6.1 day 1: Antiderivatives
and Slope Fields
Greg Kelly, Hanford High School, Richland, Washington
First, a little review:
Consider:
then:
y  x2  3
y  x2  5
or
y  2 x
y  2 x
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: y   2 x
y  x2  C
find
y
We don’t know what the
constant is, so we put “C” in
the answer to remind us that
there might have been a
constant.

If we have some more information we can find C.
Given: y   2 x and y  4 when x  1 , find the equation for y .
y  x2  C
4 1 C
2
3C
This is called an initial value
problem. We need the initial
values to find the constant.
y  x2  3
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.

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.

y  2 x
x y y
Draw a segment
with slope of 2.
Draw a segment
with slope of 0.
Draw a segment
with slope of 4.
0
0
0
0
0
1
2
3
0
0
0
0
1 0
1 1
2
2
2 0
4
-1 0
-2
-2 0 -4

y  2 x
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.

For more challenging differential equations, we will use the
calculator to draw the slope field.
dy
2 xy

2
dx
1

x
On the TI-89:
MODE and change the Graph type to DIFF EQUATIONS.
Push MODE
Go to:
Y=
Press
I
Go to:
and make sure FIELDS is set to SLPFLD.
Y= and enter the equation as:
y1  2  t  y1/ 1  t ^ 2 (Notice that we have to replace
x with t , and y with y1.)
(Leave yi1 blank.)

y1  2  t  y1/ 1  t ^ 2
Set the viewing
window:
WINDOW
Then draw the graph:
GRAPH

Be sure to change the Graph type back to FUNCTION
when you are done graphing slope fields.

Integrals such as

4
1
x2 dx are called definite integrals
because we can find a definite value for the answer.

4
1
2
x dx
4
1 3
x C
3
1
1 3
 1 3

  4  C    1  C 
3
 3

The constant always cancels
when finding a definite
integral, so we leave it out!
64
1
63
 21
C  C 
3
3
3
Integrals such as
 x dx
2
are called indefinite integrals
because we can not find a definite value for the answer.
2
x
 dx
1 3
x C
3
When finding indefinite
integrals, we always
include the “plus C”.

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).
x
y   t sin t dt
0
or
y1  NINT  x sin x, x,0, x 
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.

To find the indefinite integral on the TI-89, use:
  x  sin x, x 
The calculator will return:
sin  x   x  cos  x 
Notice that it leaves out the “+C”.
Use
in the
COPY and
PASTE to put this expression
Y= screen, and then plot the graph.

[-10,10] by [-10,10]
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