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taiga

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Mathematical Methods CAS Resources
« on: December 24, 2010, 07:07:20 pm »
+28
AN Mathematical Methods CAS Resources

Guides
Guide to using TI-Nspire for METHODS - b^3
How to Solve Normal Distribution questions without using calculator syntax - Stonecold
General Solutions to Circular Functions - TrueTears
Trinon's guide to Sketching Trig Graphs
Trinon's guide to Antiderivatives through derivatives
All you need to know about inequalities! - TrueTears
Guide to Probability Notation - luken93
The foolproof guide to transformations - Ancora_Imparo
Everything you need to know about Related Rates! - Zealous
Introduction to Trigonometry - cosine
How to Sketch Circular Functions Easily! - cosine
Introductory Probability - hamo94
HOW TO: Statistical Inference (New study design topic) - evandowsett
HOW TO: Probability Guide (When and how to use each type of probability) - evandowsett

Tips
Kyzoo's Compilation of Tricky Points
Mao's Exam Tips
CAS Techniques - hargao
Methods Exam Checklist - paulsterio
Paul's Mathematical Methods - Pre-Exam 1 Advice for 2012 - paulsterio
Paul's Mathematical Methods - Pre-Exam 2 Advice for 2012 - paulsterio
5 Simple Tips for Success in VCE Mathematics - Zealous
Methods raw 42 TIPS AND STUFF I REGRET NOT DOING - BlinkieBill
How I Got a Raw 44 in Methods - Tips, Tricks and Regrets - CookieDream
101 Days Before VCE Maths Exams (Methods/Specialist) [Guide] - Sine
MY TIPS ON SUCCEEDING IN MATHS - PolySquared

Worksheets
Miscellaneous SAC Resources - Practice SACS/tests/worksheets - TrueTears
Methods 3/4 Preparatory Paper - Taiga
EulerFan101's Knowledge Assessment Questions - EulerFan101
VCE PAST PAPER BOOKLETS! - EEEEEEP

Trial Exams
Puffy (Paulsterio + Luffy) 2011 - Maths Methods - ATARNotes Trial Examination
2012 - Math Methods Exam 1 Trial 1 - Hancock
2012 - Math Methods Exam 1 Trial 2 - Hancock
Methods 3&4 Trial Examination 1 - leslieeee
Student Written Methods Exam - lMathMethdz99-R

Notes
ATAR Notes Mathematical Methods CAS Notes
Methods Log Book / Revision - tony3272
Methods 3/4 Notes and Questions - psyxwar

Other
The Matrix Cookbook
How to choose a CAS calculator? - pi
Generalised Textbook Summaries - pi
List of difficult questions from past VCAA exams - insanpi
VCE Methods 2006-2015 Study Design - Which questions are still relevant? - AlphaZero
« Last Edit: August 02, 2019, 12:57:42 pm by Sine »
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Re: Mathematical Methods Guides and Tips
« Reply #1 on: December 24, 2010, 07:08:21 pm »
+7
How to Solve Normal Distribution questions without using calculator syntax
Stonecold

As everyone knows, VCAA doesn't like it, and it would be a BIG risk to use it as a part of your workings for a question worth more than a mark.  This is what Derrick Ha told us to do at his lectures, and I just wanted to share it with some in depth examples.

Question 1.  Normally distributed variable X has a mean of 5 and s.d. of 2.  Find Pr(3.3<X<7.1) correct to 4 decimal places.

Normally, you would just type the following into the calc:  normcdf(5,2,3.3,7.1) and hit execute. Answer is 0.6555.  For a 1 mark question or multi choice question, this is fine.  However for questions worth more marks, this should be avoided, as VCAA has stated that recording calculator syntax is not relevant working.  By using this method, you could jeopardize marks.  Here is the calculator syntax free alternative.

I'll let you know now, that you are going to need to memorize the two equations of the standard normal distribution and the transformed normal distribution.  It is no big deal, as you can have them in your reference book, but the last thing you want to be doing is fiddling around with that, so in my opinion, it is best to memorize it.

Standard normal distribution is $f(z)=\frac{1}{\sqrt{2\pi}}e^-^{\frac{1}{2}z^2$

Transformed normal distibution is $f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^-^{\frac{1}{2}(\frac{x-\mu }{\sigma })^2}$

Here are the workings:

$Pr(3.3

No calculator syntax, yields the same answer, which is exactly what you want.

Question 2.  For a standard normally distributed variable, what is the value of z for Pr(Z<z)=0.6 correct to 3 decimal places?

Normally, in the CAS you would just type in invnorm(0.6,0,1) and press enter.   Answer is 0.253.  However, once again, this is calculator syntax and not appropriate working.  This is how you should set out your answer:

$Pr(Z

Solving for z gives $z=0.253$

Whilst you have still used you calculator to get the answers in both of these questions, you have shown the examiner that you understand how the question would be completed using mathematics related to the course, and that you understand that a normal distribution is no different to any other continuous random variable, and that the probability over a given interval is calculated in the very same way.  i.e. By calculating the definite integral over the specified interval.

These workings use mathematical syntax rather than calculator syntax, so you can rest assured  that workings such as these will guarantee you the marks on short answer questions based on the normal distribution.

I would also like to point out the you should still do your calculator workings using the normcdf and invnorm functions, as this is far less risky as it will minimize keying in wrong data, and will also save time.  However in your written workings, complete something similar to the above to avoid losing precious marks on what are rather simple questions.
« Last Edit: December 09, 2011, 08:39:18 pm by Rohitpi »
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Re: Mathematical Methods Guides and Tips
« Reply #2 on: December 24, 2010, 07:10:49 pm »
+8
General solutions to circular functions.
TrueTears

Many people just simply apply a formula (similar to the one provided in the Essentials 3/4 text), some knows where it comes from but most just mindlessly apply the formulas without knowing where it comes from and if the question gets tricky then they will get stuck. Hopefully this tutorial help you understand how we go about finding general solutions to circular functions in a more systematic fashion.

We will consider the general solutions to each of our 3 main circular functions, $\sin$, $\cos$ and $\tan$. First we will look at the $\sin$ function.

Example 1.

Find the general solution to $\sin\left(3x - \frac{3\pi}{4}\right) = \frac{1}{2}$

First notice why does this question say find the GENERAL solution? This is because no domain is specified, if no domain is specified then there are infinitely many solutions to the above equation. [What this means is that if you sketch the graph of $y = \sin\left(3x - \frac{3\pi}{4}\right)$ and draw the horizontal line $y = \frac{1}{2}$, then there are infinitely many x values which gives a y value of $\frac{1}{2}$]

So here is how I would go about solving this question.

Let $X = 3x - \frac{3\pi}{4} \implies x = \frac{1}{3}\left(X+\frac{3\pi}{4}\right) \cdots [1]$

Now we have the equation $\sin(X) = \frac{1}{2}$ so let us find the 2 basic solutions to $\sin(X) = \frac{1}{2}$ then we will use $[1]$ to use the 2 basic solutions to $X$ to find the 2 basic solutions for $x$.

Solving $\sin(X) = \frac{1}{2}$ is quite trivial. This is an "exact value" question.

$X = \frac{\pi}{6}, \pi - \frac{\pi}{6} = \frac{5\pi}{6}$

Now if you do not know how I solved the above equation, you need to review your circular function fundamentals ASAP.

Substituting the 2 basic $X$ value into [1] yields:

$x = \frac{1}{3}\left(\frac{\pi}{6}+\frac{3\pi}{4}\right), \frac{1}{3}\left(\frac{5\pi}{6}+\frac{3\pi}{4}\right)$

$x = \frac{11\pi}{36}, \frac{19\pi}{36}$

Now the next step is finding the GENERAL solutions for x.

Look at the following graph of $y = \sin\left(3x - \frac{3\pi}{4}\right)$

The red line is the line $y = \frac{1}{2}$, as you can see, since a domain is not specified, it crosses the sin graph infinitely many times.

Now why did we find TWO basic solutions and not just one? As you can see the purple lines represent the solutions obtained from $x = \frac{11\pi}{36}$ and to get the other purple line solutions we simply have to add and subtract periods away from our basic solution of $x = \frac{11\pi}{36}$.

But as you can see from the graph, no matter how many periods we add or subtract we will never end up on the green lines and this is what the other basic is for!

If we add and subtract periods away from $x = \frac{19\pi}{36}$ then we reach all the other green solutions.

So what is the period of the graph? Well it's $\frac{2\pi}{3}$, again go back and review your fundamentals if you don't know how to calculate periods.

So our general solution is:

$x = \frac{11\pi}{36} + \frac{2\pi}{3} \times n, \frac{19\pi}{36} + \frac{2\pi}{3} \times n$ where $n \in \mathbb{Z}$

Now notice some of you might go, "wait what? Didn't you say we must SUBTRACT periods as well as adding them?"

This is another common mistake students often make, look at my definition of $n$ in my answer. I said $n$ is an INTEGER which means $n$ ITSELF can take on negative values, eg, n = ...-3,-2,-1,0,1,2,3...

So for example say n = 1

Then we have $x = \frac{11\pi}{36} + \frac{2\pi}{3} \times 1, \frac{19\pi}{36} + \frac{2\pi}{3} \times 1$ So here we are adding periods.

But if n = -1 then we have:

$x = \frac{11\pi}{36} + \frac{2\pi}{3} \times (-1), \frac{19\pi}{36} + \frac{2\pi}{3} \times (-1)$

Which is equivalent to:

$x = \frac{11\pi}{36} - \frac{2\pi}{3} \times 1, \frac{19\pi}{36} - \frac{2\pi}{3} \times 1$ So here we are subtracting periods.

