Professor: I don’t quite understand problem 18 in section 1.3. I have estimated a value for a0 based on the slope field. The question asks us to use Euler’s method to estimate a0, but I am not sure what I am looking for in the results of the Sage calculation.

Thank you for any help you can provide.

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The idea is similar to the one we talked about today in class. However, instead of visual inspection, you use Euler’s method. If a function is divergent and you repeat Euler’s method enough times, the y-values will become extremely large. On the contrary, if the function is convergent, the y-values will approach a limit.

Bottom line: Repeat Euler’s method many times and observe the behaviour of the y-values. This will give you the distinction between the convergent and divergent solutions.

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I keep trying to plot number 14 and end up getting an error message every time:

var(‘x y’)

implicit_plot(-1*y^(-2) == 2*(sprt(x^2 + 1)) – 3),(x,-5,5),(y,-5,5))

(x, y)

Error in lines 0-2

Traceback (most recent call last):

File “/projects/c121aa61-ffaa-4fae-aa67-074955507648/.sagemathcloud/sage_server.py”, line 873, in execute

exec compile(block+’\n’, ”, ‘single’) in namespace, locals

File “”, line 1

implicit_plot(-Integer(1)*y**(-Integer(2)) == Integer(2)*(sprt(x**Integer(2) + Integer(1))) – Integer(3)),(x,-Integer(5),Integer(5)),(y,-Integer(5),Integer(5)))

^

SyntaxError: invalid syntax

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I see two problems:

1) The square root of x is sqrt(x), not sprt(x).

2) There is an extra parenthesis after x^2+1.

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