2. SLOPE FIELDS By Ali and Ethan

3. OBJECTIVES  To learn how to construct slope fields from differential equations.  To learn how to find the equation of a slope field.  To be able to graph solution curves and understand their relationships with the slope field and equations.

4. REQUIRED BACKGROUND KNOWLEDGE  Know how to graph using an equation.  Know how to identify slopes by looking at a graph  Know how to read and understand graphs

5. We’ll start by reviewing how to find the slope using a derivative of a differential equation.  Take the equation  Find the slope at the point (0,1) by plugging in the respective values in the equation.  That gives us  at point (0,1)

6. Which of the following tables looks correct the equation ? (x,y) (-1,1)1 (-1,0) (0,0)0 (1,0)1 (1,1)3 (x,y) (-1,1)0 (-1,0)-2 (0,0)0 (1,0)2 (1,1)4 (x,y) (-1,1)0 (-1,0) (0,0)0 (1,0)1 (1,1)2 (x,y) (-1,1) (-1,0)-2 (0,0)0 (1,0)2 (1,1)3

7. CORRECT If you plug the x and y values into the derivative, you should get the number in the left column.

8. INCORRECT Try plugging in the x and y values into the derivative.

9. CREATING THE SLOPE FIELD Now we will learn how to create a slope field by using the equation. The points on a slope field are all plotted with the slope of the point that they are at the same way all the values were found for the table

10. For example, consider the slope field below. It is a slope field of the differential equation. The lines shown on the graph represent the slope of the original equation that is a derivative of at that point.

11. SOME PRACTICE Find the slope fields of the following equations.

12. Identify the correct slope field for A. B.

13. CORRECT! ( ͡ ° ʖ ͡ °)

14. INCORRECT Try taking another look at the points where the slopes are different between the graphs and seeing which of those points correspond to the equation.

15. What is the equation of the given slope field? A.B.C.D.

16. CORRECT! If you look at points that are unique to certain equations, it is easier to find the slope field that matches.

17. INCORRECT Try taking points that are unique to certain equations and testing to see if they appear in the slope field.

18. SOLUTION CURVES Draw the solution curve for that passes through the point (0,2).

19. WALKTHROUGH Draw the solution curve for that passes through the point (0,2). Find the point Follow the slope lines next to it to make a solution curve.

20. SOLUTION CURVE PRACTICE Find the solution curve that passes through (0,1) for the differential equation A. B.

21. CORRECT!

22. INCORRECT Take another look at the slopes leading up to the specified point and make sure that they travel through it.

23. COMPREHENSION QUIZ You will answer questions to test your understanding of each section. If you want to review a section, click on it. PLUGGING IN POINTS IDENTIFYING SLOPE FIELDS SOLUTION CURVES

24. QUESTION 1 For, find the slope at the point (4,1). A. 11B. 14C. 4 D. 1

25. INCORRECT Plug in the x and y values from the point into the equation to get the answer.

26. CORRECT!

27. QUESTION 2 Find the correct slope field for A.B.

28. INCORRECT Try plugging in values from points with a distinguishable slope to determine which slope field matches.

29. CORRECT! An example of where the slope fields have different slopes is at (0,1). A. B.

30. QUESTION 3 Identify the correct solution curve for that passes through the point (-1,0) A. B.

31. CORRECT!

32. INCORRECT The solution curve should represent the different slopes throughout the derivative, not just at one point.

33. CONGRATULATIONS! Now that you know how to use slope fields, go and get ‘em tiger!

34. RETURN TO A SECTION PLUGGING IN POINTS IDENTIFYING SLOPE FIELDS SOLUTION CURVES