So that is why we don't write our solution as:

$x = \frac{11\pi}{36} \pm \frac{2\pi}{3} \times n, \frac{19\pi}{36} \pm \frac{2\pi}{3} \times n$ where $n \in \mathbb{Z}$

Because the subtracting periods is already taken into account due to the restriction on $n$

However some of you like to have the $\pm$ in the middle and another way of writing the answer is this: (Note the difference!)

$x = \frac{11\pi}{36} \pm \frac{2\pi}{3} \times n, \frac{19\pi}{36} \pm \frac{2\pi}{3} \times n$ where $n \in \mathbb{N} \cup \{0\}$

Why do we need $\pm$ in the middle here? This is because n is now an element of NATURAL numbers or 0, which means n = 0, 1, 2, 3 ...

So the 'subtracting' periods is NOT taken into account from our restriction on n, that is why we need to put $\pm$ in the middle since we need to 'manually' take into consideration ADDING and SUBTRACTING periods.

Both way of presenting the answer is fine, pick one and stick to it

Example 2.

Find the general solution to $\cos\left(3x - \frac{3\pi}{4}\right) = \frac{1}{2}$

NO DIFFERENCE, APPROACH THIS QUESTION THE EXACT SAME WAY AS EXAMPLE 1, TRY IT YOURSELF!

Example 3.

Find the general solution to $\tan\left(x\right) =\sqrt{3}$

Now the tan function is a tiny bit different in that we only need to find ONE basic solution and not TWO. The rest of the principles of adding and subtracting periods is all the same.

Here is how I would solve this question:

Let us first sketch the graph of $y = \tan(x)$ below:

The red line is the line $y = \sqrt{3}$ and the purple lines are the solutions to the equation.

As you can see from the graph, by finding any value of the x value that corresponds to the purple line and then adding and subtracting periods from that x value we get, we will be able to find all the solutions! So we don't need to solve for TWO basic solutions, ONE will be enough! (You might ask why do we need to solve for just one basic solution, graphically I have explained it, but algebraically this is because $\tan(x)$ is a one to one function while sin and cos are not. You don't really need to know this though)

So let us solve it!

$\tan(x) = \sqrt{3}$

$x = \frac{\pi}{3}$

Thus the general solution is:

$x = \frac{\pi}{3}+\pi \times n$  where $n \in \mathbb{Z}$ (Note the period of $\tan(nx)$ is $\frac{\pi}{n}$ and not $\frac{2\pi}{n}$)

OR another way of writing it is:

$x = \frac{\pi}{3} \pm \pi \times n$ where $n \in \mathbb{N} \cup \{0\}$

Well that's it folks, enjoy and post any questions if you don't understand!
« Last Edit: May 06, 2012, 05:58:17 pm by VegemitePi »
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Re: Mathematical Methods Guides and Tips
« Reply #3 on: December 24, 2010, 07:11:44 pm »
+4
Trinon's guide to Sketching Trig Graphs
Trinon

N.B. Images are down, but the information should still be able to be interpreted without them Have a go at making the images yourself as a learning exercise

$f(x) = -2sin(3(x-\frac{\pi}{4})), x \in [-\pi,\pi]$

Before even attempting to sketch a graph, you want to write down the key components of the graph.

1. $Central axis, y=0$

2. $Amplitude = 2$

3. $Period = \frac{2\pi}{n} = \frac{2\pi}{3}$

4. $Scale Factor = \frac{Period}{4} = \frac{\pi}{6} = \frac{2\pi}{12}$

5. $Horizontal Translation: + \frac{\pi}{4} = \frac{3\pi}{12}$

6. $Reflection about x-axis$

Now we can start to draw the graph.

1. Put in dotted lines where the central axis is and where the top value will be and the bottom value will be. To find the top and bottom lines, we simply plus/minus the amplitude from the central axis.

3. Next we are going to put in the marks of the scale factor. The scale factor is a quarter of the period and lies on the central, top and  bottom axis. This will help us later sketch the graph. Carefully mark in a small cross where each minimum, maximum and central axis intercept occurs.

4.Now we need to find the axis intercepts.
Y-Axis:

$Let x = 0$

$f(0) = -2sin(\frac{-3\pi}{4}) = \sqrt{2}, \therefore (0, \sqrt{2})$

Find the domain for which we will solve for x:

$-\pi \leq x \leq \pi$

$\frac{-5\pi}{4} \leq x - \frac{\pi}{4} \leq \frac{3\pi}{4}$

$\frac{-15\pi}{4} \leq 3(x - \frac{\pi}{4}) \leq \frac{9\pi}{4}$

X-Axis:

$Let y = 0$

$0 = -2sin(3(x-\frac{\pi}{4}))$

$3(x-\frac{\pi}{4}) = -3\pi, -2\pi, -\pi, 0, \pi, 2\pi$

$x-\frac{3\pi}{12} = \frac{-12\pi}{12}, \frac{-8\pi}{12}, \frac{4\pi}{12}, 0, \frac{4\pi}{12}, \frac{8\pi}{12}$

$x = \frac{-9\pi}{12}, \frac{-5\pi}{12}, \frac{\pi}{12}, \frac{3\pi}{12}, \frac{7\pi}{12}, \frac{11\pi}{12}$

$x = \frac{-3\pi}{4}, \frac{-5\pi}{12}, \frac{\pi}{12}, \frac{\pi}{4}, \frac{7\pi}{12}, \frac{11\pi}{12}$

5. Next we are finding the endpoints.

$f(\pi) = f(-\pi) = -2sin(\frac{9\pi}{4}) = -\sqrt{2}$

$\therefore (-\pi, -\sqrt{2}) and (\pi, -\sqrt{2})$

6. Now we can mark in the end points and sketch the graph. Be mindful of the intercepts.

Sketching a tan graph

The steps are much the same as sketching a sin or cos graph. The only differences are the period and the introduction of asymptotes.

For a standard $\tan{(x)}$ graph the period is $\pi$. The formula for finding the period is now $\frac{\pi}{n}$.

To find the asymptotes, you first need to find the center of the tan graph on the x-axis. This is done by noting the translations. Lets say have the following equation:

$f(x) = \tan{(n(x-h))} + k$

The center of the tan graph will be at $(h, k)$. Now to find the asymptotes, you simply mark in half of the period (in this case $\frac{\pi}{2n}$) from either side of the center point, then continue every period until the end of the domain.

The center of each tan curve will be directly in between each asymptote. It is now possible to mark in where the other curve centers are, and draw a rough line for what the graph will look like. When doing this, I like to use dotted lines to signify that it isn't actually the curve, but what it may look like.

Last but not least, you let the equation equal 0, and find the x-axis intercepts. This part is identical to finding the x-axis intercepts for the sin and cos graph.

I've included an example below that I did by hand:

And we're done!
There you have it. I hope this guide has been helpful. Do realise that once you get the hang of this technique, you can start to leave steps out. For example, on this graph I didn't even bother to find the x-axis intercepts because I could see that they would fall on the scale points that I had already drawn.
« Last Edit: May 28, 2012, 03:13:47 pm by VegemitePi »
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Re: Mathematical Methods Guides and Tips
« Reply #4 on: December 24, 2010, 07:12:19 pm »
+4
Trinon's guide to Antiderivatives through derivatives
Trinon

Ever have an equation you want to derive, but couldn't because you plain don't know how? Well than this is the guide for you!

As a side note, this is actually covered under the Methods study design and a question like this will most probably be asked on either of the two exams.

So without further Apu (hehe, Simpsons related joke):

I'm only going to run through the fundamental method, because there isn't much else to it. It only starts getting hard in Specialist Maths when they start throwing things like differentiate $xCos^{-1}(3x)$ and hence anti-differentiate $Cos^{-1}(3x)$ and things like that.

We start off with an equation that we can't anti differentiate with any method that has been covered in the methods study design.

$y = log_e(x)$

We first multiply this equation by $x$ so that we get $xlog_e(x)$.

Next we find the derivative via the product rule:

$\frac{d}{dx}(xlog_e(x)) = log_e(x) + \frac{x}{x} = log_e(x) + 1$

Next we re-arrange the new equation:

$log_e(x) = \frac{d}{dx}(xlog_e(x)) - 1$

If we now anti-derive both sides we get:

$\int log_e(x) dx = \int\frac{d}{dx}(xlog_e(x)) dx - \int1 dx$

$\implies \int log_e(x) dx = xlog_e(x) - x + C$

Now you can Anti-derive the in-anti-derivable!

Hope this helps guys. If you've got any questions just ask.
« Last Edit: May 06, 2012, 05:38:51 pm by VegemitePi »
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Re: Mathematical Methods Guides and Tips
« Reply #5 on: December 24, 2010, 07:13:21 pm »
+5
Compilation of Tricky Points
Kyzoo

Here is my collection of those "finer details" that catch people out. And this is for Non-CAS so some stuff may either be irrelevant or missing.

EXAM TECHNIQUE

• Don’t assume you know what a question is asking for just because it looks familiar. Make the effort to read and interpret everything carefully, as if it were something you had never seen before.

GENERAL STUFF

• Whenever you are referring to a graph curve, write $y = f(x)$ not just $f(x)$. There's a difference between $f(x)$ and $y=f(x)$. To be safe always include the $y=...$ part

• When the question asks for 1/2/3/4 decimal places, and the last decimal place is a 0, you have to include that 0 anyway. For example 2.1 to two decimal places is 2.10, not 2.1.

