Here are a couple of questions to look at before our next lesson.
Page 533–535 question 5, 14, 15, 26
Page 548–549 questions 5, 8, 9
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 \(f(x) = 2x^2 + 4x -16\).
- Express \(f\) in factored form.
- Express \(f\) in vertex form.
- Describe a sequence of transformations that would produce the graph of \(y = f(x)\), starting from the graph of \(y = x^2\).
- 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 \(y = x^2\).
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?
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 \(f(x) = e^x \sin x\).
- Show that \(f”(x) = 2e^x \sin \left(x+\frac{\pi}{2}\right)\).
- Obtain a similar expression for \(f^{(4)}(x)\).
- Suggest an expression for \(f^{(2n)}(x), n \in \mathbb{Z}^+\), and prove your conjecture using mathematical induction.
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.
\[f(x)=x^3-2x\qquad g(x)=x^2+4x\qquad h(x)=2x^2+5x-1\]
- Form the six possible rational functions using \(f\),\( g\), and \(h\).
- 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.
- How can you tell when a rational function will have a horizontal asymptote? How can you determine the equation of that asymptote?
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
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.
Solids of Revolution
Today we saw how to use a definite integral to calculate the volume of a solid of revolution. (Some textbooks will refer to this as finding a “volume of revolution.”)
The solid we studied today is shown below, and the equation used to generate this solid was \(y=\cos x +2\), with \(x\) running from 0 to 5. Can you use your knowledge of solids of revolution to derive the formulas for
- the volume of a cone with height \(h\) and a base of radius \(r\), \(V=\frac{1}{3}\pi r^2h\)
- the volume of a sphere of radius \(r\), \(V=\frac{4}{3}\pi r^3\)
Transformations of Functions Test [Updated]
On Tuesday, October 18th we’ll have a test on transformations of functions.
The following questions will be useful in preparing for the test.
Pages 84–85 questions 1, 2, 4, 19
Pages 86–89 questions 5, 6, 13, 14, 23
Update: Not that calculators will not be permitted for this test.
Differentiation and Integration Test
On Sunday, October 16th we’ll have a test on differentiation and integration (not including integration by parts and integration by substitution).
The following questions will be useful in preparing for the test.
(Pearson Textbook)
Pages 751–752—pick any two questions that you have not yet completed
Pages 759–760 2, 3, 13
Pages 762–770 questions 1, 2, 5, 9, 15, 26, 36, 40, 52, 60
Pages 846–853 questions 1, 3, 7, 11 (a and b only), 16