1. 6.1: Antiderivatives 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. 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.

5. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

6. If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

7. Constructing a Slope Field Construct a slope field for 2x – y.

8. Constructing a Slope Field Construct a slope field for 2x – y.

9. Go to: and enter the equation as:Y= For more challenging differential equations, we will use the calculator to draw the slope field. (Notice that we have to replace x with t, and y with y1.) (Leave yi1 blank.) On the TI-89: Push MODE and change the Graph type to DIFF EQUATIONS. MODE Go to:Y= Press and make sure FIELDS is set to SLPFLD. I

10. Set the viewing window: Then draw the graph: WINDOW GRAPH

11. Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

12. 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!

13. 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”.

14. 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 (NINT). or This is extremely slow and usually not worth the trouble. A better way is to use the calculator to find the indefinite integral and plot the resulting expression.

15. To find the indefinite integral on the TI-89, use: The calculator will return: Notice that it leaves out the “+C”. Use and to put this expression in the screen, and then plot the graph. COPYPASTE Y=

16. [-10,10] by [-10,10] 

17. 6.1 day 2 Euler’s Method Leonhard Euler 1707 - 1783 Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind. (When this portrait was made he had already lost most of the sight in his right eye.)

18. Leonhard Euler 1707 - 1783 It was Euler who originated the following notations: (base of natural log) (function notation) (pi) (summation) (finite change)

19. There are many differential equations that can not be solved. We can still find an approximate solution. We will practice with an easy one that can be solved. Initial value:

22. Exact Solution:

23. It is more accurate if a smaller value is used for dx. This is called Euler’s Method. It gets less accurate as you move away from the initial value.

24. Euler’s Method Use Euler’s Method with increments of ∆x =.1 to approximate the value of y when x = 1.3 and y = 3 when x = 1. (x,y) ∆x (x + ∆x, y + ∆y)

25. Euler’s Method Use Euler’s Method with increments of ∆x =.1 to approximate the value of y when x = 1.3 and y = 3 when x = 1. (x,y)∆x (x+∆x,y+∆y) (1,3)2.1.2 (1.1,3.2) 2.2.1.22 (1.2,3.42) 2.42.1.242 (1.3,3.662)

26. The TI-89 has Euler’s Method built in. Example: We will do the slopefield first: 6: DIFF EQUATIONS Graph….. Y= We use: y1 for y t for x MODE

27. 6: DIFF EQUATIONS Graph….. WINDOW t0=0 tmax=150 tstep=.2 tplot=0 xmin=0 xmax=300 xscl=10 ymin=0 ymax=150 yscl=10 ncurves=0 diftol=.001 fldres=14 not critical GRAPH We use: y1 for y t for x Y= MODE

28. t0=0 tmax=150 tstep=.2 tplot=0 xmin=0 xmax=300 xscl=10 ymin=0 ymax=150 yscl=10 ncurves=0 diftol=.001 fldres=14 WINDOW GRAPH

29. While the calculator is still displaying the graph: yi1=10 tstep =.2 If tstep is larger the graph is faster. If tstep is smaller the graph is more accurate. I Press and change Solution Method to EULER. WINDOW GRAPH Y=

30. To plot another curve with a different initial value: Either move the curser or enter the initial conditions when prompted. F8 You can also investigate the curve by using. F3 Trace

31. t0=0 tmax=10 tstep=.5 tplot=0 xmin=0 xmax=10 xscl=1 ymin=0 ymax=5 yscl=1 ncurves=0 Estep=1 fldres=14 GRAPH Now let’s use the calculator to reproduce our first graph: We use: y1 for y t for x I Change Fields to FLDOFF. WINDOW Y=

32. Use to confirm that the points are the same as the ones we found by hand. F3 Trace Table Press TblSet Pressand set:

33. This gives us a table of the points that we found in our first example. Table Press

34. The book refers to an “Improved Euler’s Method”. We will not be using it, and you do not need to know it. The calculator also contains a similar but more complicated (and more accurate) formula called the Runge-Kutta method. You don’t need to know anything about it other than the fact that it is used more often in real life. This is the RK solution method on your calculator. 