• Distinguish between when it is asking for a certain number of decimal places, or an exact solution. With decimal places you can just use the calculator.

• In explanations you can use diagrams as well as words

• You need to simply equations fully to gain full marks

• Take care to use correct units in answers. Avoid the mistake of just writing a number when it should be accompanied by a unit

• Always label axis-intercepts with co-ordinates rather than a single number. Label y-intercept as $(0, y)$ and x-intercept as $(x, 0)$

• Be careful of whether it asking for an actual time, or a value of t. Because it if is asking for a “time after…” then it will not be a value of t.

• 1.25 hours = 1 hour 15 minutes

$x^{16/7} = y$
=>$x = \pm y^{7/16}$

Whenever there is an even denominator present when "powering" the equation, the $\pm$ sign is neccesary

DRAWING GRAPHS

• Always label curves with its corresponding equation $y =$...

• Whenever you have to sketch a graph. Always, always look out to see if there is any restricted domain indicated in the question wording.

• Within the domain of a derivative function, any endpoints involved are always open ones ○. There are no closed endpoints • on a derivative function within its domain. This is because the derivative does not exist where there are two possible values. For example, for the derivative function of $y = |x|$, there are two open endpoints at (0,1) and (0, -1)

• Whenever you have to sketch a weird function that you are unsure about, always get the calculator graph first then sketch. Especially do this when you have to draw two functions on the same set of axis. Otherwise you may get the shapes wrong.

• When drawing a curve approaching an asymptote, make sure the curve never touches or bends away from the asymptote whilst approaching.

• Do not assume the domain to always magically be the maximal domain. You must interpret the situation and restrict the domain accordingly.

• Whenever part of the graph you need to curve overlaps with a line that is already there, you must clearly indicate this (probably best by using some colour other than black)

• When drawing graph lines, put arrows on the end of the lines to indicate that they go on ►

• When you need to draw several functions for a question, look out to see if it says to “show on one graph” or “show on one set of axis”. Otherwise you lose marks for drawing each curve on individual graphs.

• With any hybrid function or functions with a restricted domain, you need to take care to indicate endpoints and whether they are open or closed

• Whenever the horizontal variable is time. (You have a function $f(t)$). It is automatically assumed that the domain does not exist for $t<0$. If you draw the function for that domain, the graph is incorrect as you have failed to interpret the question.

• The horizontal and vertical axis are not always labelled “x” and “y”. Interpret the situation, then label with the correct variables. For example when the function is $T(e)$, the vertical axis is labelled $T(e)$, and the horizontal axis is labelled $e$. Furthermore in probability the vertical axis is labelled $p(x)$.

This also applies to the asymptotes. Don’t label $y =$... when $V =$... is more accurate

FUNCTIONS

• Whenever there is a question asking for the factors of a polynomial expression, be careful of whether they state linear factors or not. Linear factors are those of degree one.

• With equations involving sinusoidal functions, you need to be especially careful about the restricted domain.

• Always make sure you don’t mistake “minimum” for “maximum” and visa versa.

$(x-a)(x-b)^2$ has 3 real solutions $(x = a, b, b)$. But 2 distinct real solutions $(x = a, b)$

$f^{-1}(x)\neq(f(x))^{-1}.$$f^{-1}(x)$ is the inverse function. $(f(x))^{-1}=\frac{1}{f(x)}$

COMPOSITE FUNCTIONS

• Be able to identify expressions such as $ln(sin(x))$ immediately as composite functions

• Regarding $g(f(x))$, make sure you go through the functions in the right order. Don’t think $g(x)$ is the base function that is subbed into $f(x)$. Carefully look at the expression and interpret, the correct order is $f(x)$ subbed into $g(x)$.

• Regarding $p(x) = g(f(x))$. The notation that is used to denote $p^{'}(x)$ in exams is always $f^{'}(x)g^{'}(f(x))$

TRANSFORMATIONS

• When dealing with the transformation of restricted function, remember that the transformation affects its domain and range as well.

• In problems make sure to look at the wording carefully to see if there is any specific order of transformations. Don’t automatically assume the default order of “Dilation --> Reflection --> Translation” is applicable to every situation. Sometimes the wording specifies otherwise.

POLYNOMIALS
• Be sure to correctly determine the degree of a polynomial degree. E.g. do not mistake $x(x+2)^2(x-1)$ as a 3rd degree function. Otherwise you will sketch the wrong shape of the graph that comes from and goes to the wrong places.

LOGARITHMS
• For $f(x) = ln (x)$, because of its domain, x cannot be 0 or a negative number. Whenever you have a question involving this, you must state that $ln(x), x\in(-\infty, 0]$ has no real solutions and thus is an invalid solution

• Whenever you have to solve a logarithm equation. Be sure to substitute the solutions back into the log brackets. Whenever the bracket expression comes up with 0 or less, than that solution is invalid.

$\int \frac{1}{ax}\, \mathrm{d}x = ln|ax|+c = ln(a) + ln|x| + c = ln|x| + c$, because c can be any number.

$0.85^n < 0.1$
$=> n>\log_{0.85} 0.1$

or alternatively

$0.85^n < 0.1$
$n>\frac{\log_{e} 0.1}{\log_{e} 0.85}$

CIRCULAR FUNCTIONS

• Whenever you have a variables in degrees for a sinusoidal function $f(x)=sin k^o$, you must convert the variable into radians to do anything with the function (e.g. differentiate, anti-differentiate, transform, find the period, etc).

$f(x)=sin k^o$
$=> f(x) = sin\left ( \frac{k\pi}{180} \right )^c$

RATES OF CHANGE

• Be careful when it asks for “rate of decrease”, if the derivative is a negative value than the "rate of decrease" has a positive value

• When you are given a rate of change, make sure to pay attention to the units so that you don’t get the wrong derivative. E.g. when the rate of change is volume is $5cm^3 /s$, the corresponding rate of change is $\frac{dV}{dt}$, not $\frac{dV}{dr}$

TANGENT

• Be careful about whether it is asking for the normal or the tangent

• When the gradient of the tangent is $\pm \infty$ , and the function is not differentiable at that point, does not mean the tangent does not exist. In fact there is a vertical tangent with a horizontal normal. Same with when the gradient is 0.

ANTI-DIFFERENTIATION and DIFFERENTIATION

• When anti-differentiating an indefinite integral, take care to include the “+ c” part. Along with the “dx” term immediately following the integral.  $\int 2x\, \mathrm{d}x = x^2 + c$

• The anti-derivative does not include "+c." The antiderivative of $2x$ is $x^2$ not $x^2 + c$

$\int \frac{3}{5x}\, \mathrm{d}x = \frac{3}{5} ln|x| + c$
$\int \frac{3}{5x}\, \mathrm{d}x \neq \frac{3}{5} ln(x) + c$

• Whenever you have an expression $\int_{a}^{b} kf(x)dx$ always transform it into $k\int_{a}^{b} f(x)dx$ as not having to multiply every term by “k” makes it a lot easier.

• Whenever it says use calculus, for differentiation it means you must provide the correct derivative expression, for anti-differentiation it means you must provide the correct integral and antiderivative

• In multiple choice where you have choose which expression evaluates the requested area, be careful about which number is on the top, and which is on the bottom. With $\int_{a}^{b} f(x)dx$, "x=a" isn't necessarily the left-limit

• Derivative does not exist at cusp points or where function is not continuous

CALCULATOR

• Do not use the ZOOM function. Instead set the window manually

• When trying to find the maximum, minimum, x-intercept, etc, do not drag the dot. Instead set the left and right boundaries by inputting numbers.

• Make sure calculator is in DEG or RAD depending on what you need

• Make extensive use of the memory system “…” STO> “…”

• Also make extensive use of 2nd --> ENTER to save time with reentering certain calculator expressions

GENERAL PROBABILITY

• Be careful in discrete and binomial probability to discern whether it’s a < or $\le$ sign

• With measures of spread and centres, it always refers to the x values. Y values are always irrelevant.

• Denote the median as “m”

• Be careful about whether it’s asking for percentage or decimal probability.

• Be sure to be able to distinguish between independent and mutually exclusive

• With tree-diagram questions, when you are drawing the tree diagram draw only the branches required to answer the question. It is much quicker,easier, and neater this way.

BINOMIAL PROBABILITY

• Be careful to distinguish between when you need to find the probability of only a single possible path, or all paths. Because the former is not compatible with the binomial probability formula.

E.g. Mark, Alan, and John have taken an exam. They have a 0.8 probability of passing. What is the probability that only Mark passes?

The answer to this is not $\binom{3}{1} 0.8^1 (1-0.8)^{3-1}$

There are three possible ways that only one of the three can pass: either only Mark, Alan, or John passes. But if Mark is only one that passes, then represents only one of those three possible ways.

Thus the answer is $0.8^1 (1-0.8)^{3-1}$

CONTINUOUS PROBABILITY

• When writing the hybrid functions of probability density function, you can write the domain for the parts defined by $y = 0$ as otherwise, or elsewhere

• For probability density functions, there are two things you need to indicate for the hybrid function

1) The endpoints where parts start and end, and whether or not they are closed or open, usually the part above the x-axis is the closed one

2) You must draw the entire function, including where $y = 0$. Take care to indicate where the function is on the x-axis with some coloured line.
« Last Edit: December 09, 2011, 08:39:45 pm by Rohitpi »
vce: english, methods, spesh, chemistry, physics, geography.

tutoring in any vce maths.

