This is the webpage for my two sections (310 and 311) of Math 1B with Prof. Nicolai Reshetikhin, Fall 2011.
office hours: Thursdays 1:30-2:30 and Fridays 11-12 [subject to change], in 741 Evans -- Berkeley time, of course.
for midterm 1: extra office hours Wed. 2/11, 5:15-6:45 (in the evening), no office hour Fri. 2/13.
- Put both your first and last names as well as your section number on all quizzes and homeworks.
- Staple the week's homework into a single packet, or alternatively you can email me your homework as a pdf file. In either case, it is due by the beginning of section on Monday.
- Homework is graded on a binary scale: you get either 0 points or 1 point, depending on whether you have visibly made a solid attempt on a randomly chosen problem.
- On quizzes, it is fine to use the back of the page for extra space. Just indicate it clearly.
- On quiz problems (which are worth 3 points each), a computational error will lose you 1 point, while a conceptual error will lose you 2 points.
- Show all your work; it will keep you from making mistakes, and can also gain you partial credit even if you do make a mistake somewhere. (For instance, if you make a careless mistake early on, if the rest of your computation is otherwise correct, then you will not lose so many points.)
- Write clearly. When drawing graphs, be sure to emphasize the relevant features. For instance, use large circles (either filled-in or not) to illustrate discontinuities.
- Writing extra stuff can lose you points; the point is to illustrate that you know how to do the problem, not that you're capable of writing down a list of formulas which might or might not be helpful in the given situation.
- Remember to include your "dx" terms. When performing trig / inverse trig substitutions, if you forget these you may very well make an incorrect substitution, and this will be considered as a conceptual error.
- Organize your work carefully on the page, and use equals signs when appropriate (instead of just putting a mess of symbols all over the page).
- There are two ways to deal with bounds of integration when doing a change-of-variables: either convert the bounds to the new coordinate, or convert the output of the integration back to the original coordinate before evaluating. In my experience, generally the first one is easier.
- A substitution must involve a new variable; if your integral is in terms of x, do not make e.g. the substitution x=sin(x). It does not make any sense.
- When solving partial fractions, if you have e.g. ax2+bx+c=A(x+1)(x+2)+B(x+1)(x+3)+C(x+2)(x+3), you can quickly find A, B, and C by setting x to be -3, -2, and -1, respectively (as opposed to setting up a system of 3 linear equations in these 3 unknowns by equating coefficients).
- When giving the bound on a function (say on the absolute value of a 2nd or 4th derivative, in order to bound the error on a midpoint/trapezoid/Simpson's rule approximation to an integral), you must justify why your indicated bound holds.
- Fractions are essentially always easier to work with than decimals (especially without a calculator).
- There is an essential difference between a function converging (say as x approaches &infty;) and its integral converging. (It is necessary for a continuous function to converge to 0 for its improper integral to have any chance of converging, but this is not sufficient.)
- For thinking about true/false questions involving "if-then" statements, it can be helpful to just block of the sentence: "IF [P=first block] THEN [Q=second block]." Then, remember that a counterexample needs to be something which satisfies the hypotheses (P) but fails the conclusion (Q). (Think back to the "If it is raining, then Aaron has an umbrella" example.)
- In order to use the theorem that a bounded monotonic sequence converges, you need to find a fixed bound; it doesn't suffice to have your sequence bounded by another sequence (consider e.g. an=n < bn=n2). Of course, if say a monotonically increasing sequence is upper-bounded by a convergent sequence, then the limit of the latter sequence can serve as an upper bound for your sequence of interest.
- When using the intergral test, it is better form to use a new, continuous variable instead of continuing to use the discrete variable indexing the series. So for instance, given a series Σan, you would want to find a function f(x) such that f(n)=an for all n (instead of calling your function f(n)).
- Given a series Σ(an+bn), you can only split it apart into Σan+Σbn if all these series converge. For instance, compare the series Σn=1∞(1-1)=0 with the difference (Σn=1∞1)-(Σn=1∞1) of divergent series. Similar remarks apply for improper integrals (i.e. those whose bounds of integration are ±∞).
- Remember that a sequence being divergent does not imply that it "diverges to ∞". The standard example is an=(-1)n.