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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.
    \[\begin{eqnarray}(A\wedge B)\to C\\ \overline{A \to (B \to C)}\end{eqnarray}\]
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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.
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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 \(\frac{dy}{dx} = 2x\).

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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]
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11 SL Quadratics: Intersecting Quadratics

Complete the following questions before our next class (tomorrow).

Exercise 1E questions 1abd, 2, 3

A harder question that you might also try is question 4. If you do try this question tonight, you’ll probably find that Example 24 in the textbook is helpful.

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Welcome to Grade 10 Philosophy!

Welcome Grade 10 Philosophers!

I hope you’re enjoying the course so far, and there’s lots of exciting things to come!

You can download the course overview document here (you’ll need to be using your Mulgrave account to access that document).

Future homework assignments will be posted on this website, along with other resources that you may find useful in the course.

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11 SL Polynomials Challenge

The questions below are optional, but if you can answer them correctly, please do show your solutions to Dr. McDonald! (Also, the equations won’t show up correctly in an email, so click to see these questions on the website if you’ve received an email notice for this post.)

Consider the quadratic equation \(ax^2+bx+c=0\), with \(a\neq 0\) for questions 1 and 2.

  1. Complete the square to find another expression for the left side of the equation.
  2. Use your answer from question 1 to isolate \(x\). What is the name of the formula you’ve just derived?
  3. Consider the quadratic function \(f(x)=3x^2+kx-4\), where \(k\) is some constant real number. Explain how you know that, no matter the value of \(k\), the graph of \(f\) will always have two \(x\)-intercepts.
  4. Consider the quadratic function \(f(x)=x^2+kx-(k+8)\), where \(k\) is some constant real number. For which value of \(k\) will the \(x\)-intercepts of the graph of \(f\) be closest together?
  5. Are there any quadratic functions that can’t be represented in factored form? Are there any quadratic functions that can’t be represented in vertex form? Explain your answers.