Calculus 6.1

1
Antiderivatives and Slope Fields
Greg Kelly, Hanford High School, Richland,
Washington
2
First, a little review
It doesnt 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
We dont 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.
4
Initial value problems and differential equations
can be illustrated with a slope field.
5
0
0
0
0
1
0
0
0
2
0
0
3
2
1
0
1
1
2
2
0
4
-1
-2
0
0
-4
-2
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
Lets Try Another
x y dy/dx
0 0
0 1
0 2
0 -1
0 -2
1 0
1 1
1 2
1 -1
1 -2
-1 0
-1 1
-1 2
-1 -1
-1 -2
2 0
2 1
2 -1
2 -2
-2 0
-2 1
-2 2
-2 -1
-2 -2
8
This is what it looks like using WinPlot
9
To Use Winplot

• Open a 2-dim Window
• Under Equations, go to Differential?dy/dx
• Type in your differential equation
• You can change the number of rows as well as the
  color and length of the segments

10
AP Test Question

• This is the actual question from the 2006 test