Category: Mathematics

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Quadratics + Even and Odd Functions

For tomorrow’s lesson complete the following questions.

Pages 110–112 questions 28, 30, 31, 43

Also, see if you can answer the following question: Is the sum of two odd functions always an even function? (And similarly, is the sum of two odd functions always an odd function?)

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More Probability

Here are a few questions that will involve all of the material that we’ve covered so far. Complete these before our next lesson on Thursday.

Page 548–552 questions 6, 12, 13, 14, 16, 18, 20, 25

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Quadratic Functions Homework

Complete the following question for our next lesson on Tuesday. Remember, you can leave a comment below if you run in to any trouble with these!

Consider the quadratic function \(f(x) = 2x^2 + 4x -16\).

  1. Express \(f\) in factored form.
  2. Express \(f\) in vertex form.
  3. Describe a sequence of transformations that would produce the graph of \(y = f(x)\), starting from the graph of \(y = x^2\).
  4. Describe a sequence of transformations different from your answer to c) that would also produce the graph of \(y = f(x)\), starting from the graph of \(y = x^2\).
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Probability: The Basics

Here’s the question we saw at the end of today’s class. Can you find the answers?

There are 25 students in a tutor group. Within the group, there are 10 students taking HL Mathematics and 12 student taking HL Chemistry. In the group, there are 8 students who take neither HL Mathematics nor HL Chemistry.

What is the probability that a student selected at random from the tutor group is taking

a) both HL Mathematics and HL Chemistry?
b) either HL Mathematics or HL Chemistry?

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Calculus + Induction

Here’s the question we considered on Thursday. Now that you’ve got a solution for parts 1 and 2, complete part 3 as a homework assignment due on Tuesday, the 25th of October.

The function \(f\)is defined by \(f(x) = e^x \sin x\).

  1. Show that \(f”(x) = 2e^x \sin \left(x+\frac{\pi}{2}\right)\).
  2. Obtain a similar expression for \(f^{(4)}(x)\).
  3. Suggest an expression for \(f^{(2n)}(x), n \in \mathbb{Z}^+\), and prove your conjecture using mathematical induction.
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Rational Functions and Horizontal Asymptotes

Try to answer these questions for our next lesson. Can you suggest an answer to question 3?

Consider the functions given below.
\[f(x)=x^3-2x\qquad g(x)=x^2+4x\qquad h(x)=2x^2+5x-1\]

  1. Form the six possible rational functions using \(f\),\( g\), and \(h\).
  2. Which of the rational functions you have produced have horizontal asymptotes? For any functions that have horizontal asymptotes, write down the equation of that asymptote.
  3. How can you tell when a rational function will have a horizontal asymptote? How can you determine the equation of that asymptote?
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Further Integration Test

On Monday, October 24th, we’ll have a test on further integration methods (integration by parts and integration by substitution, there will be no questions on solids of revolution).

The following questions will be useful in preparing for the test (all taken from the Cambridge book, Chapter 19).

Page 644 questions 3–7
Page 648 questions 6, 7, 8
Pages 653–654 questions 1, 3, 5–9
Page 658 questions 2 and 3

 

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Rational Functions

Here’s the function that was introduced at the end of the last lesson.

\[f(x) = \frac{5x-6}{x+2}\]

Can you determine the \(x\)-axis intercepts and any asymptotes for the graph of \(y=f(x)\)?

We’ll discuss this question in today’s lesson.