Category: Mathematics

Now that we’ve looked at the unit circle definitions of the trigonometric functions, complete the following questions.

Complete the following questions for our next lesson.

Pages 286–288, questions 1, 2, 3, 8, 10, 12, 25, 28, 29, 34, 35, 38, 40

Use any of the techniques we have discussed in class to determine the convergence/divergence of the following series.

1. \(\displaystyle{\sum_{n=1}^\infty \frac{1}{\sqrt{n}}}\)
2. \(\displaystyle{\sum_{n=1}^\infty \frac{1}{5n}}\)
3. \(\displaystyle{\sum_{n=1}^\infty \frac{n}{2n-\sqrt{n}}}\)

Having now looked at the definition of the limit of a sequence and some associated terms, see if you can show that the sequence defined by

\[u_n=\sqrt{n+1}-\sqrt{n}\]

converges.

Here’s a short question that will give you an opportunity to practice using Euler’s Method. Complete this before our lesson tomorrow.

Use Euler’s Method to find the approximate value of \(f(3)\) if \(y=f(x)\) is the solution to
\[\frac{\textrm{d}y}{\textrm{d}x} = y^2-x\]passing through \((2,1)\). Use a step length of \(0.5\).

(Notice that this is a first-order nonlinear DE that is neither separable hor homogeneous, so we won’t be able to use any methods available in the course to find an explicit solution for this DE.)

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?