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Leaving Cert Project Maths: The Average Vs The Expected Value

March 29, 2011 in General, Grinds, Leaving Cert | Leave a comment

The Average

The average or the mean of a finite set of numbers is, well, the average. For example, the average of the (multiset of) numbers \{2,3,4,4,5,7,11,12\} is given by:

\text{average}=\frac{2+3+4+4+5+7+11+12}{8}=\frac{48}{8}=6.

When we have some real-valued variable (a variable with real number values), for example the heights of the students in a class, that we know all about — i.e. we have the data or statistics of the variable — we can define it’s average or mean.

Definition

Let x be a real-valued variable with data \{x_1,x_2,\dots,x_n\}. The average or mean of x, denoted by \bar{x} is defined by:

\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum_{i=1}^nx_i}{n}.

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Leaving Cert Maths: Differentiation from First Principles

March 29, 2011 in General, Grinds, Leaving Cert | 1 comment

In Leaving Cert Maths we are often asked to differentiate from first principles. This means that we must use the definition of the derivative — which was defined by Newton/ Leibniz — the principles underpinning this definition are these first principles. You can follow the argument at the start of Chapter 8 of these notes:

https://jpmccarthymaths.com/wp-content/uploads/2010/07/lecture-notes.pdf,

to see where this definition comes from, namely:

f'(x)\equiv \frac{dy}{dx}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}. (*)

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LC Project Maths Financial Maths: Inflation Adjustment and Mortgage Amortisation

February 10, 2011 in Grinds, Leaving Cert | Leave a comment

Inflation Adjustment

Suppose that we want to invest €P over t years with an interest rate of i (if the interest rate is 4%, then i=0.04).

Initially we have €P – called the principal. After 1 year however, we will have the principal plus the interest: which is i of P. Hence the value after one year will be:

P+Pi=P(1+i).

So to get the value of the investment you multiply the previous years value by 1+i. So after three years, for example, the investment has value:

P(1+i)(1+i)(1+i)=P(1+i)^3,

and we can generalise to any number of years.

The final value of an investment of a principal P after t years at an interest rate of i, F, is given by:

F=P(1+i)^t.

p. 30 of the tables – Compound Interest

But what about inflation?

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Leaving Cert Maths: Preparing for the Mocks

January 18, 2011 in Grinds, Leaving Cert | 1 comment

This note is primarily about LC Project Maths but also contains details on the Old Syllabus

The Leaving Cert mocks are coming up very fast but don’t fret. There is a lot of time in between February and June to learn maths – but there will be no more chances to test your skills as an examinee. This short note will focus on five areas which you must address to get the maximum benefit from your mocks.

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Leaving Cert: Integration

November 22, 2010 in General, Grinds, Leaving Cert, MA 1008 | Leave a comment

A short note covering integration for Leaving Cert maths.

 

(Please note that the proof of the Fundamental Theorem of Calculus inside isn’t quite correct. We need the Mean Value Theorem to prove it but the one in here is just for illustrative purposes.)

Leaving Cert Project Maths

November 15, 2010 in Grinds, Leaving Cert | Leave a comment

For those doing Project Maths at Leaving Cert, an initial stumbling block will be on how to revise. The best way to revise mathematics is by doing exercises and a ready supply of these is traditionally found in past exam papers:

http://examinations.ie/index.php?l=en&mc=en&sc=ep&formAction=agree

However you might be under the impression that these past papers are useless with the new syllabus: not necessarily. For those doing Project Maths in 2011, the syllabus is to be found at:

http://ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Review_of_Mathematics/Project_Maths/LCM_str1-4_sep09_ex11.pdf

Strand 5, “Functions”, doesn’t become examinable until 2012. However this topic includes differentiation and integration, which will certainly be examinable as per the old syllabus.

Essentially the past papers may be done partially. Paper 1, Q. 6-8 is definitely fair game. Figuring out which of the other questions on past papers are still relevant involves cross-checking the new syllabus with the old.

Mathematical Analysis

November 9, 2010 in Grinds | Leave a comment

Some fairly comprehensive notes for introductory Mathematical Analysis.

(By a Dr. Kin Y. Li from the Hong Kong University of Science & Technology)

The Quotient Rule for Differentiation

November 3, 2010 in Grinds, Leaving Cert, MA 1008, MS 2001 | 5 comments

Here we present the proof of the following theorem:

Let f,g:\mathbb{R}\rightarrow\mathbb{R} be functions that are differentiable at some a\in\mathbb{R}.  If g(a)\neq 0, then f/g is differentiable at a with

\left(\frac{f}{g}\right)'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}

Quotient Rule

Remark: In the Leibniz notation,

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}

Proof: Let q=f/g:

q(a+h)-q(a)=\frac{f(a+h)}{g(a+h)}-\frac{f(a)}{g(a)}

=\frac{f(a+h)g(a)-f(a)g(a+h)}{g(a+h)g(a)}

=\frac{f(a+h)g(a)\overbrace{-f(a)g(a)+f(a)g(a)}^{=0}-f(a)g(a+h)}{g(a+h)g(a)}

=\frac{g(a)[f(a+h)-f(a)]-f(a)[g(a+h)-g(a)]}{g(a+h)g(a)}

\Rightarrow \frac{q(a+h)-q(a)}{h}=\frac{g(a)\left[\frac{f(a+h)-f(a)}{h}\right]-f(a)\left[\frac{g(a+h)-g(a)}{h}\right]}{g(a+h)g(a)}

Letting h\rightarrow 0 on both sides:

q'(a)=\left(\frac{f}{g}\right)'(a)=\frac{g(a)f'(a)-f(a)g'(a)}{[g(a)]^2} \bullet

Making sense of Algebra

October 26, 2010 in Grinds, Leaving Cert, MA 1008, MS 2001 | Leave a comment

Hopefully. The following note (in progress) might help you understand the power and proper functioning of basic real algebra Short_note_on_algebra

The Chain Rule

October 26, 2010 in General, Grinds, Leaving Cert, MA 1008, MATH6037, MS 2001 | 1 comment

MATH6037 please skip to the end of this entry.

The sum, product and quotient rules show us how to differentiate a great many different functions from the reals to the reals. However some functions, such as f(x)=\sin 2x are a composition of functions, and these rules don’t tell us what the derivative of \sin 2x is. There is, however, a theorem called the chain rule that tells us how to differentiate these functions. Here we present the proof. In class we won’t prove this assertion but we will make one attempt to explain why it takes the form it does. In general only practise can make you proficient in the use of the chain rule. See http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf or any other textbook (such as a LC text book) with exercises.

Proposition 4.1.8 (Chain Rule)

Let f, g:\mathbb{R}\rightarrow\mathbb{R} be functions, and let F denote the composition F=g\circ f (that is F(x)=g(f(x)) for each x\in\mathbb{R}). If a\in\mathbb{R} such that f is differentiable at a and g is differentiable at f(a), then F is differentiable at a with

F'(a)=g'(f(a))f'(a)

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