, slant, etc. 5.Sketch a possible solution curve through your point 6.Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? 7.What is the value of d 2 y/dx 2 at your point?

27. Slope Field Activity Given 1.Find the slope for your point 2.Sketch a tangent segment across your point. Now do the same for the rest of the points 3.Are you on an equilibrium solution? 4.Find your isocline. Is it vertical, horizontal, slant, etc. 5.Sketch a possible solution curve through your point 6.Is your point an extremum or point of inflection? Is the graph of y increasing/decreasing, CU or CD? 7.What is the value of d 2 y/dx 2 at your point?

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