Title: Calculus 6.1 day 1
16.1 day 1 Antiderivatives and Slope
Fields
Greg Kelly, Hanford High School, Richland,
Washington
2First, 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.
3If we have some more information we can find C.
4Initial value problems and differential equations
can be illustrated with a slope field.
50
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
6If 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.
7For 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.)
8Set the viewing window
Then draw the graph
9Be sure to change the Graph type back to FUNCTION
when you are done graphing slope fields.
10(No Transcript)
11Integrals such as are called
indefinite integrals because we can not find a
definite value for the answer.
12Many 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).
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.
13To find the indefinite integral on the TI-89, use
The calculator will return
Notice that it leaves out the C.
14p