1. 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.

2. 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.

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

4. 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”.

5. Many of the integral formulas are listed on page 307. The first ones that we will be using are just the derivative formulas in reverse. On page 308, the book shows a technique to graph the integral of a function using the numerical integration function of the calculator (fnInt). or This is extremely slow and usually not worth the trouble. Formulas like 2, 3, and 4 might not make sense at first but you will still need them for the homework assignment. How they work will be explained in section 6.2.

6. Many of the integral formulas are listed on page 307. The first ones that we will be using are just the derivative formulas in reverse. With these formulas, you can now handle the problems in Assignment 6.1… Formulas like 2, 3, and 4 might not make sense at first but you will still need them for the homework assignment. How they work will be explained in section 6.2.

7. With these formulas, you can now handle the problems in Assignment 6.1…

8. Some differential equations can be expressed as the product of a function of x and a function of y. Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume for now that y 2 is never zero.) for reasons you’ll see in a minute. How would we solve for y if they are are part of the same derivative?

9. Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume for now that y 2 is never zero.) Separation of Variables comes down to this: Get the y ’s with the dy ’s and the x ’s with the dx ’s.

10. Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Combined constants of integration

11. Separable Differential Equations This is called a general solution of the differential equation. If you were given an initial value and asked you to solve for y, then you’re being asked to find a particular solution. For example, if (0, 1) were an initial value, find the particular solution. particular solution Answer:

12. Separable Differential Equations This is called a general solution of the differential equation. If you were given an initial value and asked you to solve for y, then you’re being asked to find a particular solution. The domain of this solution would be… But why? …only an open interval that contains the initial point (0,1) This is where a slope field graph can help you understand this.

13. Below is the graph of Since we were given the initial value (0, 1), the only solution we know for sure exists is on an interval that contains that point. You can think of this in the same way you learned what made a function continuous: When you draw the solution on a slope field, you should never have to pick your pencil up off the graph. So the graph should look like…

14. −1 < x < 1 …this …because this is the interval that contains the point (0, 1) And since this function is undefined at x = ±1

15. Separable Differential Equations This is called a general solution of the differential equation. If you were given an initial value and asked you to solve for y, then you’re being asked to find a particular solution. So the domain of this solution would be… −1 < x < 1

16. 000 010 12 14 1 2 20 0 214 228 0 0 1-2 Sketch an approximate curve for y given the initial value (0,1). Now sketch an approximate curve for y given the initial value (–1, –1). Remember this differential equation?

17. Find a general solution for y. Remember this differential equation?

18. We can do better than ± e c Since e is a constantand c is a constant…

19. Find a general solution for y. Remember this differential equation? Since

20. Solve for y given the initial value (0,1). Remember this differential equation?

21. We can do better than ± e c Since e is a constantand c is a constant… Now let’s compare methods: A to ±e c Solve for y given the initial value (0,1). We’ve seen how it was done with A Which is the answer we got before

22. Solve for y given the initial value (0,1). Now solve for y given the initial value (–1, –1). Remember this differential equation? On your calculator, graph the slope field and both solution curves that you found.  What would be an approximate sketch of these two graphs?