•
  
 y


•



•
•




•
 
We’ve plotted points on graphs before…
Plot the following points on the empty grid at the top of your worksheet.
 x
 x
2
–1
–2
0
1 y
 x
2
4
1
0
1
4


•
  
 y


•



•
•




•
 
Now, instead of plotting points on a graph, try plotting slopes. Plot the slopes of y on top of the points you just plotted.
How would you do that?
 x
 x





0
1
2
–1
–2 y
 x
2
4
1
0
1
4 y
 
2 x
0
2
4
–2
–4


Suppose all that you knew was y

. You don’t know what y is but you do know it’s derivative. Use what we just did to plot the slope of y in the following problem that is on your worksheets…


   







 y y
 
2 x

Draw a segment with slope of 2.
  
Draw a segment with slope of 0.
Draw a segment with slope of 4.
x x y y

0
0
0 0 0
0 1 0
2
3
0
0

1 0 2
1 1 2
2 0 4
-1 0 -2
-2 0 -4
Plug 0 into y
 for x
Plug 1 into y
 for x


   






 y

  y
 
2 x
If you know an initial condition, such as (1,

2), you can sketch the curve.
By following the slopes, you
 
In this case, it is a parabola.
How could we have seen this coming?
A parabola is exactly what you would get when you integrate y
 
2 x






What would the
    graph of these
  

 like?

 y '

2 xy x y y

1
1
0 0 0
0 1 0
1
2
2
4
2 0 0
2 1 4
2 2 8
-1 0 0
-1 1 -2
Plug 0 in for x and y
Plug 1 y and
0 for x
1 for x
…and so on…


    
• 







 •
 
Sketch an approximate curve for y given the initial value (0,1).

Now sketch an
 for y given the initial value (–1,–1).
y '

2 xy x y y

1
1
0 0 0
0 1 0
1
2
2
4
2 0 0
2 1 4
2 2 8
-1 0 0
-1 1 -2


A
Problems that begin with the derivative are called differential equations . When you are given an initial point, they are called initial value problems . Plotting these slopes gives you what is called a slope field .
Match the slope field with the function that you think it is modeling. y
 ln x ____
(#4 on worksheet) y
 e x
____
 
B
 
 
 
            











Problems that begin with the derivative are called differential equations . When you are given an initial point, they are called initial value problems . Plotting these slopes gives you what is called a slope field .
A
Match the slope field with the function that you think it is modeling.
(#4 on worksheet) y y y
 y
  

 ln e
1 x e x x x
 
B
 
 
 
            











Problems that begin with the derivative are called differential equations . When you are given an initial point, they are called initial value problems . Plotting these slopes gives you what is called a slope field .
A
Match the slope field with the function that you think it is modeling.
(#4 on worksheet) y y y
 y
  

 ln e
1 x e x x x B
A
 
B
 
 
 
            











Notice that the curves for each function follow the slopes like a boat following a river current.
A
  








    y
 ln x y
 e x
B
  








   


Initial value problems, differential equations, and slope fields are often used to solve problems where only the rate of change (the derivative) is known. In advanced math, physics, and engineering classes, slope fields are also called direction fields or vector fields. These problems are so common and have so many applications that there are entire sequences of college courses dedicated only to different types of differential equations. Much more to come on this topic in the days to come… p