Calculus 6.1 day 1

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Slope Fields and Eulers Method
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• When taking an antiderivative that is not dealing
  with a definite integral, be sure to add the
  constant at the end.

We dont know what the constant is, so we put C
in the answer to remind us that there might have
been a constant.
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If we have some more information we can find C.
Given y 2x and f(1) 4, find the equation
for y.
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Initial value problems and differential equations
can be illustrated with a slope field.
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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
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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.
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Integrals such as are called
indefinite integrals because we can not find a
definite value for the answer.
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Eulers Method
Leonhard Euler made a huge number of
contributions to mathematics, almost half after
he was totally blind.
(When this portrait was made he had already lost
most of the sight in his right eye.)
Leonhard Euler 1707 - 1783
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It was Euler who originated the following
notations
Leonhard Euler 1707 - 1783
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There are many differential equations that can
not be solved. We can still find an approximate
solution.
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Steps for using Eulers Method

• Begin a the point specified by the initial
  condition.

• Use the d.e. to find the slope dy/dx at that
  point.

• Increase x by a small amount ?x. Increase y by a
  small amount ?y where ?y (dydx)?x. This
  defines a new point (x ?x, y ?y) that lies
  along the linearization.

• Use this new point, return to step 2. Repeating
  the process constructs the graph to the right of
  the initial point.

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Exact Solution
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Examples 9 and 10 also show Eulers Method I also
have a program that will do an Euler Table and
graph.