Common student mistakes/misconceptions in a first year calculus course

What are the common mistakes and misconceptions students make in a first year calculus course?

More importantly:

What can I do to prevent/rectify them?

Context: Soon I will be doing some calculus lecturing. As this is the first time I've been entrusted with this responsibility, I've been thinking a lot about what I can do beyond regurgitating the material. I've had some experience doing tutorials (I imagine this would be equivalent to what a T.A. does in the U.S.) but lecturing is different as I will be introducing the material as opposed to reinforcing it. Obviously becoming a good (or even average) lecturer takes time and experience, and can not be obtained via a single answer to any question I could possibly ask here. Instead, I chose to ask the questions above.

I asked the first question because I don't think I can accurately answer it myself until I've taught the course at least once - I'd rather be able to address these issues the first time around. The second question is more general. There are many well-known mistakes students make when first learning mathematics, but they are well-known because they occur frequently and continue to do so over time. The fact that these mistakes/misconceptions continue to occur means that these particular issues haven't been resolved.

The topics that will be covered in the course are:

• Differential Equations (separable, linear second order constant coefficients)
• Applications of Calculus (volume of revolution)
• Limits (not including $\epsilon - \delta$ definition)
• Continuity
• Taylor Series

I know that this post may be too general/not suitable for this site. If this is the case, I apologise.

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Your question is really ambitious: it is almost impossible to prevent mistakes. There's something I can't understand: you are teaching limits without definitions? – Siminore Dec 20 '12 at 14:00
I can't remember off the top of my head despite just completing a first year calculus course, but here are a few. 1. The constant term +c does not magically appear at the end of an equation (so you can't just add it at the end), and any equation manipulations will also apply to this constant term. So if you have $\frac{y}{x} = e^x + c$, for example, then this simplifies to $y = x e^x + cx$, as opposed to $y = x e^x + c$. 2. $\int \! \frac{1}{x} \, \mathrm{d}x$ being $\ln|x|$, not $\ln{x}$. 3. If you get a hyperbola after solving a D.E. then the solution is only one branch – Sp3000 Dec 20 '12 at 14:23
Make sure you press that Taylor series only works locally and some applications for it like approximating $(27.1)^{1/3}$ from expanding around 27. I also think in doing continuity you could use a simplification of the closure definition to give more intuition so you're not stuck saying "if you take the pencil off the paper..." The definition I mean is $f$ is continuous if $f(\overline{A})\subset \overline{f(A)}$. – toypajme Dec 20 '12 at 15:19
I'm shocked! In Italy, we teach $\epsilon$ and $\delta$ even to students of humanities. – Siminore Dec 20 '12 at 16:00
We only learned $\epsilon-\delta$ in multivariate calculus, since the single-variable case conforms to immediate intuition so well. I assume the philosophy on that varies wildly. For Taylor series I would make sure to use $\frac{1}{1+x^2}$ as an example. Some students will assume that the radius of convergence is $\infty$ because the function "behaves nicely". Of course, when you look into the complex numbers their intuition is totally justified, but for the real case I find it's a good way to emphasize why properly solving the problem (rather than being heuristic and intuitive) is important. – Robert Mastragostino Dec 20 '12 at 17:10

I don't feel I could do justice to all the possibilities in a single post. However, I do know a very informative website that covers several such errors, and moreover covers non-technical problems that students encounter.

Here is the site: I hope you find it useful!

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Yes, as I skimmed that in 15 seconds, I noticed several common mistakes I have seen my students make... like $\sin (3x) = 3\sin x$ and $\sqrt{x+y} = \sqrt x + \sqrt y$ and missing parentheses that are actually needed. – Graphth Dec 20 '12 at 15:37
@Graphth I think the "instructor mistakes" and the "failing to seek help" mistakes are also very important, at least for those reflective enough to take them to heart :) – rschwieb Dec 20 '12 at 15:42
Sure, I tell my students over and over about my office hours and they should ask for help if they need it. I even emailed all of the students with a low C or worse and told them to work hard and ask for help. I had the students work on worksheets in class several times and I walked around to answer questions. Yet, most of the students who really needed help never asked. – Graphth Dec 20 '12 at 16:07
@MichaelAlbanese One final really good piece of advice I heard. Never ever "write off" student difficulties to things like laziness. Sure, sometimes that is the main contributor. However, if that habit becomes too ingrained, you will overlook some simpler difficulties they are having, which when fixed will set them on a successful trajectory. Try to exercise the art of patience in figuring out what their misconception is, no matter how stupid it seems! (Within reason!) – rschwieb Dec 21 '12 at 14:28
@MichaelAlbanese No, you didn't misrepresent yourself: I just feel like the advice is too important to left unsaid :) I'm sure you'll do great! – rschwieb Dec 21 '12 at 14:54

