9 E Mathematics

Complete the following questions before our next class. We’ll also have a brief quiz on Sets during that class, so answering these questions will be a good way to prepare!)

Exercise 2E questions 1–5
Exercise 2F.1 questions 1–8

12 HL Integration by Parts (Part 2)

On Tuesday we’ll be talking about the Exploration, but we can also discuss solutions to the questions below.

From the Cambridge book, Chapter 19, complete Exercise 19F.

Have a look at these, and also start thinking about an application of mathematics, or a particular topic in mathematics, that you might consider for your Exploration. You can also have a look at the resources posted here.

10 Philosophy: Logic Quiz

On Thursday, September 27th we’ll have a quiz on Logic.

For this quiz you should know how to symbolize the logical structure of a statement, how to show that an argument is valid/invalid, and how to show that a statement is a tautology/contradiction. You should also know the definitions of the relevant logical terminology.

Here are the questions we discussed in class today. If you didn’t get a chance to complete them in class, work on these before the quiz. If you can answer these questions, you should be well prepared for the quiz!

  1. Symbolize the argument below and use truth tables to show that it is invalid.
    P1: If it’s raining outside then Dr. McDonald will have his umbrella.
    P2: It is not raining outside.
    C: Dr. McDonald does not have his umbrella.
  2. Prove that the following argument form is valid.
    \[\begin{eqnarray}A\to B\\ B\to C\\  \overline{A \to C}\end{eqnarray}\]
  3. Is the argument form below valid or invalid? Prove your answer.
    \[\begin{eqnarray}(A\wedge B)\to C\\ \overline{A \to (B \to C)}\end{eqnarray}\]

11 SL Applications and Optimization of Quadratics

Complete the questions below before our next class on Monday. (These questions will also help you to prepare for the test on Thursday next week.)

Exercise 1F questions 1–5, 7, 9, 12, 15
Exercise 1G questions 2, 3, 7

On Monday we’ll discuss

  • the solutions to these questions, as well as
  • any difficulties you may have had with the review questions that were suggested here.

12 SL Derivatives

Complete the following questions before our next class.

Exercise 14E questions 1 i and ii, 2, 3ab, 4b, 5ab, and 6.

Note that in some of these questions you’ll see an alternative notation for the derivative, \(\frac{dy}{dx}\). Whether you use this, or \(f'(x)\), usually depends on how the original function is given to you. So, both \(f(x)=x^2\) and \(y=x^2\) describe the same function (with derivative \(2x\), as we saw in class), but if we’re starting with \(f(x)=x^2\), then we’d write \(f'(x)=2x\), while if we’re start with \(y=x^2\), we’d write \(\frac{dy}{dx} = 2x\).

12 HL Integration by Parts

Complete the following two questions for next class (Friday).

  1. Find \(\int x \sin x \; dx\), and verify that your answer is correct.
  2. Find \(\int x^2 \sin x \; dx\), and verify that your answer is correct.
[spoiler title=’Hint for question 2′ style=’blue’ collapse_link=’true’]You may find it helpful to use integration by parts twice.[/spoiler]