assignments
All assignments are due at 3pm on the given date.
due Mon Sep 10
Assignment 1
- From Strang: 1.1.30, 1.1.52, 1.2.44, 1.5.14, 1.5.36.
And to get the problem-solving juices flowing (taken from the website of Tim Gowers; I'm not sure if they're original to him but I think the first one is not):
- You are walking from one end of an airport terminal to the other. The airport has several moving walkways, and you need to stop to tie your shoelace. Assuming you want to get to the other end as quickly as possible, is it better to tie your shoelace while you are on a moving walkway or while you are between walkways?
- Six cards have different numbers written on them and are then laid face down on a table so that you can’t see what the numbers are. You are allowed to select any two cards and ask which has the bigger number. How many questions of this kind do you need to ask before you can put the cards in order?
PRIZE QUESTION! The best answer will win one of a luxury holiday, a sportscar or a small amount of credit at the science building snack bar. Details to follow.
- Create a new mnemonic for SOHCAHTOA.
due Wed Oct 3
Assignment 2
- From Strang: 2.3.16, 2.3.24, 2.4.18, 2.5.4, 2.5.20, 2.5.32, 2.6.6, 2.7.16, 2.7.38.
Bonus question (very hard!). Let f(x) be defined to be 0 when either x is 0 or when x cannot be written as a fraction ("aka is irrational") and be 1/q whenever x = p/q written as a fraction in lowest terms (with positive q). So for example, f(3 / 5) = 1 / 5)
and f(π) = 0
. Investigate the function. What does it look like? Where is it continuous/discontinuous?
due Mon Oct 8
Quiz 1
due Fri Oct 26
Assignment 3
From Strang: 3.2.38, 3.2.40, 3.3.18, 3.4.8 (by hand), 3.4.12 (by hand), 3.5.12 (by hand), 3.8.20, 3.8.28.
due Mon Nov 5
Assignment 4
From Strang: 4.1.24, 4.1.26, 4.2.12, 4.2.21, 4.2.26, 4.3.36, 4.4.32, 4.4.42.
due Fri Nov 16
Quiz 2
due Wed Nov 21
Assignment 5
- Show by induction that \[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \] for \(n \geq 1\). Find the area under the parabola \(f(x)=x^2 \) between \(x=0\) and \(x=1\) using the limit-of-sum-of-rectangles method. (Hint: the two parts of this question are not entirely unrelated.)
- Find the mistake in the proof of the following theorem (this is importantly different from explaining why the theorem isn't true).
Theorem: All horses are the same colour.
Proof. We use induction to show that any set of \(n\) horses is monochromatic (i.e. they are all the same colour as each other). There are only finitely many horses in the world and so this is sufficient.
Base case: Any set of one horse is monochromatic and so the statement holds at \(n=1\).
Induction hypothesis: Assume that all sets of \(k\) horses are monochromatic.
Induction step: Consider a set of \(k+1\) horses. Put them in some arbitrary order. The first \(k\) horses are monochromatic, say they are colour X, by the induction hypothesis. The last \(k\) horses are monochromatic too, also by the induction hypothesis. These must also be colour X because a horse in both lists of \(k\) horses is colour X because it is in the first and thus forces all of the second list to be colour X too. And so our set of \(k+1\) horses is monochromatic.
The result now follows by induction: all horses are the same colour.
- From Strang: 5.3.4, 5.4.10, 5.4.12, 5.5.16, 5.6.32, 5.6.36, 5.7.34.
due Wed Dec 5
Assignment 6
From Strang: 6.1.20, 6.2.16, 6.2.18, 6.2.44, 6.4.10, 6.4.18 6.4.26, 6.4.42.
due Thu Dec 6
Quiz 3
Available electronically from Wednesday 5th December. Remember, your best two quizzes from the three count.
due Tue Dec 11
Bonus Assignment!
Answer some or all of these questions to improve your participation/attendance adjustment. Or ignore them with no detrimental effect to your grade.
- Strang 7.1.4, 7.1.32 (integration by parts).
- Strang 13.1.1, 13.2.6, 13.2.10 (functions of two variables).
due Tue Dec 11
Final Exam
Sci 217 at 10am.
due Wed Dec 12
Final Grade