2010: Melbourne High School (VCE)
2011 - 2016: Monash University BComm/BEng (Hons)

If you guys have any concerns/suggestions for making ATARNotes a better place, don't hesitate to PM me.

taiga

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Re: Mathematical Methods Guides and Tips
« Reply #6 on: December 24, 2010, 07:14:09 pm »
+1
EXAM TIPS
Mao

It's been a year since I've done MM, which I completed without much knowledge in relationship to the assessment structure and what examiners really expected. Just coming from the "correct" way of doing things [and by god I really hope examiners go by "correctness" than the VCELand bullshit], hopefully these will point you in the right direction of achieving an awesome score which I'm sure all of you will! GOOD LUCK

This guide is written according to the study design, and is not specific to either of the exams. However, since both exams are placed very close together, that shouldn't have much of an impact.

Functions and graphs

- power functions/polynomials sketching. When given a factorised polynomial, such as 'p(x)=(x-1)(x-2)(x-3)', its x intersepts will be the opposite of those numbers, i.e. '1,2 and 3'. That is a VERY common mistake [I've made it on the specialist paper on Monday], so make sure that you don't rush and write "-1, -2 and -3" as I have done.
- the above is only the case when the coefficient of x is positive. In cases such as p(x)=(1-x)(2-x)(3-x) the x-intercepts is not the opposite of the number as the coefficient of x is negative. [See NULL FACTOR below in the algebra section]
- When given a graph with intercepts clearly labeled, always write it in the factorised form, and don't forget to put a common factor "k" out the front to allow for verticle dilations, and find its value by using the y coordinate. e.g. if given a graph that has x intercepts at '-1, 1 and 3' and the y intercept is at '6', then $p(x)=k(x+1)(x-1)(x-3)$, subbing in 'x=0', we find that 'k=2'.
- as a simple check, ALWAYS look to the RHS of the graphs. if it is going up, it is a positive polynomials, if it is going down, it is a negative polynomial. ALSO remember that these positives and negatives aren't necessarily given out the front, they may put it inside a linear factor which is in reversed order, such as $p(x)=(1-x)(x-2)(x-3)$
- much of these go for exponential functions, but also keep in mind that when the exponential function is expressed as $y=ka^{bx}+c$, its asymptote is ALWAYS the value of 'c', regardless of what the rest are. it does not have an x intercept when 'c' is 0, and when 'b' is negative, it approaches 0, not negative infinity.
- logarithms are similar to exponents, except where you worry about horizontal asymptotes, you worry about verticle now. $y=k\log_e (b(x-a)) + c$ will have an asymptote at 'x=a'. Remember that this function is only defined if the term inside the log is positive, so make sure you always check the domain when curve sketching or solving equations.
- circular functions, sin, cos and tan are quite tricky. But remember that they are all of the form $y=af(b(x-c))+d$, the 'b' term controls the period, $\mbox{period}=\frac{2\pi}{b}$ [or for the case of tan, $\frac{\pi}{b}$], and 'a' controls the amplitude.
- the modulus function is probably the most difficult of these, but the way to deal with them is simple: treat it as if the modulus wasn't there, then flip whatever is neccessary to positive.
- when the modulus is on the outside $|f(x)|$, we reflect anything which are below the x axis to the positive y direction. If the modulus is on the inside $f(|x|)$, we draw the right hand side of the graph first, then reflect in the y axis. The distinction is that recognising the second type is hard, but the graph is terribly easy to draw.
- reinstated many times on the forums, transformations can be understood, explained and questioned in many different ways. A few key things to remember is that along an axis = parallel to the axis = away from the other axis, and the order of transformation does matter, i.e. $f(b(x-c))$ and $f(bx-c)$ are completely different things. It is very important that you keep to the prior format so you can determine the horizontal dilation.
- graphs of sums and differences of graphs can be done by drawing both graphs, then using addition of ordinates. Remember that when doing this, you are in no way required to be dead accurate, you are not from I, Robot, examiners don't expect a masterpiece in an exam. But key features are must: when one function is 0, the sum is expected to be at the other graph's value; when the two functions intersect, the value would be approximate double; when the two functions are opposites, the value would be 0; etc.
- graphs of inverses, the easiest way to do this is by drawing 'y=x' and flip along this [basically the act of switching x and y coordinates]. Try to make both sides as identitcal as possible, turn the paper on 45 degrees and do this, squint your eyes, look like a retard... no one should judge how you choose to do these important papers.

Algebra
- the null factor theorem: $p(x)q(x)=0\implies p(x)=0\; \mbox{or}\; q(x)=0$. DO NOT FORGET THIS. This is not only useful for solving polynomials $(x-1)(x-2)=0\implies x-1=0 \mbox{ or } x-2=0$, but also some circular function and even exponentials - $e^{2x}\cdot (x^2 - 2x + 1) = 0\implies x^2 - 2x + 1 = 0$ since the exponential term is always positive
- REMEMBER THE QUADRATIC FORMULA. REMEMBER WHAT THE DISCRIMINANT MEAN. REMEMBER HOW TO FACTORISE. These are very important. and the discriminant can be a quick calculation checker to see if you have got the right number of solutions.
- the log laws, these are in your textbook. Try to know them fairly well [maybe don't stress on the change of base rule as much], but knowing how to manipulate a logarithm will definitely be on exam 1.
- When solving for exponential equations such as $e^{2x}-4e^{x}+3=0$, don't forget your index laws [go over them again]. These are typical quadratic equations, and a substitution of $A=e^x$ is often best [the term is chosen from the middle term], you will get $A^2-4A+3=0$, now proceed to solve for A, then substitute back and solve for x.
- UNDER NO CIRCUMSTANCES, AND I STRESS, NO CIRCUMSTANCES, should you log or square root a negative number [or divide by 0], if you did that, you have done something wrong, restart.
- When solving trigonometric equations: Simple case, one function, one value, such as $2\sin (2x)=1$ for the domain $[0,2\pi ]$, the easiet way is to substitute $\theta = 2x$, and manipulate the domain accordingly, $\theta \in [0,4\pi]$, now solve for $\sin \theta = \frac{1}{2}$, and find the first two solutions, then add $2\pi$ until you go over the limit, $\theta = \frac{\pi}{6},\frac{5\pi}{6},\frac{13\pi}{6},\frac{17\pi}{6}$, and then solve for x. The complicated case is where you have trig functions on BOTH sides of the equation, where you have to divide by one of those trig functions. THIS IS DANGEROUS. Two things may happen: $\sin(x)=\sqrt{3}\cos(x)\implies \frac{\sin(x)}{\cos(x)}=\sqrt{3}$, dividing by cos in this case doesn't really affect what happens, as you did not cancel anything out on the LHS. $\cos^2(x)=\frac{\cos(x)}{2}\implies \cos(x)=\frac{1}{2} \ or \cos(x)=0$, in this case, since you have cancelled something on both sides, you HAVE TO include the second "cos(x)=0".
- when finding the inverse function, remember the only condition is that it passes the horizontal line test. This means one-to-one and many-to-one relationships both can have inverse functions. Restrict domain as necessary where required. Remember that sometimes, the question may not give you the maximal of this domain, but a section of it. In this case, it is still correct.
- when algebraically rearranging for inverses, there are a few difficult types of functions: $y=x^2+2x+3$, in this case, complete the square - $x=y^2+2y+3 \implies x=(y+1)^2+2 \implies y= \pm\sqrt{x-2}-1$; $y=\frac{x+1}{x-1}$, in this case, try to make the numerator a number - $x=\frac{y+1}{y-1} = \frac{y-1+2}{y-1}=1+\frac{2}{y-1}\implies y=\frac{2}{x-1}+1$; $y=\log_e x + \log_e (x+2)$, in this case, remember your log laws and merge the log laws together - $x=log_e (y^2+2y)\implies e^x = y^2+2y + 1 - 1 =(y+1)^2-1\implies y=\sqrt{e^x+1}-1$
- The modulus function is also very difficult to deal with algebraically. You can arrive at modulus functions by $\sqrt{x^2}=(\sqrt{x})^2 = |x|$. When needing to solve this, ALWAYS break it down into the two halves, and solve them separately. e.g. $|x-2|>3\implies \begin{cases} \mbox{for }x<2 & -(x-2)>3 \implies x-2<-3 \implies x<-1 \\ \mbox{for }x>2 & (x-2)>3 \implies x>5\\ \end{cases}$

Calculus
- it is VERY EASY to mix differentiation and integration, so TAKE YOUR TIME in the exam, make sure you KNOW what you are doing.
- the various laws are all explained in your textbook, as well as the derivatives of a few of the common transcendentals (log, cos, sin, etc). Make sure that you recognise the REVERSE of this process, as these are the only functions you will be asked to antidifferentiate. so if you arrive at $\int \log_e(x)\; dx$, you KNOW that you've done something wrong, because you are unfamiliar with ANYTHING which has a derivative that is log. This means that you need to recognise $\int \sec^2(x)\; dx = \tan (x)+C$. The only exception to this is when you have a "differentiate this HENCE antidifferentiate that".
- When doing one of these questions, ALWAYS copy down the differentiation equation you had, such as $\frac{d}{dx}(xe^x) = e^x+xe^x$, and then integrate both sides (remember that integration and differentiation cancel, and that you need a +C), you arrive at $xe^x+C=\int e^x + xe^x\; dx$, now you can split the integral and antidifferentiate whatever you can, and move to the other side to get the integral you want.
- when differentiating, ALWAYS check if the chain rule should be used to the function/part of the function. too many times have people lost marks for forgetting that negative or 2x on the inside.
- when differentiating a function within a domain, make sure that THE END POINTS ARE NOT INCLUDED. OPEN CIRCLES, ROUND BRACKETS. A tangent cannot be drawn through end points [included or not], hence it is not differentiable there. Discontinuous functions cannot be differentiated where they are not continuous, cusps and vertical tangents also need to be taken care of [not differentiable]
- the linear approximation formula, $f(x+h)\approx f(x)+hf'(x)$, will most probably be examined. make sure you know what it is [textbook]
- know how to manipulate the definite integral, such as if $\int_a^b f(x)\; dx = 4$, then $\int_{a/2}^{b/2} f(2x)\; dx = 2$ (shrank everything by 2)
- when finding area, it is ALWAYS positive, and unit squared.
- when finding areas enclosed, make sure that you check where the functions intersect, and use definite integrals that are appropriate.

