Math 22b Spring 2008:

  • Fact sheet
  • HW#1 due Wed 1/30. Chapter 6, p318: 2a,2b,6; p330: 7,13,17,20,22.
    In 13, the domain of the function x^y is the right half of the Euclidean plane, i.e. x>0 and y is arbitrary.
    In 7b, apply the Two-Path Test. In 7c, show that if A is any nonzero vector, the function f(tA) has a limit as t goes to 0.
    In 17, you can assume that sin(x) and square root function x^(1/2) are continuous functions of one variable in their appropriate domain. Hint: Note that |sin(xy)| is less than or equal to |xy|. Try to use the Squeeze Theorem.
    In 20, 22, you must prove that your answers are correct. Understanding precisely the definition of "interior points" is essential here.
    The complement of a set S in R^n is the set of points in R^n which are NOT members of S.
  • Your grader will be Max Margolis (margolis@brandeis.edu). His office hours will be Mon 2-3:30pm.
  • lecture notes on functions
  • HW#2 due Wed 2/6. Chapter 6, p338: 2,7; p346: 1j,2j,9c,10a,14,15.
    When computing directional derivatives, be sure to normalize the direction vector at the end.
    In 7, the notation "bracket X,e bracket" means the dot product of X with e.
  • lecture notes on derivatives
  • HW#3 due Wed 2/13. Chapter 6, p355: 2b, 5, 6a, 6b, 7a; p365: 7, 9, 14, 15f
    In 9, use Chain Rule I. In 15f, use the function x^y=exp(y ln(x)), which you considered in HW#1.
  • Test #1 will be on Thu 2/28 in class. Material covered up to Taylor's theorem (including sec 7.2.) You are allowed a letter size 2-sided aid sheet. The test is 1 hour long, but everyone will be given one extra hour.
  • HW#4 due Mon 2/25. Chapter 7, p371: 20,29; p377: 1h, 5, 8; p387: 1d, 4, 7.
    In 20,29, use picture to decide; you don't need to give a formal proof.
    In 5, write the distance square between (x,y,z) on the surface and (0,0,0). This is a function of TWO variables. Find its critical points, and compare the values of your function at those points.
    In 8, you need to use the limit definition to find partials of f at O.
    In 7, apply the 2nd order Taylor's theorem to the points X=(x,y) and X_0=(x_0,y_0).
    Solutions will be discussed on Wed 2/27.
  • lecture notes on extrema problems
  • HW#5 due Wed 3/12. Chapter 7, p398: 1f-1i, 2n, 11; p404: 4; p413: 1f, 2e, 5.
    In 5, a stated constraint for the volume f(x,y,z)=xyz is that x+2y+2z cannot exceed 6. You need a bit of common sense to decide what are the other constraints on the variables x,y,z. For example, can you
    have a box with a negative length? You should find that the correct region on which to consider the function f(x,y,z) is a certain solid region D in 3-space bounded by 4 triangles. Your task is the find maximum of f in D.
  • Solutions to test 1.
  • HW#6 due Wed 3/19.
  • HW#7 due Wed 3/26. p446: 1c, 1e, 2c, 2e, 3c, 7; p452: 1a, 1f, 4, 5c.
    Before doing 4, you should go through 3, the problem right before it on p452.
  • lecture notes on vector-valued functions.
  • HW#8 due Thu 4/3. p470: 3f,7,13a,16,17a; p491: 1c,6e,7b,11a,17.
    In 3f, use just one rectangle for your first estimate. Then divide the rectangle into two along y=1 and get a second estimate.
    In 7, write the volume as an integral over a disk D in R2. Find the lower and upper bounds: m Area(D) and M Area(D) for the integral. To find the radius of D, a picture will help.
    In 13a, note that if the graph of a function over a region is below the xy-plane, then its integral over that region is negative.
    In 16, interpret M(t+h)-M(t) as the integral of g(y) over a certain rectangle, and then estimate that integral using the fact that g(y) is an increasing function.
    In 1c,6e,7b,11a, be sure to draw a picture of the region of integration before doing any computations.
  • Test #2 will be on Thu 4/10 in class. All material covered since Test #1 will be included: from section 7.3 up to section 9.3. You are allowed a letter size 2-sided aid sheet. The test is 1 hour long, but everyone will be given one extra hour.
  • lecture notes on integrations
  • HW#9 due Wed 4/16.
  • lecture notes on Stoke's theorem (prelim.)
  • HW#10 due Wed 4/30. p540: 7, 10, 11, 16; p553: 1h, 2, 7
  • Solutions to test 2.