Following our discussion of improper integrals, complete the questions below prior to the start of our lesson on Tuesday.
Pages 1354–1355 questions 33, 34, 36, 3, 4, 5, 7, 11, 14
Induction and the Binomial Theorem Test
We’ll have a test on mathematical induction and the Binomial Theorem on Wednesday, February 8th.
To prepare for the test, complete p. 203–205 questions 20, 22, 24, 28, 26, 45, 47
We can discuss these questions in Tuesday’s lesson.
Proof by Induction
Here’s a short homework assignment on proof by indication, to be collected on Monday, February 6th.
Show that \(6^n+4\) is divisible by \(10\) for all \(n \in \mathbb{Z}^+\).
You can download the template from class here.
L’Hôpital’s Theorem
L’Hôpital’s Theorem can be used when evaluating limits that have a certain sort of “indeterminate form” (either \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\)). Subject to some other conditions (What are they? Make sure you check that they’re satisfied!), we can use L’Hôpital’s Theorem to calculate limits like
\[\lim_{x\to 0}\frac{\sin x}{x}\]
Use L’Hôpital’s Theorem to find the value of this limit for tomorrow’s lesson. Also, what do you think about the result shown below?
\[\lim_{x\to 0}\frac{\cos x}{x}=\lim_{x\to 0}\frac{-\sin x}{1}=0\]
The Binomial Theorem
Find the simplified expansion of \((a+b)^7\). You can use any method you prefer, but the Binomial Theorem may save you some work!
Mock Exams
If you have any questions concerning the mock exam material, you can post them in the comments section below.
Our exams are on Monday Period 4 (paper 1, no calculator) and Tuesday P1 (paper 2).
Good luck!
Sequences and Series Test
We’ll have our test on Sequences and Series on Tuesday, January 17th.
In order to prepare for the test, complete the following questions.
Pages 200–205 questions 1–3, 5, 6, 9, 13, 19, 23, 25, 27, 29, 35
Infinite Geometric Series
For tomorrow’s lesson, try to complete page 173 questions 5, 6, 23, 27, and 28.
Continuous Probability Distributions
Complete the following questions for our lesson tomorrow.
Pages 898–901 questions 1, 2, 7, 11, 19, 20
Series Break Homework
Complete the following questions during the break. Remember, if you get stuck you can leave a comment below. If you see a question in the comments and you have a suggestion to make, post that too!
Pages 172–173 questions 1, 2, 3, 11, 12, and 20
Additionally, see if you can complete all of the question below.
Consider the series shown here.
\[\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\cdots\]
- Find the formula for \(S_n\)
- Write down the value of
- \(S_3\)
- \(S_{10}\)
- \(S_{100}\)
- What do you notice happening to the value of \(S_n\) as \(n\) increases? Can you use the formula for \(S_n\) to explain why this is happening?