Probability, the part that everyone hates.
- remember that E(x) is the mean, it is the sum of x*Pr(x)
- go over the definition of var(x) and sd(x), $var(x)=(sd(x))^2=E(x^2)-(E(x))^2$
- read over your year 11 probability stuff, ESPECIALLY REMEMBER THE SIMPLE THINGS LIKE VENN DIAGRAMS. This give a very nice visual image about how addition law in probability works, which is very important when you need to do conditional probability.
- sampling [very simple ones] can be tested, these are normally just tree diagrams, or drawing two balls out of a bag of 5.
- KNOW conditional probability.
- BINOMIAL DISTRIBUTIONS. Remember your formula for this, remember how to use pascal's triangle [or how to expand combinatorics], knowing how to do this is very important.
- for binomial distributions, mean = np, variance = npq
- know how to use the calculator functions for binomial distributions, binomcdf is one of the most useful calculator functions.
- for continuous random distributions, remember that $\int_a^b p(x)\; dx = 1$
- $\mu=\int_a^b x\cdot f(x)\; dx$, $\sigma^2 = \int_a^b x^2 \cdot f(x)\; dx - \mu^2$
- to find the median, evaluate and rearrange for m: $0.5=\int_a^m p(x)\; dx$
- For normal distributions, ALWAYS do two things: find the Z score, and draw the bell curve and shade in the probability (the area). The mathematics with these are not hard (and most of the calculations are done on the calculator using normcdf and invnorm), you just need to think visually and manipulate the area, add bits where necessary, etc.
- When using invnorm, ALWAYS check that the probability you are feeding in is from negative infinity. that's how invnorm works.
- FOR CAS ONLY. When modelling repeated events with karnaugh tables/transition matrices, remember they must follow this pattern: $\left[\begin{array}{cc}A\cap A & A \cap A' \\ A'\cap A & A'\cap A' \\ \end{array}\right]^n \cdot \left[\begin{array}{cc}A_s\\ A'_s\\ \end{array}\right]$, what is important is that the rows should match in what they mean.
- FOR CAS ONLY, steady-state matrices haven't been tested in any papers I've seen, but you should have an okay grasp on it just in case. Though I doubt it will be on the paper.

okay, this is probably it.... i would have missed some stuff, but I think the majority is covered. Enjoy.

PS: someone sticky this pl0x
ALSO, tribute to trinon, who is in the process of writing another guide and could not do this one

Other Acknowledgements to:
ell, fredrick, hamtarofreak

Mod edit (VegemitePi): Fixed up LaTeX issues
« Last Edit: May 06, 2012, 05:36:28 pm by VegemitePi »
vce: english, methods, spesh, chemistry, physics, geography.

tutoring in any vce maths.

2010: Melbourne High School (VCE)
2011 - 2016: Monash University BComm/BEng (Hons)

If you guys have any concerns/suggestions for making ATARNotes a better place, don't hesitate to PM me.

taiga

• Honorary Moderator
• ATAR Notes Legend
• Posts: 4104
• Respect: +586
Re: Mathematical Methods Guides and Tips
« Reply #7 on: December 24, 2010, 07:28:37 pm »
+3
Guide to using TI-Nspire for METHODS
b^3

Version 1.5
Ok guys and girls, this is a guide/reference for using the Ti-nspire for Mathematical Methods CAS. It will cover the simplest of things to a few tricks. This guide has been written for OS Version 3.1.0.392. To update go to http://education.ti.com/calculators/downloads/US/Software/Detail?id=6767.

Any additions or better methods are welcomed. Also let me know if you spot any mistakes.

Guide to Using the Ti-nspire for SPECIALIST - The more intricate & complex but enjoyable: http://www.atarnotes.com/forum/index.php?topic=125433.msg466856#msg466856

NOTE: There is a mistake in the printable version. Under normal distribution for pdf functions it should read "For the height of the probability curve at a certain point use [Menu] [5] [5] [1] (Pdf)"
Also under the shortcut keys the highlighting should read "Copy: Ctrl left or right to highlight, [SHIFT (the one with CAPS on it)] + [c]"

Simple things will have green headings, complicated things and tricks will be in red.
Firstly some simple things. Also Note that for some questions, to obtain full marks you will need to know how to do this by hand. DON’T entirely rely on the calculator.

Solve, Factor & Expand
These are the basic functions you will need to know.
Open Calculate (A)
Solve: [Menu] [3] [1] – (equation, variable)|Domain
Factor: [Menu] [3] [2] – (terms)
Expand: [Menu] [3] [3] – (terms)

Matrices
Matrices can be used as an easy way to solve the ‘find the values of m for which there is zero or infinitely many solutions’ questions. When the equations $ax+by=c$ and $dx+ey=f$ are expressed as a matrix $\begin{bmatrix}
a & b\\
d & e
\end{bmatrix}*\begin{bmatrix}
x\\
y
\end{bmatrix}=\begin{bmatrix}
c\\
d
\end{bmatrix}$
, letting the determinate equal to 0 will allow you to solve for m.
E.g. Find the values of m for which there is no solutions or infinitely many solutions for the equations 2x+3y=4 and mx+y=1
Determinant: [Menu][ 7] [3] Enter in matrix representing the coefficients, solve for det()=0

Remember to plug back in to differentiate between the solutions for no solutions and infinitely many solutions.

Modulus Functions
While being written as || on paper, the function for the modulus function is abs() (or absolute function). i.e. just add in abs(function)
For example y=|x| and y=|x^2-4|

Defining Domains
While graphing or solving, domains can be defined by the addition of |lowerbound<x<upperbound
The less than or equal to and greater than or equal to signs can be obtained by pressing ctrl + < or >
e.g. Graph $y=x^{2}$ for $x \in (-2,1]$
Enter $f2(x)=x^2 |-2 into the graphs bar

This is particulary useful for fog and gof functions, when a domain is restriced, the resulting function’s domain will also be restricted.
E.g. Find the equation of $fog(x)$ when $f(x)=x^2,x \in(-2,1]$ and $g(x)=2x+1,x\in R$
1. Define the two equations in the Calulate page. [Menu] [1] [1]

2. Open a graph page and type, f(g(x)) into the graph bar

The trace feature can be used to find out the range and domain. Trace: [Menu] [5] [1]
Here $fog(x)=(2x+1)^{2}$ where the Domain = (-1.5,1] and Range =[0,4)

Completing the Square
The easy way to find the turning point quickly. The Ti-nspire has a built in function for completing the square.
e.g. Find the turning point of $y=2x^2+8x+9$

So from that the turning point will be at (-2,1)

Easy Maximum and Minimums
In the newer version of the Ti-nspire OS, there are functions to find maximum, minimums, tangent lines and normal lines with a couple of clicks, good for multiple choice, otherwise working would need to be shown. You can do some of these visually on the graphing screen or algebraically in the calculate window.
Maximums: [Menu] [4] [7] – (terms, variable)|domain
Minimums: [Menu] [4] [8] – (terms, variable)|domain
E.g. Find the values of x for which $y=2x^{3}+x^{2}-3x$ has a maxmimum and a minimum for $x\in [-\frac{3}{2},2]$

Tangents at a point: [Menu] [4] [9] – (terms, variable, point)
Normals at a point: [Menu] [4] [A] - (terms, variable, point)
E.g. Find the equation of the tangent and the normal to the curve $y=(x+2)^{2}$ when $x=1$.

Finding Vertical Asymptotes
Vertical Asymptotes occur when the function is undefined at a given value of x, i.e. when anything is divided by 0. We can manipulate this fact to find vertical asymptotes by letting the function equal $\frac{1}{0}$ and solving for x.
e.g. Find the vertical asymptotes for $y= \tan(x) ,x \in [0,\pi]$ and $y=\ln(2x+1)-2$

So for $y= \tan(x), x \in [0,\pi]$ there is a vertical asymptote at $x=\frac{\pi}{2}$ and for $y= \ln(2x+1)-2$ at $x=\frac{-1}{2}$
Don’t forget to find those other non-vertical asymptotes too.

The x-y Function Test
Every now and then you will come across this kind of question in a multiple choice section.
If $f(x)+f(y)=f(xy)$, which of the following is true?
A. $f(x)=x^2$
B. $f(x)=\ln(x)$
C. $f(x)= \frac{1}{x}$
D. $f(x)=x$
E. $f(x)=(x+2)^2$
You could do it by hand or do it by calculator. The easiest way is to define the functions and solve the condition for x, then test whether the option is true. If true is given, it is true otherwise it is false.

So option B is correct.

The Time Saver for Derivatives
By defining, f(x) and then defining df(x)= the derivative, you won’t have to continually type in the derivative keys and function. It also allows you to plug in values easily into f’(x) and f’’(x).
E.g. Find the derivative of $y=2x^3+3x^2-4x+2+ \frac{1}{x}$
Define f(x), then define df(x)

The same thing can be done for the double derivative.