LIMITS

When introducing the concept of limits and/or whenever referring to limits, especially $\lim_{x\to \infty}f(x)$ or $\lim_{x\to 0}f(x)$:

Be very careful to explicitly state "the limit of $f(x)$ as x approaches infinity" or "the limit as x approaches $0$", or variations of this (e.g. the limit as $x$ gets extremely close to $0$...)

The point being: try to avoid referring to (even if casually) "the limit AT infinity" or "the limit AT zero." The problem, of course, is that the evaluation of a limit as, say, $x \to a$ usually/often coincides with the evaluation of the function at point $a$. And so it's easy for students to misunderstand what taking a limit actually means.

This of course applies to limits in general:

• for $\lim_{x\to a} f(x)$, read "the limit of $f(x)$ as $x$ approaches $a$.

Perhaps others can comment on good ways to introduce limits without introducing too much conceptual dissonance for students.

ALGEBRA errors:

"Calculus is $90$% algebra, and $10$% strictly calculus."

It might help to begin the course with an "assessment" of students' algebraic competence, and revisit, or assign homework addressing the areas in which students floundered (based on the assessment). Doing so early on will help sharpen students' algebraic competence, before needing it later in the course, and permit you and the students to focus on the newly introduced conceptual material, and less on sloppy algrebra!

• One additional thought: You might want to include Spivak's "A Hitchhiker's Guide to Calculus" as either required reading, or as a recommended supplement to the text you will be working with. (It's a paperback, and relatively cheap, as far as textbooks go. Also, it is only 122 pages in length).

• Once again Michael Spivak has produced a wonderful mathematical treasure. This time he has left behind the analysis type approach of 'Calculus' (which is a gem) that he wrote many years ago and had written a book for the beginner. It covers the basics of calculus, and gives the reader a better first introduction than many of the standard textbooks would ever muster. - Review
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Thanks for your answer. Having done some tutorials, I am aware that students get confused with the concept of a limit, and your suggested use of language will certainly help. – Michael Albanese Dec 21 '12 at 14:08

Here are some common pitfalls, and what I like to do about them. This is of course not meant to be comprehensive, but I hope it will be helpful.

Integration by substitution: Students forget to change the bounds/substitute the original variable back in. I like to write $$\int_a^b 2xe^{x^2}\ dx=\int_{x=a}^be^u\ du$$ to emphasize that the bounds are still in terms of $x$ and not $u$. On a similar note, I often label my bounds on multiple integrals, even when the order of integration is understood from the order of the $dx_i$.

For improper integrals, students tend to think that $\infty-\infty$ will "cancel out." It might not hurt to compute $\int_{-1}^1\frac{dx}{x}$ by taking the limits simultaneously in a few different ways and show that you get different values.

When using the integral test for series, students run into trouble with singularities because they don't chose the correct lower bound. To prevent this sort of error, I actually write $$\int_\text{who cares?}^\infty$$ and I encourage my students to do the same. Not only does this help them avoid mistakes, it emphasizes that convergence of series is a property of long-term behavior unaffected by the first finitely many terms.

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I really like your notation in the last part. It reminds me of when I first learnt about summation notation. The lecturer said the index was irrelevant, it's just a name, it didn't matter if it was $i$, $j$, or anything else. He then did an example of the form $$\sum_{\textrm{Bob} = 1}^{10} a_{\textrm{Bob}}.$$ – Michael Albanese Dec 23 '12 at 13:17

Students think that you are asking them to solve a differential equation when in fact you are only asking them to check that a putative solution is indeed a solution.

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I've run into this problem before in tutorials. Do you think that this occurs because the student has misread the question, or is it because they don't really understand what it means for a given function to be a solution? – Michael Albanese Dec 23 '12 at 13:19
Actually I have no idea what causes this! It's true that some students who are fine at solving, say, separable ODEs have no idea what a differential equation is. But other students are fine at solving separable ODEs, do have an idea what a differential equation is, and still have this problem. – user29743 Dec 23 '12 at 17:58