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Werner_E
Werner_EAnswer
25-Diamond I
Aug 18, 2024 4:09 PM UTC

The equation u=cos(a x) has no effect, neither have the ranges a and u you define.

u=1 has to be avoided because of the term ln(sqrt((1-u^2)/u) (-> minus infinity).

 

You could simply plot h, but you will have to manually edit the ranges for the input variables so that the calculation does not fail for any grid point.

Werner_E_0-1723994194617.png

Its more convenient to use the "CreateMesh" function and plot the created nested arrays

Werner_E_5-1723997798191.png

 

You could also create an array with values corresponding to the range variables you had defined and look for it minimum value

Werner_E_2-1723997208930.png

 

or you could use "minimize" to find the minimum

Werner_E_4-1723997290952.png

The values found do not exactly match the solution you name.

 

Mathcad file attached

 

 

 

 

 

 

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3 replies

Werner_E
Werner_E25-Diamond IAnswer
25-Diamond I
Aug 18, 2024 4:09 PM UTC

The equation u=cos(a x) has no effect, neither have the ranges a and u you define.

u=1 has to be avoided because of the term ln(sqrt((1-u^2)/u) (-> minus infinity).

 

You could simply plot h, but you will have to manually edit the ranges for the input variables so that the calculation does not fail for any grid point.

Werner_E_0-1723994194617.png

Its more convenient to use the "CreateMesh" function and plot the created nested arrays

Werner_E_5-1723997798191.png

 

You could also create an array with values corresponding to the range variables you had defined and look for it minimum value

Werner_E_2-1723997208930.png

 

or you could use "minimize" to find the minimum

Werner_E_4-1723997290952.png

The values found do not exactly match the solution you name.

 

Mathcad file attached

 

 

 

 

 

 

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    AlfredFlaßhaar15-MoonstoneAuthor
    15-Moonstone
    Aug 18, 2024 6:06 PM UTC

    Of course, I solved the problem in a different way beforehand, using paper and pen. It is unusually demanding for high school graduates and appealing even at my advanced age. I want to arouse the interest of young people who are willing to learn by graphically representing the minimum point of u and a in 3D graphics, in addition to the written calculation. Unfortunately, I am not very good at using Mathcad14. My area of ​​expertise is more analysis. In any case, I have learned something here. Thank you very much for that.

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    Werner_E
    Werner_E25-Diamond I
    25-Diamond I
    Aug 18, 2024 9:07 PM UTC

    I agree that its a very demanding problem for high school graduates (and not only for them).

    I wonder how many pupils of that age in our area would be able to provide a solution.

     

    BTW, using CreateMesh its not to difficult to "zoom in" for better clarity.

    Werner_E_0-1724015260894.png

     

    You may also define the function as its given originally.

    Werner_E_0-1724033765489.png

    I don't think that Mathcad is capable enough to provide an exact symbolical solution, but we can get a numerical one either by using "minimize"

    Werner_E_1-1724033797174.png

    or by using the Nabla operator and a solve block with "Find"

    Werner_E_2-1724033851628.png

    The auxiliary function "nabla" was necessary because using the built-in function directly did not work.

     

    And of course you could use the function for plotting as already shown

    Werner_E_3-1724033956460.png

    You may also demonstrate the periodicity, but you will have to copy with non-real z-values. Unfortunately unlike the 2D plot, the 3D plot does not accept (and ignore) NaN's, so I set non-real values to 23 and limit the z-axis in the plot menu to 22. Just to be on the safe side I also do the same for values which may make the function fail (zeros of sin or cos which make the log fail)

    Werner_E_1-1724041518143.png

     

     

     

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