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Assignment 1

There are more questions below than what you need to hand in. You choose, which ones to hand in.

Submit for marking a total of 15 solutions to questions (not more). Of the questions that you hand in, 13 need to be analytically solved (handwritten or typeset) and 2 need to be Mathematica based questions.

Each analytic question weights 6pts/100; each Mathematica based question weighs 11pts/100.

In your hand-in, please try and order the questions in ascending order. Also, for each question please try to note both the number of the question as appearing below as well as the example number. It is best if you place the Mathematica based questions last, making sure to submit both Mathematica code and output.



From Unit 1:
  1. Example 6.
  2. Example 8.
  3. Example 9.
  4. Example 10.
  5. Example 11.
  6. Example 12.
  7. Example 13.
  8. Consider the imaginary number i taken to powers of integer n. What is i^n for any integer n?
  9. Example 16. Investigate this example using Mathematica.
  10. Example 17 (c).
  11. Example 18.
  12. Example 19.
  13. Write 2^n as a sum of (n+1) binomial coefficients. Explain why this is true.
  14. Example 21.
  15. Explain (prove) with words (without using factorials) why Pascal's formula is true.
  16. Example 22.
  17. Search the web for a proof of why the set of rational numbers is countable. Understand and reproduce this proof.
From Unit 2:
  1. Example 30.
  2. Example 31.
  3. Example 32.
  4. Example 34.
  5. Example 35.
  6. Example 36.
  7. Example 37.
  8. Example 38.
  9. Example 41.
  10. Example 43.
  11. Example 44.
  12. Example 45.
  13. Example 46.
  14. Example 47.
  15. Example 51.
  16. Example 52.
  17. Example 54.
  18. Example 55.
  19. Example 56.
  20. Example 59.
  21. Search for Euler's formula and see how it can be used to obtain some well known trigonometric identities. Explain what you found.
  22. Example 62.
  23. Example 63.
  24. Example 65.
  25. Example 67.
  26. Example 68.
  27. Example 70.
  28. Example 72.
  29. Example 73.
  30. Example 74.
  31. Example 75.
  32. Example 77.
  33. Analyse the arcsin function. Plot it. Find its domain and range.
  34. Example 78.
  35. Example 79.
  36. Example 80.
  37. Example 81.
  38. Example 82.
  39. Example 83.
Mathematica based exercises: 6, 7, 8, 12, 17, 24, 31, 34, 37, 38, 41, 47, 48, 49, 50, 51.