The below is a collection of observations by the various graders concerning common errors made by students in Michael Hutchings's 2006 1B exams.  Of course, there was much that was creditable about students' work, but these comments, by their very nature, focus on flaws -- please do not think us unduly grumpy!  It is hoped that by examing the solutions and these comments students will be able to see what constitutes good and bad work (respectively) and improve their understanding and presentation of mathematics.

The remarks at the bottom come from a similar exercise undertaken at Oxford (after an abstract algebra exam, but most of the points are still valid) which this editor pinned to his desk while preparing for any future exams.  The fact that he went on to graduate study at Berkeley suggests that they may have been useful.

Midterm 1:

Q1: Some students attemtped to  use partial fractions to decompose 1/(squareroot(u)(u-1)).  Others erred by saying that the above function is equal to u^{-3/2} - u^{-1/2}.

Q2:  Many students messed up evaluating a definite integral by substitution because after substituting and integrating, they evaluated the antiderivative at the wrong limits.  This happens because either they don't change the limits of integration when they substitute, or they have confusing or nonexistent notation for the limits of integration and what variable they are integrating over and consequently do the wrong thing at the end.

Q3: A few common errors included:

a)  Confusing midpoint and trapezoidal rule (in (a) and (b));
b)  Not differentiating -1/x^2 properly;
c)  Having the inequality the wrong way round and getting, say, n=<8 rather than n>=9;
d)  Saying n=[some non-natural number] works.

Q4:  (no comments made)

Q5: Some students set up the integration by parts incorrectly -- for example, by choosing dv=arctan x dx and then saying v = 1/(1+x^2).  Others did not know what to do with \int x^3/(x^2+1) dx.  A substantial number did not distribute the -1/3.

Q6: By far the most common mistake was to take the second integral and integrate by parts to get infinity + (another improper integral) and then conclude that the whole thing is divergent.  It is important to note that this only works if you then prove the second improper integral converges (it does in this case, but if no one who did this showed that).  For example, using this technique, it is not difficult to "prove" that int_0^1 (e^x)/(x^{1/2}) dx diverges (e.g. u=x^{-1/2}, dv=e^x dx, etc.).  Some other common errors were:

a) Saying  x < x^{3/2} on [0,1],
b) Confusing the "p-rule" at 0 with the "p-rule" at infinity
c) Being confused about which direction inequalities need to go for convergence / divergence
d) Stating that e^x > e, which is not true on the interval [0,1].
 

Midterm 2:

Q1: (no comments made)

Q2: (no comments made)

Q3: Some common mistakes:

a) multiplying the series for sin x by x^3 or x^2 and calling that the series for sin(x^3)
b) integrating the series for sin x (thereby obtaining the series for -cos x) and then substituting x^3
c) introducing a +C when doing the definite integral, and not in an innocuous way where it goes away when you evaluate the integral, but where it's still there and then they try to solve for it by setting x=0.  all except for one concluded that C=0 since sin 0 = 0.
d) poorly placed inequalities involving 10^{-3}, such as asserting that the estimate itself is less than 10^{-3}.
e) I also noticed that a significant number of people decided how many terms were necessary before evaluating the integral, based on inequalities such as (x^16)/1920 < 1/1000.  of course this method "worked"  because the integral was from 0 to 1.

Q4: A number of people thought they could simplify \ln(n)/\ln(n+1) to \ln(n/(n+1)) or \ln(n-(n+1)) or \ln(n)/(\ln(n)+\ln(1)).  There was a lot of improper manipulation of the equation 2|x+1| < 1, leading to the wrong interval.  (And the people who did this tested the endpoints with the ratio test, seemingly not realizing that the ratio test will never work if you are testing the correct endpoints.)  Some people implicitly asserted that convergence of a series is equivalent to the terms in the series limiting to zero.  There were also a few comparison tests with inequalities going the wrong way.

Q5:  Of those who tried to find a Taylor series, many took a power series for x/(1+x^3) and then "squared" it term-wise to get a power series for x^2/(1+x^3)^2.  Of those who found the correct power series, a great majority assumed that the 17th derivative corresponded to the constant for n=17, rather than to the coefficient of x^17.

Q6: (no comments made)

Postlude: Comments on an Oxford algebra exam.  (a) and (c) especially apply across the board:

"a) A few of the candidates wrote clear, efficient and stylish answers. One or two of those three adjectives could be applied to a good many more scripts. But a disappointing number, even of scripts that had earned good marks, were very poorly written. They contained telegraphic writing of the kind "R ring A ideal" with or without punctuation. It could usually be decoded without too much trouble, but why should examiners be expected to decode?
(b) Many candidates changed notation, sometimes consciously, more often unconsciously, from that of the examination paper to that which, presumably, they had memorised from their notes. As examiner, one adapts. But why should one be expected to? And it undermines any confidence one may have in our students' deeper understanding of their subject.
(c) Similarly, many candidates distorted what was asked in the question to something familiar, but different. Sometimes that was more extensive than had been intended by the examiner--so these candidates were including a significant amount of irrelevant material. We as a Faculty have expressed the hope that Oxford undergraduates acquire `transferable skills'. If those are what I suppose they may be (which is, I'm afraid, not certain) then they should include the skill of recognising what is relevant, what is not. "