MA2104 2006: Exercises

Older exercises end up here. The week number on the exercise refers to the week when the exercise is, or should have been, posted. The exercises usually refer to material lectured on during that week, or earlier.

• Week 34: solution
  1.2: 12, 23*, 24, 38*
  1.3: 34, 36*, 43, 48*, 51, 53
  1.4: 24*
  1.5: *Find the image f[S] of the set S={x+iy: x≤0 og −π≤y≤0} where f(z)=e^z.
• Week 35: solution (updated: error in 1.7.31 corrected)
  1.5: 4, 10*, 12, 17*, 29
  1.6: 22, 40
  1.7: 2, 14*, 19b*, 31*, 32a (sin⁻¹ is arcsin)
  2.1: 4, 8*, 17* (show first that a set and its complement have a common boundary), 21
  2.2: 6, 9, 23*
• Week 36: solution
  2.3: 5, 9, 15*, 21, 27, 28
  2.4: 6, 31*, 33, 35, 38*
• Week 37: solution
  3.1: 3, 6, 7*, 18, 23*
  3.2: 3, 6, 11, 16*, 17, 21*, 26*, 31, 40*
  3.3: 8*, 11
  Plus: Find all the values of iⁱ. (Leonhard Euler (1707–1783) noticed the remarkable fact that all these values are real.)
• Week 38: solution
  3.3: 18*, 19, 27, 29, 32*, 35
  3.4: 1-6, 10, 17, 29*, 32, 33*, 36, 37.
• Week 39: solution
  3.6: 1, 2*, 3, 17, 20, 21*, 28*, 31*
  3.7: 3, 9*, 11*.
• Week 40: solution; see also an animation illustrating uniform and nonuniform convergence for problems 4.2.1–2
  3.7: 14, 15*, 19*, 20
  4.2: 1(a–c), 2(a–c), 13*, 18, 25*
  4.4: 21*.
• Week 42: solution; see also animations for problems 7.2.5, 7.2.9, 7.2.13 and note how the latter overshoots at the discontinuity: This is an example of the Gibbs phenomenon.
  7.1: 8, 12*, 15
  7.2: 5, 9*, 13, 17
  7.5: 11.
• Week 43: solution; the integrals in this problem are known as the Fresnel integrals.
  Consider the function exp(−x²). Integrate it over the triangular path [0,M,M+iM,0] and let M→∞. Note that then the integral along the x axis converges to a known limit. Prove that the integral along the vertical bit goes to zero, and by considering the integral along the hypothenuse, find the integrals of cos x² and sin x² from 0 to ∞.
• Week 44: solution – with an unusually high number of errors. Sorry, no time to fix them now.
  4.4: 22* (hint: partial fraction decomposition), 23
  4.5: 1, 13* (check out example 4 first), 25*
  7.5: 15*, 17 (look at Example 3 first)
  8.2: 5(a)*, 9(a).
• Week 45:solution
  5.1: 15, 20*, 25
  5.2: 7*, 11, 13* (Hint: the “double angle trick” may be useful)
  8.2: 8*
  Exam of 2003-12-15: 1*, 4.
• Week 46:solution
  From old exam problems:
  MNFMA213 1998-12-10: 5
  MNFMA214 2002-05-16: 2
  MNFMA214 2003-05-19: 4, 5
  MA2104 2004-12-13: 3, 5.