Math 103X.01 Homework Assignments

Day 01: 27 August

  1. Reading: Sections 1.1, 1.2
  2. Homework Exercises:

Day 02: 29 August

  1. Read: Sections 1.3, 1.5
  2. Exercises: Section 1.3: 2(a,b), 5, 10, 15(a,b), 16a, 17, 23.

Day 03: 03 September

  1. Exercises: Section 1.5: 1, 2, 8, 10, 12, 13, 15
  2. Extra Credit: Let P denote the polynomials in a variable x with real coefficients. Define linear maps D: P-->P and I: P-->P by the rules D(p) = p' (i.e., the derivative of p with respect to x) and I(q) = Q where Q is the antiderivative of q with respect to x that satisfies Q(0)=0. Explain why D(I(p)) = p for all p in P. Does I(D(p)) = p for all p in P? Why or why not? Can you give a simpler formula for I(D(p)) (one that doesn't involve integration or differentiation)?
  3. Read: Sections 2.1 and 2.2

Day 04: 05 September

  1. Exercises: Section 2.1: 1b, 4, 9, 10, 18, 22, 28, 31
  2. Exercises: Section 2.2: 1
  3. Read: Sections 2.3 and 2.4

Day 05: 10 September

  1. Exercises: Section 2.2: 5cde, 6ce, 10, 15
  2. Exercises: Section 2.3: 1, 2, 5, 9
  3. Read: Sections 2.4 and 2.5

Day 06: 12 September

  1. Exercises: Section 2.3: 4bd, 8ab, 12a, 13a, 16, 18
  2. Read: Sections 2.4 and 2.5 (really, this time)

Day 07: 17 September

  1. Exercises: Section 2.4: 2, 6, 11, 18
  2. Exercises: Section 2.5: 2de, 3a, 5ad, 8, 11, 12, 17c
  3. Read: Sections 2.6, 3.1, and 3.2 (we are skipping Section 2.7)

Day 08: 19 September

  1. Exercises: Section 2.6: 2ac, 3c, 4a, 9
  2. Exercises: Section 3.1: 1cd, 2, 5, 15, 17
  3. Exercises: Section 3.2: 2, 4, 6
  4. Read: Section 3.3

Day 09: 24 September

  1. Exercises: Section 3.3: 3, 4, 12, 19, 25, 29, 42
  2. Extra Credit:
    1. Show that if u1, u2, and u3 are unit vectors in the plane such that u1 + u2 + u3 = 0, then the angle between any two of these unit vectors is 120 degrees. Hint: Use dot products.
    2. Recall that we showed that, if a, b, and c are points in the plane and we want to minimize the function f on the plane given by letting f(x) be the sum of the distances from x to each of a, b, and c, then critical points of f are a, b, c, and any point p (other than a, b, and c) such that the sum of the unit vectors pointing from p to each of a, b, and c is zero. Determine conditions on a, b, and c that will allow/disallow such a point p to exist. Hint: This is really a question about the shape of the triangle whose vertices are a, b, and c, and the answer should be in this form.
    3. Show that, when a, b, and c satisfy the condition so that such a p exists, then p can be constructed by ruler and compass once the points a, b, and c are given. Hint: What you should first try to do is figure out, for a and b given, what the set of points p is such that the triangle spanned by a, b, and p should have an angle of 120 degrees at the vertex p. Once you know this, you'll be most of the way there.
      4. Point of interest: An actual algebraic formula for p turns out to be somewhat complicated.
  3. Read: Section 3.4

Day 10: 26 September

  1. Exercises: (Due 3 October, after the Test) Section 3.4: 1, 5, 10, 20, 21
  2. Read: Sections 4.1 & 4.2

Day 11: 01 October

  1. First Test

Day 12: 03 October

  1. Exercises: Section 4.1: 3, 13, 14
  2. Exercises: Section 4.2: 1,6
  3. Extra Problem: Compute the length of the path c:[0,2pi] --> R2 that is defined by the rule c(t) = ( cos3t ,  sin3t ).

Day 13: 08 October

  1. Exercises: Section 4.2: 15, 17, 18
  2. Exercises: Section 4.3: 4, 6, 10, 11, 15, 17, 18
  3. Read: Section 4.4

Day 14: 10 October

  1. Exercises: Section 4.4: 3, 8, 16, 24, 25, 27, 31, 32

Day 15: 15 October

  1. Fall Break (no class)

Day 16: 17 October

  1. Exercises: Section 5.1: 1ad, 2ad, 5, 6b
  2. Exercises: Section 5.2: 1ad, 2d, 4, 6
  3. Exercises: Section 5.3: 1ad, 2df, 6, 10
  4. Extra Credit: Find the volume of the region defined as the intersection of the two open, solid cylinders x2 + y2 < r2 and x2 + z2 < r2 where r>0 is a constant. (Hint: First, figure out what each x-cross-section looks like and compute its area. Then use Cavalieri's Principle.)
  5. Read: Sections 5.4 and 5.6 (skip 5.5)

Day 17: 22 October

  1. Exercises: Section 5.4: 1ab, 2cd, 7, 11
  2. Exercises: Section 5.6: 1, 4, 5, 10, 14, 23, 24
  3. Read: Sections 6.1 and 6.2

Day 18: 24 October

  1. Exercises: Section 6.1: 1-6, 8-11
  2. Exercises: Section 6.2: 1, 2, 3, 6, 8, 9
  3. Read: Section 6.3

Day 19: 29 October

  1. Exercises: Section 6.2: 13, 15, 19, 23
  2. Exercises: Section 6.3: 1, 2, 4, 8, 9, 12
  3. Read: Section 7.1

Day 20: 31 October

  1. No homework. Study for the test.

Day 21: 05 November

  1. The second in-class test!

Day 22: 07 November

  1. Exercises: Section 7.1: 2, 3, 5, 9, 13
  2. Exercises: Section 7.2: 1ad, 2bc, 14, 15, 16, 18
  3. Read: Sections 7.2 and 7.3

Day 23: 12 November

  1. Exercises: Section 7.3: 2, 3, 5, 8, 13
  2. Read: Section 7.4
  3. Exercises: Section 7.4: 1, 3, 5, 15, 17
  4. Extra Credit: Prove Archimedes' Theorem: Consider the sphere x2 + y2+ z2  = r2 and the 'tangent' cylinder x2 + y2 = r2. Show that, for any two constants a and b with -r < a < b < r, if we look at the piece of the sphere between the two planes z = a and z = b and the piece of the cylinder between these two planes, then the areas of these two pieces are the same. (Hint: Figure out how to parametrize the sphere and the cylinder so that you can compare the two necessary integrals.)
  5. Read: Section 7.5

Day 24: 14 November

  1. Exercises: Section 7.5: 5, 7, 10, 11, 19
  2. Exercises: Section 7.6: 3, 4, 7, 9, 16

Day 25: 19 November

  1. Read: Sections 8.1, 8.2
  2. Exercises: Section 8.1: 3bc, 7, 9, 12, 15
  3. Read: Section 8.3

Day 26: 21 November

  1. Exercises: Section 8.2: 1, 3, 5, 6, 7, 8, 21
  2. Exercises: Section 8.3: 2, 4, 7, 12, 16, 21
  3. Read: Section 8.4

Day 27: 26 November

Day 28: 28 November

  1. Thanksgiving (no class)

Day 29: 03 December

Day 30: 05 December

  1. Third in-class test!