Category: Mathematics

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11 HL: Rational Functions

Complete the following questions before our next class. (Most of this should be review from last year, but if you find any unfamiliar material we can look through that in our next class.)

Exercise 6D.1 questions 2adgj, 4

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11 HL: Absolute Values in Equalities

Complete the following questions before our next class.

Exercise 6C.3 1be, 2ade, 3bc, 4afg

If you’re interested in looking in more detail at the \(\leq\) relation, here’s a fun puzzle to consider.

The real numbers are totally ordered by the \(\leq\) relation, that is, \[\text{(1) for any } a,b \in \mathbb{R},\text{ we have } a \leq b \text{ or } b \leq a.\] We also have the following properties, which are true for any \(a,b,c \in \mathbb{R}\)

\[\text{(2) if } 0 \leq a \text{ and } b\leq c, \text{ then } ab\leq ac, \text{ and}\]

\[\text{(3) if } b\leq c, \text{ then } b+ a\leq c+a.\]

What happens if we try to extend the \(\leq\) relation to the complex numbers? If we can, we would want to do so without “breaking” properties 1–3. Can we extend our ordering relation to the complex numbers without running into trouble in this way?

Hint
If our ordering is total, then we must have either \(0\leq i\) or \(i\leq 0\). Show that both of these options lead to a problem. (Property (2) might be particularly useful here.)
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11 SL: Graphs of Quadratic Functions

Complete the following questions before our next class.

Exercise 2B.2 questions 1ad, 2c
Exercise 2B.3 questions 1ae, 2d (you don’t need to complete the square for these questions)
Exercise 2C question 3, 4c, 7
Exercise 2D question 1a, 2b, 3bce, 5

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11 SL: Graphs of Quadratic Functions

Complete the following questions before our next class.

(Red book for these questions)
Exercise 4E questions 4ab and 5a
Exercise 4F question 3ad

(Black book for the questions below)
Exercise 2A questions 2ab, 4ac, 5ac
Exercise 2B.1 questions 1ace, 2, 4