Advanced Calculus (Math 142B, Winter 2010)


Justin Roberts

Lecture times. MWF 1-1.50, in Warren WLH 2112.

My office hours are Wednesday and Friday 11-12 in room 7210, AP&M.

How to contact me. This webpage (math.ucsd.edu/~justin/142W10.html) should answer most basic questions. My email is justin@math.ucsd.edu and my phone is 534-2649.

Discussion section. The TA is Tom Laetsch, and his sections are on Mondays at 4pm in AP&M room B412 and 5pm in  WLH 2112. His office hours are Tuesday 10-12 and Wednesday 12-1 and 2-3 (changed, as of Jan 21)  in AP&M room 6452. His email is tlaetsch@math.ucsd.edu. 

The book is Victor Bryant's "Yet another introduction to analysis".  I recommended this book because it seems unusually readable, has some quite nice problems, solutions at the back, and is cheap. My lectures probably won't follow it too closely but I will be setting some homework problems from it. I will also use a supplemental free online textbook by William Trench called "Introduction to real analysis". This book is more formal in style than Bryant's (and it goes further, covering vector calculus stuff that is not part of 142AB) but it also has lots of good problems in it (without solutions) which I will be setting for the graded part of homework. Here is the link: Introduction to real analysis, by William Trench.

A rough course outline for 142A and 142B. (This isn't exactly the same as the order the books do things.)

1. The real numbers
2. Sequences
3. Series
4. Continuous functions

We did the above in 142A. (Old webpage here.) So now we move onto:

5. Differentiation: max and min, Mean Value Theorem, L'Hopital's rule, derivatives of standard functions
6. Taylor polynomials and Taylor series
7. Integration
8. Sequences of functions, power series
9. Fourier series
10. Possibly further topics: perhaps special functions, distributions, Fourier transform...

Homework will be assigned weekly on Mondays. It will generally consist of problems from both the books above. It will be due in a "homework box" in the AP&M building at 5pm on Tuesday the week after; the discussion sections are the afternoon on Monday, so you'll have a bit of time after section to finish things up. Probably only one of the questions will actually be graded (though there may be extra marks for completeness of homework).

Midterms There will be one in-class midterm, most likely on Wednesday 27th January (let me know if that is  bad choice for any reason!) 

Final The final exam is on TBA. 

Grading policy. Homework is worth 20%, the midterm 20%, and the final 60% of the total grade. But you can discard the midterm score if you do better on the final; that means your final score is the higher of
20% homework + 20% midterm + 60% final or
  • 20% homework + 80% final.
    (This calculation will be done automatically for you at the end of term. If for some reason you miss the midterm, you will not be allowed a make-up, but automatically are covered by the second formula.)
    •

  • Homework #1, for Tuesday Jan 12th.

  • Do your best to continue to write "good proofs" in the style of Math 190! The more English words, the better (well, up to a point!).

  • Not too much for this week: 

    • Bryant page 157 #4

    Trench page 84 #1,3,6.

    Homework #2

    Is a problem sheet here


    Homework #3 (again a shorter one, due next Tuesday 26th as usual).

    1. Use L'Hopital's rule to work out the limit, as x->0, of f(x)=log ((1+1/x)^x). By considering g(x)=e^f(x),
    show that the limit as x->0 of (1+1/x)^x is equal to e.

    2. Consider Taylor polynomials for cos(x), centred at the value a=pi/4.
    (a). Find an interval around a=pi/4 on which cos(x) is approximated by its fifth Taylor polynomial to within one millionth.
    (b). Suppose we want a Taylor polynomial T_n(x) which will approximate cos(x) to within one millionth on the interval
    (pi/4-0.1, pi/4+0.1). How large do we need to make n?

    3. Work out the first three non-zero terms of the Taylor series for f(x)=x/sin(x).

    4. Write down general formulae for the Taylor polynomials T_n(x) for the functions f(x)=e^x, g(x)=log(1-x), centred at x=0. On the interval [0,1/2], if we want accuracy to within one billionth, how big does n have to be in each case?

    Midterm will be on Friday 29th (not Wednesday, as originally planned) in class, and will cover all the stuff we've done so far, up to  and including these kinds of problems on Taylor polynomials.

    Homework #4 (This is really a bunch of revision problems for the midterm. Due Tuesday Feb 2nd.)

    Sheet4

    Homework #5 (for Tuesday Feb 9)

    Sheet5

    Homework #6 (for Tuesday Feb 16)

    Sheet 6

    Homework #7 (for Tuesday Feb 23)

    Sheet 7

    Homework #8 (for Tuesday March 2)

    Sheet 8

    Homework #9 (for Tues March 9)

    Sheet 9

    Final exam Friday March 19th, 11.30-2.30, in the usual room

    Here is a sheet of review problems...

    Review problems