Calculus 6.1 day 2

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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
Greg Kelly, Hanford High School, Richland,
Washington
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It was Euler who originated the following
notations
Leonhard Euler 1707 - 1783
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Recall a differential equation problem
Find the particular solution
that satisfies that initial condition (1, 0)
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There are many differential equations that can
not be solved. But we can still find an
approximate solution at particular points.
Euler came up with a method that we can use to do
this.
Essentially what we will do is 1) Start with an
initial value, 2) Use the slope to find a
tangent line at that point, 3) Then use the
tangent line to approximate another value on the
curve. We can repeat this process as many times
as necessary.
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Ex. 1 Use Eulers Method to find the first three
approximations of the differential equation
passing through (0, 1).
Use a step of h 0.1
Tangent line at (0, 1)
Since step 0.1,
Use tangent line to get
Tangent line at (0.1, 0.9)
Since step 0.1,
Use tangent line to get
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Tangent line at (0.2, 0.82)
Since step 0.1,
Use tangent line to get
Third approximation (0.3, 0.758)
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Ex. 2
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Exact Solution
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Ex. 3
The table gives selected values of the derivative
of a function g at certain x-values. If
and Eulers method with a step size of
1.5 is used to approximate then what is
the resulting approximation?
x g(x)
-1.0 2
-0.5 4
0.0 3
0.5 1
1.0 0
1.5 -3
2.0 -6
Initial value (-1, -2)
Slope there 2
At (0.5, 1) Slope there 1
It took us 2 iterations to get there!