Common student mistakes/misconceptions in a first year calculus course

Solution 1:

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!

Solution 2:

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

Solution 3:

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.

Solution 4:

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.