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I am teaching precalculus/basic calculus to my class (high schoolers of around 18 years of age), and I'm always searching for nice functions to plot and study (finding the domain, the function's sign, the zeros, derivatives, limits...), that are not too easy and not too hard on the kids with the computation of the derivatives. What are in your opinion and/or experience some good (possibily general) examples of functions that give students some head scratching but not really a headache? In my limited experience, coming up with nicely plottable and nicely differentiable functions is not really straightforward.

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    $\begingroup$ I have the feeling that you're not aware that "elementary" has a particular meaning in mathematics: "In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or x^1/n" en.wikipedia.org/wiki/Elementary_function Also, "derivable" and "differentiable" mean very different things. $\endgroup$ Apr 27 at 21:20
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    $\begingroup$ Piecewise functions leave a great variety and possible graphs. Bonus they are a foundation concept essential for Calculus. $\endgroup$
    – nickalh
    Apr 28 at 8:22
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    $\begingroup$ @Acccumulation I understand your comment, but I assumed that the context of my question made clear the level of the students: at best high schoolers, so obviously they will not delve into technicalities. I'll clarify the question. $\endgroup$ Apr 28 at 8:46
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    $\begingroup$ @Acccumulation Related question "derivable" doesn't exist in English? $\endgroup$
    – Stef
    Apr 28 at 9:51
  • $\begingroup$ @Stef I probably absent-mindedly wrote "derivable" in lieu of "differentiable". I'll edit the question. I'm sorry but English is not my mother tongue! $\endgroup$ Apr 28 at 9:58

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I'm personally a fan of simple examples:

  • $x e^x$ (has nice critical point, point of inflection)
  • $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics)
  • $\frac{x}{x^2+1}$ (a serpentine curve)
  • $3x^5-5x^3$ and other differences of two monomials.
  • $x^n + y^n = 1$ (you can solve for y as a function of x. Students should have seen that tangents to circles are perpendicular to radii, you can talk about squircles in the case that n = 4, and n = 3 is also interesting. A patient student who knows about conjugates can work the derivative of $\sqrt{1-x^2}$ from the limit definition.)
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  • $\begingroup$ $e^{-x^2}$ also has the nice property of having a rescaled version of $\mathrm{erf}(x)$ as an antiderivative, which gives you an excuse to talk about that once you get to integral calculus. But the error function might be too advanced for the precalculus level that OP is asking about. $\endgroup$
    – Kevin
    Apr 27 at 21:52
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Don't forget purely graphical problems. Give a graph of $f$ or $f'$, ask for the student to sketch the other one.

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Suggestions. I took your requirement "not elementary" in the technical sense.

Bessel functions, say $J_0(x)$, and other special functions of mathematical physics. But since your students have not studied differential equations, maybe not a good choice.

Gamma function $\Gamma(x)$. How to do plotting without knowing more advanced mathematics?

Lambert W function. Here the inverse function $xe^x$ is elementary, so can be used for plotting.

Sine integral function $\operatorname{Si}(x)$. It is not elementary, but its derivative $\frac{\sin x}{x}$ is, so that can be used for plotting.

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    $\begingroup$ Not a bad answer to the question if the question's subject line is construed the way mathematicians construe it. But taken as a whole, I think the intent of the question was something else. $\endgroup$ Apr 28 at 3:14

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