Loci

## Calculus

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Of course Wolfram|Alpha computes limits, calculates extrema (and constrained extrema), and evaluates definite integrals.

But "1/(2 + cos x)" provides outputs that can promote interesting discussion in an introductory calculus class.

From the graphs or from the fact that both the numerator and denominator of our input are positive, the calculus class can recognize that our input describes a function which is always positive. Wolfram|Alpha provides a somewhat complicated expression for the indefinite integral, essentially the same expression we would get from Mathematica, Maple, Sage, or other Computer Algebra System that uses the Risch algorithm.

Once the class is convinced that this antiderivative is a periodic function, you can ask, "How can 1/(2+cos x), a positive and continuous function, be the derivative of a periodic function that repeats values and hence cannot be strictly increasing?"

This is an opportunity to caution students that computers can give answers that are incorrect or at least incomplete. Differentiating Wolfram|Alpha's antiderivative does indeed yield 1/(2+cos x), but the antiderivative is undefined at odd integer multiples of pi. (The singularities are removable.)

Thus a weakness of the technology can itself provide teachable moments!

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