1. Slope Fields

2. What are slope fields? Graphical representation of the slope of a function at different points in the plane. Since we are given the differential equation, the graph that’s sketched through the points is called the solution curve (an approximation of the real solution)

3. Differential Equation A differential equation tells us that the slope of the curve at any point x is the x-value at that point. In slope field problems, they always give us the differential equation or it’s graph. 1. Let’s say: f’(x)=2x/y So at point (6,4), f’(x) or the slope = 2(6)/4 = 3. 2. Let’s say: f’(x)=3xy So at point (-2, -1), f’(x) or the slope = 3(-2)(-1) =6

4. We have differential equation… now what? 1.Select a coordinate (x,y) and plug it into the diff. equation. 2.Evaluate at that point (x,y) and you will get the slope at (x,y). 3.Draw a short line segment at point (x,y) on the coordinate plane with that slope. 4.Do this until you have enough points to draw a solution curve through all the points. 5.If given an initial point, then draw graph with the initial point on the graph.

5. Some helpful tips before we start Make a chart of points (x,y) and their respective slopes to make it easier for you to graph. Draw line segments in all 4 quadrants. Draw enough line segments so you can predict the slopes of other points without evaluating.

6. Let’s draw a slope field Let’s draw a slope field 1.Select a coordinate (x,y) and plug it into the diff. equation. 2.Evaluate at that point (x,y) and you will get the slope at (x,y). 3.Draw a short line segment at point (x,y) on the coordinate plane with that slope. 4.Do this until you have enough points to draw a solution curve through all the points. Ex. 1: f’(x)=2x. Sketch a slope field for the given differential equation at the 9 points indicated. Initial point f(2)=0 http://www.math.lsa.umich.edu/courses/116/slopefields.html

7. Let’s draw another slope field ;) Ex. 2: f’(x)=1/x. Sketch a slope field for the given differential equation at 9 the points indicated. Initial point: f(1)=1 1.Select a coordinate (x,y) and plug it into the diff. equation. 2.Evaluate at that point (x,y) and you will get the slope at (x,y). 3.Draw a short line segment at point (x,y) on the coordinate plane with that slope. 4.Do this until you have enough points to draw a solution curve through all the points. http://www.ies.co.jp/math/java/calc/DiffEqu/DiffEqu.html

8. Quiz time! Multiple Choice: Choose the slope field of the following graph with particular solution (0,0) to the correct differential equation. a)y’=y+x b)y’=y-x c)y’=x2 d)y’=7-x Graph of differential equation:

9. And the answer is… A) is the correct answer! Congrats on those who figured it out.

10. Quiz time a)f’(x)=y b)f’(x)=y+x c)f’(x)=sin(x+y) d)f’(x)=-x/y

11. And the answer is… 1. b) 2. a) 3. d) 4. c) Yeah!

12. More Quiz! FREE RESPOSE QUESTION: a)Sketch the slope field for the differential equation y’=6x2 b)Explain why a solution could not have the graph shown to the right. c)Draw the exact solution curve through the points with initial value (0,1) using antidifferentiation.

13. And the answer is… A) I will show on the board. B) I will show on the board. C) I will show on the board.

14. Slope field websites http://www.sosmath.com/diffeq/slop e/slope1.html http://www.sosmath.com/diffeq/slop e/slope1.html http://archives.math.utk.edu/visual. calculus/4/slope_fields.3/ http://archives.math.utk.edu/visual. calculus/4/slope_fields.3/ http://www.math.montana.edu/frank w/ccp/calculus/des/slope- fields/learn.htm http://www.math.montana.edu/frank w/ccp/calculus/des/slope- fields/learn.htm http://www.calculusapplets.com/slop efields.html http://www.calculusapplets.com/slop efields.html

15. A cool Thing !