What are the most common errors in math exams? 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
 A: 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.
A: 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 $$\begin{align*}x^2-4&=0\\\Rightarrow x^2&=4\\\Rightarrow x&=2\end{align*}$$
A: *

*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. 
A: 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 $(c ,f(c))$. 
A: ${1\over 3}+{2\over 5}={3\over 8}$.
And: "if $f'(a)>0$, then $f$ is increasing near $a$".
A: The following are not only common mistakes that students make, but are often glossed over.
1: For all $x$ and all $r,s$ $(x^r)^s=x^{rs}$
This is false since if we define $x=-1,r=2,s=\frac{1}{2}$ Then this "rule" gives us
$$((-1)^2)^{\frac{1}{2}}=(-1)^1=-1.$$ On the other hand, $$(((-1)^2)^{\frac{1}{2}})=1^{\frac{1}{2}}=1.$$ 
2: Dividing ratios of functions and forgetting the domain. For instance consider the following computation, $$\frac{x}{x}=1.$$ This is false since we are stating that this is true for all $x$. However if $x=0$ we have a zero in the denominator and so the expression $\frac{x}{x}$ is undefined when $x=0$. What should be written is $$\frac{x}{x}=1, x\neq0.$$ 
A: 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. 
A: $${\cos ^{ - 1}}x \ne \frac{1}{{\cos x}}$$and
$$\begin{array}{lcl}\cos \left( {10} \right) =  - 0.839071529076 & \hspace{0.5in} & {\mbox{in radians}}\\ \cos \left( {10} \right) = 0.984807753012 & \hspace{0.5in} & {\mbox{in degrees}}\end{array}$$
