Have a look at the following questions from option topic (Series and Differential Equations) past papers shown below.
November 2011, questions 2 and 4.
November 2012, question 3.
We will discuss these in our next lesson.
The Integral Test
One nice feature of the integral test is that—when it can be applied to show that a given function converges—we also get an easy way to find a bound on the size of the error term.
In other words, we can use a partial sum to find an approximation of the actual value of a series while at the same time knowing how far off we are (at worst) from the true value of the sum of that series.
Complete the question below for tomorrow’s lesson.
Consider the \(p\)-series with \(p=3\).
- Find a general expression for a bound on the error term when \(S_k\), for some \(k \in \mathbb{Z}^+\), is used to approximate the sum.
- Find the number of terms required to approximate the value of this series to within 0.0005.
- Find the value of \(\displaystyle{\sum_{n=1}^\infty \frac{1}{n^3}}\), accurate to three decimal places.
The Limit Comparison Test
Use any of the techniques we have discussed in class to determine the convergence/divergence of the following series.
- \(\displaystyle{\sum_{n=1}^\infty \frac{1}{\sqrt{n}}}\)
- \(\displaystyle{\sum_{n=1}^\infty \frac{1}{5n}}\)
- \(\displaystyle{\sum_{n=1}^\infty \frac{n}{2n-\sqrt{n}}}\)
Calculus Option Test 1
We’ll have our first test on the Calculus Option topics on Thursday, March 9th.
This test will cover material that includes differential equations, important theorems and definitions in Calculus (Rolle’s Theorem, the Mean Value Theorem, continuity and differentiability, etc.), and l’Hôpital’s Rule.
The following questions will be useful.
page 1471–1474 questions 1, 4, 10, 16, 19, 22–27
page 1436 questions 9, 17
page 1354 questions 4, 6, 14, 15
Limits of Sequences
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.
Euler’s Method
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.)
Mock Examination Review
In our next lesson we’ll review the mock examination questions, so please remember to bring your mock examinations with you to class.
Separable Variables Differential Equations
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.
Improper Integrals 2
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?
Improper Integrals
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