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6.1 day 1:
Antiderivatives
and Slope Fields
Greg Kelly, Hanford High School, Richland, Washington
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View Full Document First, a little review:
Consider:
2
3
y
x
=
+
then:
2
y
x
′
=
2
y
x
′
=
2
5
y
x
=

or
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:
2
y
x
′
=
find
y
2
y
x
C
=
+
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:
and
when
, find the equation for
.
2
y
x
′
=
y
4
y
=
1
x
=
2
y
x
C
=
+
2
4
1
C
=
+
3
C
=
2
3
y
x
=
+
This is called an initial value
problem
.
We need the initial
values to find the constant.
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.
→
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View Full Document Initial value problems and differential equations can be
illustrated with a slope field.
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This note was uploaded on 02/12/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith
 Antiderivatives, Derivative, Slope

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