abMathematics

In today’s lesson we determined that \(y=\pm\sqrt{x^2+C}-1\), for \(C\in\mathbb{R}\), is the general solution to the differential equation

\[\frac{\textrm{d}y}{\textrm{d}x}=\frac{x}{y+1}\]

As a quick exercise tonight, verify that functions of this form are indeed solutions to the differential equation.

One of the questions that we had a brief look at today is posted below. Our discussion in class dealt with the case for \(p\leq1\), so all we need now to consider is the case where \(p>1\). A comparison test may prove to be difficult, so can you think of another way to establish this result?

For which values of \(p\) does \(\displaystyle{\int_e^\infty \frac{\ln x}{x^p}\textrm{d}x}\) converge?

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

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 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?

\[\lim_{x\to 0}\frac{\cos x}{x}=\lim_{x\to 0}\frac{-\sin x}{1}=0\]

Find the simplified expansion of \((a+b)^7\). You can use any method you prefer, but the Binomial Theorem may save you some work!