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I need to plot $2^{x^{2}}$ without using calculus. I would like to know how to explain why that function is smooth at $x=0.$

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Do you want to plot on paper or a graphing calculator? The former can be done by computing several points (in this case, around $x=0$). –  Argon May 28 '12 at 18:53
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Maybe you can argue that since both $2^x$ and $x^2$ are smooth at $x=0$ then so is the composition (that's the most "without using calculus" I get since smooth is a term in calculus ;)) –  Belgi May 28 '12 at 18:53
    
@Argon: On a paper. But I failed to notice how it is "smooth". –  spohreis May 28 '12 at 19:28
    
you know that $x^2$ and $2^x$ are smooth and behave 'reasonably'. Why would their composition not? If you want a proof, not just intuition, then you'll need calculus, since that's where the definition of smooth comes from. –  Robert Mastragostino May 28 '12 at 19:29
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Why anyone would want to plot this without the tools of calculus is beyond me.But I guess it's a standard pre-calculus grind-it-out problem.One question I WOULD have for such a problem before answering is what does "smooth" mean in this particular problem-that derivatives of all orders exist and are continuous? I wouldn't think so given how the question is asked. –  Mathemagician1234 May 28 '12 at 19:54

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