Just remember to redefine the equations or use a different letter, e.g. g(x) and dg(x)

Solving For Coefficients Using Definitions of Functions
Instead of typing out big long strings of equations and forgetting which one is the antiderivative and which one is the original, defined equations can be used to easily and quickly solve for the coefficients.
E.g. An equation of the form $y=ax^3+bx^2+cx+d$ cuts the x-axis at (-2,0) and (2,0). It cuts the y-axis at (0,1) and has a local maximum when $x=-1$. Find the values of a, b, c & d.
1. Define $f(x)=ax^3+bx^2+cx+d$ (Make sure you put a multiplication sign between the letters)
2. Define the derivative of the f(x) i.e. df(x)
3. Use solve function and substitute values in, solve for a, b, c & d.

So $a=\frac{-1}{2}, b=\frac{-1}{4}, c=-2$ and $d=1$ and the equation of the curve is $f(x)=\frac{1}{2}x^3-\frac{1}{4}x^2-2x+1$

Deriving Using the Right Mode
The derivative of circular functions are different for radians and degrees. Remember to convert degrees to radians and be in radian mode, as the usual derivatives that you learn e.g. $\frac{d}{dx}(\sin(x))=\cos(x)$ are in radians NOT degrees.

Getting Exact Values On the Graph Screen
Now for what you have all been dreaming of. Exact values on the graphing screen. Now the way to do this is a little bit annoying.
1. Open up a graph window
2. Plot a function e.g. $f(x)=\sqrt[3]{x}$
3. Trace the graph using [Menu] [5] [1]
4. Trace right till you hit around 0.9 or 1.2 and click the middle button to plot the point.
5. Press ESC
6. Move the mouse over the x-value and click so that it highlights, then move it slightly to the right and click again. Clear the value and enter in $\frac{1}{2}$
.

Using tCollect to simplify awkward expressions
Sometimes the calculator won’t simplify something the way we want it to. tCollect simplifies expressions that involves trigonometric powers higher than 1 or lower than -1 to linear trigonometric expressions.

Streamlined Markov Chains
For questions that require the use of the T transition matrix more than once, the following methods can be used to save time so that the T matrix does not need to be repeatedly inputted or copied down.
1. Define the T matrix as t.
2. Define the initial state matrix as s.
3. Evaluate by substituting t and s in with the appropriate powers.
E.g. For the Transition matrix $T=\begin{bmatrix}
0.6 & 0.85\\
0.4 & 0.15
\end{bmatrix}$
and initial state $s=\begin{bmatrix}
1\\
0
\end{bmatrix}$
, find S2 and S3

Binomial Distributions
For a single value of x e.g. Pr(X=2) = [Menu] [5] [5] [D] (Pdf)
For multiple values of x e.g. Pr(X<2) = [Menu] [5] [5] [E] (Cdf)
e.g. Probability of Success = 0.4, Number of trials =10, i.e. X~Bi(10,0.4)
Find the probability of two successes and less than two successes

Pr(X=2)=0.1209
Pr(X<2)=0.0464

Normal Distributions
The probability will correspond to the area under the Normal distribution curve.
For the height of the probability curve at a certain point use [Menu] [5] [5] [1] (Pdf)
From lower value to higher value = [Menu] [5] [5] [2] (Cdf) (for -∞ use ctrl + i)
e.g. The probability of X is given by the Normal Distribution with $\mu =0,\sigma =1$ i.e. X~N(0,1)
Find Pr(X<1) and Pr(0<X<1)

Pr(X<1)=0.2420, Pr(0<X<1)=0.3413

Integrals
Using the integral function and solve function for probability distributions. The area under a probability distribution function must equal 1, so if we are given a function multiplied by a k constant, we can antidifferentiate the function and solve for k.
E.g. If f(x) is given by $f(x)=\begin{cases}
kx^{2}+1 & \text{ if } 00 & \text{ } otherwise
\end{cases}$
, find the value of k if f(x) is to be a probability density function.

Shortcut Keys
Copy: Ctrl left or right to highlight, [SHIFT (the one with CAPS on it)] + [c]
Paste: [Ctrl] + [v]
Insert Derivative: [CAPS] + ["-"]
Insert Integral: [CAPS] + ["+"]
∞: [Ctrl] + ["i"]

Thanks to Jane1234 & duquesne9995 for the shortcut keys. Thanks to Camo and SamiJ for finding the errors.
« Last Edit: July 28, 2012, 02:06:07 pm by VegemitePi »
vce: english, methods, spesh, chemistry, physics, geography.

tutoring in any vce maths.

2010: Melbourne High School (VCE)
2011 - 2016: Monash University BComm/BEng (Hons)

If you guys have any concerns/suggestions for making ATARNotes a better place, don't hesitate to PM me.

taiga

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Re: Mathematical Methods Guides and Tips
« Reply #8 on: December 24, 2010, 07:28:58 pm »
+3
CAS Techniques
hargao

When I started my methods and spesh practice papers some time in November, these really anal questions kept on coming up ):
So I defined these functions in my brand spanking new Ti-nspire cas which made life a little bit more bearable.  I’m new to posting on atarnotes forum, so please be nice haha (:

Anal question type 1 (simultaneous linear equation 0,1,infinite solution questions)

$(m+3)x + my = 7$
$(m+2)x + 4y = 13$

Find m such that there are
i) no solutions
ii) one solution
iii) infinite solutions

We can write a function in the ti calculators which automatically finds the solution set to such a question.

So for the general problem of
$ax+by=c$
$dx+ey=f$

where c,f are constants and a,b,d,e may be constants or linear factors in terms of the variable “g”

input $define \; linearsoln(a,b,c,d,e,f,g) = \{solve(a \cdot e=b\cdot d \; and \; b\cdot f\neq c\cdot e,g), solve(a\cdot e\neq b\cdot d,g),solve(a\cdot e=b\cdot d \; and \;b\cdot f=c\cdot e,g)\}$

output $\{1st \; element, 2nd \; element, 3rd \; element\}$

1.   The first element of the set in the ouput tells us the requirements for no solutions to the set of linear equations
2.   The second element of the set tells us the requirements for one solution to the set of linear equations
3.   The third element of the set tells us the requirements for infinite solutions to the set of linear equations

Example
Solve the original question above

we let:

input $linearsoln(m+3,m,m+2,7,m+2,4,13,m)$ instead of $linearsoln(a,b,c,d,e,f,g)$
output is $\{ m=1\pm \sqrt{13},m \neq 1\pm \sqrt(13),false\}$

1.   For there to be no solutions, $m=1+\sqrt{13} \; or \; m=1-\sqrt{13}$
2.   For there to be one solution, $m\neq 1+ \sqrt{13} \; or \; m \neq 1-\sqrt{13}$
3.   For there to be infinite solutions, there is $no \; m \in R$ that satisfies such a condition

Anal question type 2 (tedious left hand right hand area estimation thingy)

Often we get asked to find the left-hand estimate or right-hand estimate of the area bounded by a curve and the x axis (stupid pointless vcaa question that can be done on a computer with much higher accuracy)

For Left-Hand Estimate
input $define \; lhestimate(a,b,c,n)=\frac{c-b}{n} \sum_{k=0}^{n-1}(a\mid b+\frac{c-b}{n}\cdot k)$

For Right-Hand Estimate
input $define \; rhestimate(a,b,c,n)=\frac{c-b}{n} \sum_{k=1}^{n}(a\mid b+\frac{c-b}{n}\cdot k)$

Parameter “a” is where you put in the function that you want to evaluate, say $41x-sin(x)$
Parameter “b” is where you put in the lower limit of the integral you are evaluating
Parameter “c” is where you put in the upper limit of the integral you are evaluating
Parameter “n” is where you put in the number of rectangles you are splitting the integral into.

Example

Use the trapezium rule to find the area bound by the curves $y=x^2, y=0, x=0 \; and \; x=100$ with rectangle width of 0.5

So
$a=x^2$
$b=0$
$c=100$
$n=200$ as there are 100/0.5 rectangles

Therefore input $\frac{1}{2}( lhestimate(x^2,0,100,200)+rhestimate(x^2,0,100,200))$ to find the trapezoidal rule estimation of $} \int_{0}^{100}x^2 \, \mathrm{d}x$

Anal question type 3 (stationary point scavenging hunt)

Pointless max-min questions (method also suited to drawing graphs and various types of spesh applications questions)

Input $define \; statpoint(a,b)=\{solve(\frac{d}{db}(a)=0 \; and \; \frac{d^2}{db^2}(a)<0,b),solve(\frac{d}{db}(a)=0 \; and \; \frac{d^2}{db^2}(a)>0,b),solve(\frac{d}{db}(a)=0 \; and \; \frac{d^2}{db^2}(a)=0,b)\}$

Usage
Parameter “a” is the function
Parament “b” is the variable
You can also limit the domain of the function (place a restriction on b)
Eg $statpoint(x^3+x^2-x,x)\mid 0

output $\{ 1st \; element, 2nd \; element, 3rd \; element\}$
1.   The first element gives the local maximum points
2.   The second element gives the local minimum points
3.   The third element gives the possible locations stationary points of inflection (saddle points)
note:  you need to use the first derivative test to verify the nature of these points. You may change the defined function to include a higher order derivative test, ie the third derivative test, but this may cause the potential unfortunate loss of local min. and max. points depending on the function evaluated.

