# Just how much can you abuse nSolve on an exam?

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Imagine an HL Paper 2 problem like this (real example):

Given a pdf f(x) = k ln(x) for 1 ≤ x ≤ 3, find the median of X.

Finding k is easy enough, and then you get a problem of the form

Integral[k ln(x) dx, 1 ≤ x ≤ m] = 1/2

I imagine that the expected solution is using partial integration to obtain Integral[k ln(x) dx] = k x (ln(x) - 1), then using that to obtain an equation which can be solved numerically. In this case, the problem tests both your knowledge of the definition of a median and your ability to partially integrate ln(x).

However, on an nSpire, you can define a function g(t) := Integral[k ln(x) dx, 1 ≤ x ≤ t] and then just do nSolve(g(t) = 1/2, t, 2) (the initial value 2 is required to keep it from converging on a solution below 1) to get exactly the same answer. Note that you really do need to define a function first, because otherwise the calc complains about multiple non-valued variables in a single expression. I believe that this might be impossible to do on calculators like the TI 84. In this case, the problem is a one-liner and only requires knowing the definition of a median, as well as having a recent calculator and being somewhat good at using it.

The latter solution feels somewhat like cheating, but it's what came to me naturally during a test today, and I don't see any actual rule that would prohibit me from using it. Do you think both solutions would be accepted as equally valid on an exam?

Edited by Aleksejs Popovs

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I used the TI-84 Plus last year and it has a similar Solve function, but I mostly just graphed y =fnInt(k * ln(x), x, 1, x) (different syntax on newer versions of the system) and read off value at y = 1/2 or intersection with y = 1/2. Most graphing calculators do not find antiderivatives instead they use some numerical analysis to approximate the answer, some are explained here.

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