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As we learnt at school, to draw the graph of a function in details it may be necessary to perform some steps of calculation such as finding domain and codomain, limits, asymptotes and sign of derivatives. These steps may require minutes of calculus if the function is "complicated".

A more "brutal" method to draw the graph is to compute some values of the function and connect them with lines: the more points are computed the better the graph will be. The problem of this approach is that it requires a lot of time to manually compute dozens or even hundreds of values.

Since a mental calculator has the ability to compute operations almost instantly, does this make him capable of drawing the graph of every function in few seconds?

Are there any references about this?

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    $\begingroup$ I think you mean "any" instead of "every" $\endgroup$
    – mathworker21
    Sep 10, 2019 at 12:06
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    $\begingroup$ What do you mean by a "mental calculator", exactly? Like, calculating in your mind? In that case, I doubt anyone can just plot any function in their mind. Take, for example, a Bessel function. Would you be able to conjure the value of a Bessel function in seconds? $\endgroup$
    – Matti P.
    Sep 10, 2019 at 12:09
  • $\begingroup$ @MattiP. Yes you are right, usually they can compute "only" the four operations and roots, Gauss also logarithms. $\endgroup$
    – sound wave
    Sep 10, 2019 at 15:27

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Most mental calculators do the four standard arithmetic operations, possibly square roots. I haven't seen one do trigonometric evaluations (besides the ones everyone knows), or exponential expressions with real exponents.

And there are even more tricky calculations, like the $\Gamma$ function, the Riemann-$\zeta$ function, and all their cousins. And those aren't even the craziest commonly used functions one might need to graph.

So I'm going to go out on a limb and say "no". It's much better to have a mental image of what the most common functions look like, and how they interact visuallywhen you add, multiply or chain them. This allows you to draw rough sketches of what the graph ought to look like, which is often almost as much of a help to start reasonging about them as an exact graph is.

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  • $\begingroup$ Oh right, so they can easily plot only those functions involving the operations they can mentally compute. I read that Shakuntala Devi, Willem Klein and Igor Shelushkov could find roots of high natural degrees, and that Gauss could compute logarithms. But I don't know about trigonometric functions. $\endgroup$
    – sound wave
    Sep 10, 2019 at 15:25

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