Example
Find the stationary points and their nature for the function $f(x)=x^4-3x^3+3x^2-x+5$

Input $statpoint(x^4-3x^3+3x^2-x+5 ,x)$
Output $\{false,x=\frac{1}{4},x=1\}$

1.   There are no local maximum points
2.   There is a local minimum point at x=0.25
3.   There may be a stationary point of inflection at x=1. Use first derivative test or third derivative test to verify that it is a stationary point of inflection.

(Not) anal (and trivial) question type 4

Finding the average rate of change of a function

$define \; avchange(a,b,c)=\frac{(a\mid x=c)-(a\mid x=b)}{c-b}$

Self-explanatory, and sort of pointless

(Not) anal (and trivial) question type 5

Finding the mean value of a function

$define \; meanval(a,b,c)=\frac{1}{c-b} \int_{b}^{c} a \, \mathrm{d}x$

Self-explanatory, and sort of pointless

Hopefully these might help you in some way, and maybe inspire you to define your own super big and awesome functions XD
« Last Edit: December 23, 2011, 01:43:00 pm by Rohitpi »
vce: english, methods, spesh, chemistry, physics, geography.

tutoring in any vce maths.

2010: Melbourne High School (VCE)
2011 - 2016: Monash University BComm/BEng (Hons)

If you guys have any concerns/suggestions for making ATARNotes a better place, don't hesitate to PM me.

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Re: Mathematical Methods CAS Resources
« Reply #9 on: May 06, 2012, 05:12:04 pm »
0
jane1234

GENERAL:

• Do ALL working written out. Do as little in your head as possible. This includes adding, subtracting, whatever. Trust me, this is one of the easiest ways to minimize errors, especially in exam 1. Write every step down (makes checking over your work easier too).
• Don't expand/simplify expressions when you don't have to. This again will cut down on the silly mistakes. Worst thing is when you have the right answer and then try and simplify it and get it wrong...
• ALWAYS SUBSTITUTE BACK! This means deriving antiderivatives, subbing in answer to an equation, etc. I did this and picked up about 3 silly mistakes in exam 1 which would have cost me the 50. Always do the 'opposite', essentially, of what the question asks to check.
• More for exam 2, but with worded questions you sometimes need to restrict domains accordingly. You have to THINK, will this answer work in real life? For example, you can't have a negative length, negative time etc.
• Make sure you read the right numbers/signs off your calculator. It takes half a second to double check, but so easy to skip a negative sign or read a number wrong.
• Check that the answer works. This may seem like common sense, but so easy to lose marks on. If you are asked for the height of a building and you get 0.01 cm, then clearly that answer is wrong.
• Be wary of long, complicated working out and answers. I can tell you this from experience of the many practice exams I did, but if you get something like 10234/2278438484 (especially for exam 1 when you have to add by hand) then 9 times out of 10 that answer is probably wrong. For exam 1 they are not testing your adding skills, so they are not likely to make you do a long, complicated sum. Honestly, most answers will be simple (though that doesn't exclude surds, some fractions and pi). So just double check if you're answer is a weird, long number. This doesn't mean it definitely ISN'T the answer, but it's not likely to be.
• READ THE QUESTION! I cannot stress this enough. READ THE QUESTION! If it asks for factors of a polynomial DO NOT GIVE SOLUTIONS!
• Also, READ THE QUESTION! If it asks for x values of intercepts, don't give them co-ordinates and vice versa.
• Use correct pronumerals. I nearly lost a mark on exam 1 for this. If it gives you an equation h = 2a then DON'T use x and y on the graph or say 'dy/dx' when it should be 'dh/da'.
• Check working AS YOU GO, especially with MC. You might not have as much time as you think to go over the paper, ESPECIALLY if there is a difficult question.
• Watch modelling questions. If Day 1 is at x=0, then Day 6 will be at x=5 NOT x=6.

GRAPHS:

• Most people don't worry about this, but it is very easy to lost marks for having a dodgy graph shape. In exam 2 always plot the graph on your calculator BEFORE sketching, and make sure the scale is the same on the page as it is on your calculator.
• With addition of ordinates, sub as many points as possible. Really easy to get the shape wrong for some of these.
• Don't neglect asymptotes. Your calculator will not show these.
• Do graphs in pencil, and then go over them with pen/highlighter as you wish.
• Make sure your stationary points are FLAT at that particular point. This is especially important for stationary points of inflection as people often miss this.

CALCULUS:

• When calculating areas, do yourself a favour and draw the graph. Do not assume it is all above the x-axis
• 'Use calculus' means USE CALCULUS. Especially for exam 2, you must show your working out by hand (though you can do the 'steps' on the calculator).
• Watch max/min problems. The maximum is NOT ALWAYS the turning point. It may infact be an endpoint on a restricted domain. Always, always plot the graph of everything where possible to ensure you don't do silly things like this.
• When differentiating/antidifferentiating don't forget to change cos to sin and sin to cos. Very easy to miss, especially if it is 221sin(23x-1183) or something like that.

FUNCTIONS:
• f(x/2) is wider than f(x). f(2x) is narrower than f(x), even though you might assume as 2x > x, the graph must be wider. This is WRONG.
• Been said before, but be careful with domains and ranges of composite functions. Remember the range of the inner function must be a subset of the domain of the outer function.

That's pretty much most of the stuff I had. Sorry about the lack of probability, I've forgotten how to do most of it... I might add to this later if I think of anything else...

Anyway, good luck guys! Just remember to be really careful when checking over your work, as you don't want to be losing unnecessary marks. Don't panic when you see a hard question, just remind yourself that if it's on the exam, it's in the study design and therefore you KNOW how to do it. I wish you guys all the very best for Tuesday & Wednesday, and I know you'll all ace it!
« Last Edit: May 06, 2012, 05:16:44 pm by VegemitePi »

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Re: Mathematical Methods CAS Resources
« Reply #10 on: May 06, 2012, 05:16:29 pm »
0
Methods Exam Checklist
paulsterio

Both Exams:
General:
Remember to read the question again, once you have finished and ensure you have answered it
Check that decimals are used where required and that they are to the correct accuracy (decimal places)
Check to see that you have transcribed all information given correctly, don't make mistakes copying equations
Check especially for adding, multiplying, subtracting and division errors

Functions:
When finding the inverse of a function, remember to write: "for inverse swap x and y"
If you are asked for find a composite function, remember to check if it exists
The composite function of some function and it's inverse will be y = x
It may be faster, when trying to find the intersection points between two inverses to find the intersection with y = x for one of them
If evaluating g(x) = a and you have the inverse, just find g^-1(a)
When giving the general solution to trigonometric equations, remember to write that "k" or whatever variable you use is an integer
For transformations where they give you a matrix, it is safer to multiply the matrix out and then substitute into the equation, but it may be faster to use recognition
Remember that similar triangles may be on the exam

Calculus:
When using the product of quotient rule, remember to state the rule
The derivative of f(x) is f'(x), the derivative of y=... is dy/dx = ...
Sometimes you're asked to find the derivative at a certain point, remember to do so, not just find the derivative function.
Remember to put the "+c" in antiderivatives, unless "an antiderivative is asked for
Remember the "dx" at the end of the integral

Probability:
When solving questions to do with conditional probability, remember to include the rule
Remember to include the statements X~Bi(n,p) and X~N(m, var) when dealing with Binomial and Normal
No Calculator syntax - No invNorm
To express normcdf in the correct way, write, for example, X~N(1, 0). Pr(X>10) = ...
To express inverse normal in the correct way, write, for example, Given that X~N(1,0) and Pr(X>a) = 0.5, a = ...

Thanks to Daliu
1. (x^2)/|x|=|x| [that is, x squared divided by mod x is equal to mod x]
2. Probabilities are always 0<p<1 (actually meant to be "0 more than or equal to p more than or equal to 1", couldn't type it though...)
3. If you log something, whatver is inside the log HAS to be above zero (and not including zero). ln(x) where x<0 doesn't exist.
4. If given a probability distribution function, you HAVE to draw the parts of the the function where f(x)=0 as well. Otherwise you get marks taken off.
6. Period of tan(nx) is pi/n, not 2pi/n

Thanks to BoredSaint
'Define the Variable in Probability' - as in' - "Let X be the number of...."

Advice for using a CAS in Extended Response Questions (Most applies to all CAS)
How to use a CAS to evaluate areas, showing full working
- Write down the integral statement for the area, for example, the integral of x^2 with respect to x from 0 to 5
- Type the function into the CAS, without the bounds, and get the antiderivative
- Now write the antiderivative and put in the correct bounds, using the square brackets
- Now, by hand, substitute the numbers into the anti-dervative, so F(a) - F(b), but don't evaluate it
- Go back to the CAS, and enter in the integral, this time with bounds, then copy the answer across to your paper
- So you've just worked out an area, supposedly showing "full working" and "using calculus" but you're assured of a right answer

How to use a CAS to find derivatives, showing full working
- Say we want to find the derivative of a complicated function, but it's worth 3 marks, this is what I'd do
- Determine the rule to be used. Say it's a quotient
- First, let u=... and v=...
- Now write down the rule dy/dx = (v.du/dx - u.dv/dx)/v^2
- Go to your CAS, and find du/dx and dv/dx
- Substitute all into the rule, but leave unsimplified - dy/dx = ( (......) x (........) - (.........) x (........))/(.......)
- Now use the CAS to find the derivative, dy/dx
- Copy it down, and voila, 100% correct derivative

Finding f(x) given f'(x) - a shortcut
- If we know a derivative and a point on the curve f(x), there is a shortcut to solving it
- It's using a command on the CAS called dSolve - for the ClassPad
- In the first column, type y'=...(derivative)...
- Independent Variable - x
- Dependent Variable - y
- Initial condition, type, for example if we had the point (1, 5) - "x=1,y=5"

Finding f(x) given f'(x) - a shortcut using definite integrals
- Similar to above but for people on TI Calcs - may be a fast way
- Type the integral sign with bounds, but instead of using x, use another letter, for example t
- So type in the integral sign, and then the derivative using t instead of x
- Now look at your initial conditions, say you have the point (0,5)
- Put the lower bound as your x-co-ordinate "0"
- Put the upper bound as the variable "x"
- Now after the integral (i.e. after the dt) put + the y-co-ordinate so here you would put +5
- Remember it's "dt" not "dx"
- Hit enter, and you should get your function of x

Finding a,b,c...etc in equations knowing the points
- You can use the regression function to check that your values are correct

pi

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Re: Mathematical Methods CAS Resources
« Reply #11 on: May 06, 2012, 05:18:27 pm »
+2
Guide to Probability Notation
luken93

So there's been a few questions regarding the proper notation to be used on the exam, so I thought it's probably time to make a thread

I'm assuming this is right, but feel free to pick me up on anything I've missed:

Normal Distribution
The range of scores on a particular test are such that they hold a mean of 60 with a standard deviation of 4.

