What are the most common errors in math exams?

Question

I'm new here and I would like to know what teachers have saw in their experience about errors in students exams;

I'm interested to know what are the most common errors in exams about "calculus", more specific about:

• studying a function with $\ln$, $e$, $x^2/x$, $\cos$, $\sin$... (domain, zeros, $f'(x)$, $f''(x)$, asymptote)
• calculating an integral (indefinite and definite) by parts and substitution
• calculating the derivative of complex/compound functions
• theorems like Lagrange, Rolle and Bolzano...

I'm asking this because I'm preparing for an exam and I would like to know what are the most common errors, from the "smallest" one to the "biggest" one..

thanks for your help

Answer 1

Errors in routine algebra, in particular problems with minus signs, often turn a very "doable" problem into one that the student cannot do. If this happens early enough in a problem, even partial credit can disappear.

Answer 2

Often, students erroneously identify $(c, f'(c))$ as the critical point (or $(c, f''(c))$ as inflection point), after finding c to be the root of $f'(c) =0$ (or $f''(x) =0$ )

In this case, the point is $(c ,f(c))$.

Answer 3

Freshman's folly is all too common, $$(a+b)^n=a^n+b^n$$ for $|n|>1$. Although this does hold in finite fields...

My other two "favourites" are, $$\begin{align*}x^2-x&=0\\\Rightarrow x^2&=x\\\Rightarrow x&=1\end{align*}$$ and $$\begin{align*}x^2-4&=0\\\Rightarrow x^2&=4\\\Rightarrow x&=2\end{align*}$$

Answer 4

The properties of logarithms and exponentials are often wrong. For example $\ln(xy)=\ln(x)+\ln(y)$, not $\ln(xy)=\ln(x)\ln(y)$. Not knowing $\sin$ and $\cos$ for the "special angles" like $0, \frac{\pi}{6}, \frac{7\pi}{4},$ etc.

Answer 5

• After finding the critical points of $f$, many students forget to verify that they correspond to maxima/minima using second (or higher order) derivatives.

  This happens especially for the "word problems" where it is intuitively clear--from both the wording of the problem and physical intuition--that the critical point we get should correspond to a minimum/maximum. (Example problem: Find the dimensions of the cuboid with a given surface area $A$ such that the volume is maximized.)

• It is possible that a function $f$ is increasing everywhere in an interval, but the derivative of the function is not strictly positive (just nonnegative). The standard example of such a function is $f(x) = x^3$ over $[-1, 1]$.

  From my experience, if students are asked to provide a counter-example for the fact, they are generally able to remember and produce one. On the other hand, if the question asks for the region where a given function is strictly increasing, then they are more likely to make a mistake out of haste.

Answer 6

${1\over 3}+{2\over 5}={3\over 8}$.

And: "if $f'(a)>0$, then $f$ is increasing near $a$".