Timeline for Why do we teach calculus students the derivative as a limit?
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10 events
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| Apr 3, 2011 at 20:14 | comment | added | Toby Bartels | How about ‘Don't use substitution in formulas that don't really make sense?’. Of course, you're right that people need to be taught what things mean so that they know what makes sense, but why do we use bastard notation like ‘$(x^2)' = 2x$’ in the first place? Substitute into $\mathrm{d}x^2 = 2x\, \mathrm{d}x$, and the result is true. | |
| Jan 23, 2011 at 20:31 | comment | added | KConrad | When my father was a judge at a high school math fair, a student gave a presentation on calculus. During the question period after the presentation, he asked the student "If f(x) = 3^2, what is the derivative of f(x)?" The student said "6". My father then asked "If f(x) = 9, what is the derivative of f(x)?" The student said "0". He asked the first question again and the student still said "6". Of course this student did not go on to the next round of the math fair. | |
| Jan 2, 2011 at 18:35 | comment | added | Anixx | I think this type of error is avoidable if you introduce $x$ not as a variable, but as a special symbol for identity function. Even more, you can use bold $\mathbf{x}$ for the identity function and normal $x$ for a value. | |
| Nov 7, 2010 at 8:41 | comment | added | Douglas Zare | This didn't stand out among the students' complaints. Other instructors flagged differentiating $\pi^2$ as a trick question, and I see their point. I still like the question, but I moved it from the quiz to classroom discussion, and revisited it as an example of the chain rule. | |
| Nov 7, 2010 at 0:09 | comment | added | Thierry Zell | @Douglas: I didn't think the derivative of $\pi^2$ was a mean thing to ask! Who thinks so? students? colleagues? Never mind, I'll be sure to borrow it for next time. (Even meaner would be to ask for the derivative of $e^2$, btw.) As for the blind symbol manipulation, I cannot believe that it took me all these years before seeing for the first time (in an exam) that the derivative of $\arctan x$ was $\mathrm{arcsec}^2 x$. In retrospect, I should have been expecting this for a long time! | |
| Nov 7, 2010 at 0:01 | comment | added | Thierry Zell | @Terry: I agree that the black-box use is an issue, but some might argue that it's possible to fix without necessarily going all the way to a limit definition, because the root problem is conceptual understanding of functions, and more precisely of the derivative as a function. A simple aphorism would suffice: "chug then plug, don't plug and chug!" More seriously, one could discuss the difference between the derivative as a function and the derived number at a point (slope) purely graphically, with no references to limits. | |
| Nov 6, 2010 at 16:43 | comment | added | Douglas Zare | People say I'm mean for asking for the derivative of $\pi^2$, but I think it's a memorable example for the students. Another place the blind symbol manipulation goes wrong is on $\sin^{-1} x = \arcsin x$. Many students are willing to assume that if $f(x)=g(x)$ then $f'(x)=g'(x)$, which is only a consequence for some meanings of the first equation. | |
| Nov 2, 2010 at 16:37 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Nov 2, 2010 at 16:30 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Nov 2, 2010 at 15:37 | history | answered | Terry Tao | CC BY-SA 2.5 |