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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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    To save some work, notice that the function $2^{x^2}$ is even (ie. symmetric wrt the y-axis), so you only need to plot the graph for positive $x$ and the rest you get by mirroring that.2012-05-28

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