Math 210, Spring 2013
Lab #4 Solutions
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Q#1
Specify the function as an expression, proceed...
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LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJi1JJGV4cEdGJDYjSSJ4R0YnIiIiRi5GLg==
Remember T3 has 4 terms
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OK, I hadn't realized this wouldn't show much. Let's look at it with y cropped a bit.
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Q#2
straightforward use of the sum, eval and limit commands
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QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRiU2JEkjU25HNiIvSSJuR0YrIiImIiIi
JCIrLic9alEiISIq
LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSZsaW1pdEc2IjYkSSNTbkdGKC9JIm5HRihJKWluZmluaXR5R0Yk
JCIrLSc9alEiISIq
Notice that something is slightly fishy above in the last digit... but this is what Maple returns
Q#3
Enter the RHS function and determine the Euler iterative map
QyQ+SSJmRzYiZio2I0kidUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIlISIiIiIiRjFGJUYlRiVGMQ==
Zio2I0kidUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIlISIiIiIiRi5GJUYlRiU=
QyY+SSJORzYiIiNdIiIiPkkia0dGJSNGJyIjREYn
IiNd
IyIiIiIjRA==
QyQ+SSRldW1HNiJmKjYjSSJ1R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSZldmFsZkclKnByb3RlY3RlZEc2IywmOSQiIiIqJkkia0dGJUYyLUkiZkdGJTYjRjFGMkYyRiVGJUYlRjI=
Zio2I0kidUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMsJjkkIiIiKiZJImtHRiVGLy1JImZHRiU2I0YuRi9GL0YlRiVGJQ==
QyQ+JkkidUc2IjYjIiIhIiIjIiIi
IiIj
Do the Euler steps - notice that the : after the end do suppresses output from inside the loop
QyQ/KEkiakc2IiIiIkYmSSJOR0YlSSV0cnVlRyUqcHJvdGVjdGVkRz4mSSJ1R0YlNiNGJC1JJGV1bUdGJTYjJkYsNiMsJkYkRiYhIiJGJkY0
LUkkc2VxRyUqcHJvdGVjdGVkRzYkJkkidUc2IjYjSSJpR0YoL0YqOyIiIkkiTkdGKA==
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
The plotting was trickier than I thought it would be
QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiM3Iy1JJHNlcUdGJjYkNyQmSSJ1R0YoNiNJImpHRigqJkYyIiIiSSJrR0YoRjQvRjI7IiIhSSJOR0YoRjQ=
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
Q#4
Enter f
QyQ+SSJmRzYiZio2I0kidUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYlRiVGJSIiIg==
Zio2I0kidUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYlRiVGJQ==
Write the commands in one line so it is easier to cut and paste for other N
Qy4+SSJORzYiIiM1IiIiPkkia0dGJS1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjKiRGJCEiIkYnPkkkZXVtR0YlZio2I0kidUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUYrNiMsJjkkRicqJkYpRictSSJmR0YlNiNGO0YnRidGJUYlRiVGJz5GNEYnRic/KEkiakdGJUYnRidGJEkldHJ1ZUdGLD5GNC1GMUYzRi9GNEYn
IiM1
JCIrKysrKzUhIzU=
Zio2I0kidUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMsJjkkIiIiKiZJImtHRiVGLy1JImZHRiU2I0YuRi9GL0YlRiVGJQ==
IiIi
JCIiIkkpaW5maW5pdHlHJSpwcm90ZWN0ZWRH
Already for N=10 we have exceeded the size of allowable floating point numbers.
We studied this problem analytically and found that the solution did not exist at time 1 (there was a singularity between t=0 and t=1) so we can't expect Euler's method to converge to a solution that does not exist.