Dr. McDonald's Mathematics Course Blog
Below is the list of questions we began in class today. Aim to finish up to the end of 13E before our lesson on Thursday.
Exercise 13A.2 questions 3cd
Exercise 13B question 5
Exercise 13C.2 questions 2h, 3c
Exercise 13D question 12
Exercise 13E questions 3ab, 5ab, 26, 27
Exercise 13F questions 3ac, 4ab
Exercise 13G question 5
Complete the following exercises before our next class (after the break).
Exercise Set 13A.3 questions 6bdij and 12
Also, have a look at the question below (which was discussed in class). It was mentioned in class that a solution can be found (to both parts!) without a calculator. Can you figure it out over the break?
\[\arctan\left(\frac{1}{2}\right)-\arctan\left(\frac{1}{3}\right) = \arctan(a), a \in \mathbb{Q}^+\]
I’ll post a hint in the comments below [update: one error in that comment has been corrected, and the hint has been extended]—you may find it useful.
In our last class we looked at how we could transform the sine and cosine functions to model periodic behaviour. The questions posted during class are listed below. Try to get through to the end of Section 12D, and we’ll continue with these questions during our next lesson.
Exercise 12C questions 1, 3
Exercise 12D questions 2–4
Exercise 12E question 1
Exercise 12F questions 1, 3, 5, 6def, 8
Complete the following exercises before our next class (you may find 5a and 6a useful when answering the other parts of those questions, but if you can answer b, c, and d without doing part a, that’s fine too).
Exercise 10C questions 4, 5bcd, 6bcd, 7, 10c
Our next class is the test, so you should have the questions below completed for our lesson on Thursday next week.
Exercise 10A 1abcdghim, 2abc, 3abcdfgh, 4abc
Exercise 10B 1, 5, 7, 8, 10, 13
Work on the following questions for tomorrow’s class. You don’t need to complete all of these, but try to complete at least one from each section.
Exercise 9B.2 1d, 2b, 6, 10, 11, 12B
Exercise 9B.3 2ab, 3
Also, make sure you see the updated test information here.
Challenge Question: Prove that an arithmetic sequence with first term \(a_1\) and common difference \(d\) is such that
\[S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)\]
Complete the following questions for our next class.
Exercise 9B.1 questions 2b and 3b
Here is a template you can use for proof by induction. (Note that the template is set up for a proof that involves a claim concerning all natural numbers. If the claim concerns, for example, all positive integers, you need to adjust the base case and other remarks accordingly.)
On Tuesday, October 12th [note the revised date + content] we’ll have a test on induction and the binomial theorem (including combinations and permutations). Graphing calculators will be required for this test.
In order to prepare for this test, have a look at the questions listed below.
Review Set 8A (all)
Review Set 8B 2, 8–10
Review Set 8C 1, 2, 7–10
Review Set 9A 1–7
Review Set 9B 2
Review Set 9C 1, 2, 5, 7
You may also find some of the additional resources on the HL Resources page useful.
After trying question Exercise 8G question 14, have a look at the quotation from Wittgenstein’s Tractatus Logico-Philosophicus.
With regard to the existence of \(n\) atomic facts there are \(K_n=\sum_{r=0}^n \left(\begin{array}{c}n\\r\end{array}\right)\) possibilities.
Show that Wittgenstein could equally have written “…there are \(K_n=2^n\) possibilities.”
Complete the following questions before tomorrow’s class.
Exercise 8G questions 1, 2, 3, 4, 6ab, 10, 15