NORMAL CDF
a) Find the probability of a student's scores lie between 55 and 62

On the TI-Nspire; $\text{normcdf(55, 62, 60, 4)}$
On the classpad; $\text{normcdf(55, 62, 4, 60)}$

On paper:
Let X be the range of scores of students on the test.
$X \sim N(60, 16)$
$Pr(55 < X < 62) = 0.5858$

NORMAL PDF
b) Find the probability that a student's score is 61

On the calculator; $\text{normpdf(61, 60, 4)}$

On paper:
Let X be the range of scores of students on the test.
$X \sim N(60, 16)$
$Pr(X = 61) = 0.0967$

INVERSE NORMAL
c) 75% of students passed the test. Find the score needed to pass the test.

On the calculator; $InvNorm(0.25, 60, 4)$

On paper:
Let X be the range of scores of students on the test.
$X \sim N(60, 16)$
Let a be the minimum value needed to pass, Find a such that $Pr(X > a) = 0.75$
$\therefore Pr(X < a) = 0.25$
$a = 57.3020$
Therefore a score of 57.3020 is needed to pass.

Binomial Distribution
BinomPDF
d) The probability that a particular student passes the test is 0.4.
i) If this student sits 3 tests that are independent to the other, what is the probability that the student passes 2 out of 3 tests.

On the calculator; $binomPdf(3, 0.4, 2)$

On paper:
Let Y be the performance of the student in the 3 tests
$Y \sim Bi(3, 0.4)$
$Pr(Y = 2) = 0.288$

BinomCDF
ii) This particular student needs to pass at least one of the tests to make his parents happy. What is the probability that his parents will be happy?

On the calculator; $binomCdf(3, 0.4, 1, 3)$

On paper:
Let Y be the performance of the student in the 3 tests
$Y \sim Bi(3, 0.4)$
$Pr(1 \leq Y \leq 3) = 0.784$

Number of Trials
iii) The student is now getting very worried about his parents. To ensure that the probability that his parents are happy is 0.95, how many tests will he have to sit if he has to pass at least one of them?

On paper:
Let Y be the performance of the student in the m number of tests
$Y \sim Bi(3, 0.4)$
Find m such that $Pr(Y \geq 1) > 0.95$
$\therefore 1 - Pr(Y = 0) > 0.95$
$\therefore Pr(Y = 0) < 0.05$
$\binom{m}{0}(0.4)^0(0.6)^{m} < 0.05$
$(0.6)^{m} < 0.05$
$m > 5.86$
$\therefore$ the student will need to sit 6 tests.
Hopefully that's all correct!
« Last Edit: May 06, 2012, 05:26:32 pm by VegemitePi »

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Re: Mathematical Methods CAS Resources
« Reply #12 on: May 06, 2012, 05:18:37 pm »
0
Paul's Mathematical Methods - Pre-Exam 1 Advice for 2012
Paulsterio

Well, there's 4 - 5 days before the Methods Exam 1 and this is my little bits of advice for anyone who doesn't know what to do 5 days out from the exam. Exam 2 advice will be up within the next two days.

Preamble

Essentially, in order to do well in Exam 1, we have to know its purpose and that is, to test low-level mechanical skills by hand. Exam 1 isn't an applications exam like Exam 2 is, it doesn't usually require much skill in the art of evaluating and dissecting applied questions, but what it requires is two main things, proficiency in the "by-hand" skills taught in Methods and good accuracy with regards to mental arithmetic.

The Big 15

If we analyse past Exam 1's, we can see that there is a certain trend, particular questions tend to crop up time and time again and certain skills need to be applied time and time again. Here are some of the important by-hand skills which you will need to have mastered by this stage:

1) Transformations - knowing how to apply transformations to an equation/graph + describe transformations using words
2) Knowing the basic derivatives on the formula sheet or knowing how to apply them from the formula sheet
3) Using the three main differentiation rules - the chain rule, the product rule and the quotient rule
4) Knowing the basic integrals on the formula sheet or knowing how to apply them from the formula sheet
5) Being able to find the tangent and normal (or their gradients only) at particular points
6) Finding the area under a curve
7) Integration by recognition
8 ) Knowing how to solve the three fundamental types of equations taught in methods - polynomial, logarithmic/exponential, trigonometric as well as addition of ordinates
9) Knowing how to find both general solutions and restricted-domain solutions for trigonometric functions
10) Find an inverse function
11) Being able to draw the graphs of the functions learnt in methods - polynomial, log/exp, trigonometric
12) Basic (foundation) probability (the stuff from Year 11)
13) Knowing fundamental probability facts for discrete and continuous distributions (e.g. probabilty sums to 1...etc.)
14) Knowing how to find the mean, mode and median for probability distributions
15) Knowing how to evaluate basic binomial distributions, finding the mean, mode and median as well as solving basic applied problems

My guess is that there won't be any normal distribution questions on Exam 1 - purely because they are usually CAS-based, but if anything, I would also learn how to normalise a variable (find the Z-score). Generally, if you are good at the above skills and can do all of them by hand, you're pretty much set for Exam 1. You'll see that the things I have listed also very commonly come up in Exam 1's over the past few years, so take note.

The Feared 3

Many good students in Methods, including myself last year, aim to get 40/40 in Exam 1 because it is perceived to be easier than Exam 2. However, what is the actual issue with Exam 1 and why is it that many students who aim to get 40/40 don't actually end up getting that score? I think it boils down to three main key issues and if you're able to nail these three key issues, you'll have a much higher chance of getting that 40/40.

1) Careless mistakes
2) Assumed mathematical knowledge from earlier years (especially geometry)
3) Lack of solid mathematical reasoning

Overcoming "The Feared 3"

1. Careless Mistakes

The only way to overcome careless mistakes is to be meticulous with your working and be careful. No matter how many times you check over your exam, nothing beats doing it right the first time. I hate taking shortcuts and I hate not writing as much as I possibly can without being ridiculous - like seriously if you can write it, why wouldn't you - here's an example of meticulous working out for an Exam 1 question which I did last year.

Example 1 - Good Setting out and Good Working out

2. Assumed Mathematical Knowledge

In order to overcome this, you need to be familiar with maths as a whole, not just the methods study design and curriculum, you need to be familiar with maths in order to be good at it, just like you need to actually understand the innards of physics and chemistry to score highly in them. Mathematics is not a mechanical subject, you're not meant to be calculators, you're meant to be mathematicians who can think.

Examples of this include:
- Example 2 - VCAA 2010 MM (CAS) EX1 - Q11 - "Cone of Death"

3. Lack of Solid mathematical reasoning

In order to attain better mathematical reasoning, you have to think mathematically and do more practice exams - the more practice and exposure you get, the better your reasoning becomes, there are no ways around this!

Examples:
- Example 3 - VCAA 2011 MM (CAS) EX1 - Q9

Last Week Before the Exam! Practice Exams?

Yes, now is the time for you (if you haven't started) to really start doing exams under perfect exam conditions. This includes doing them to exact time and making the marks count. I know that most of the time when I did practice exams, I didn't really care, so I just rushed, looked at the answers...etc. Don't do this! Set yourself an aim and reward yourself - "if I get 40/40 for this exam, I will get Maccas" - if you're able to really simulate exam conditions, you'll know how well you can work under pressure and under adrenaline.

Now, after you've done all the practice exams you've wanted to do, it's important you go over your mistakes. Redo the questions, make sure you can do them, if not, look at the solutions and keep trying until you can do them. You haven't finished a practice exam and gotten everything out of it until you can do ALL of the questions and get 100% on that exam, if you're not there, you have heaps to improve on, so why are you doing more practice exams when the ones you've been doing aren't perfect? Review them first, then move on.

Other Resources
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods CAS Resources

There comes a time during our preparation where we become obsessed with marks, the moment where getting a 40/40 on our Exam 1 becomes more important than enjoying the maths we do and enjoying the learning and the applications that mathematics provides. Remember that no matter what happens on the exam, we have spent two years learning Methods, in many ways, what's even more important than doing well on these exams is what we've been able to gain over that period of time.

I wish you all well, and I hope you all perform to your desired potentials come the 7th of November.
« Last Edit: November 17, 2012, 11:25:08 pm by pi »

pi

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Re: Mathematical Methods CAS Resources
« Reply #13 on: May 06, 2012, 05:18:54 pm »
0
Paul's Mathematical Methods - Pre-Exam 2 Advice for 2012
Paulsterio

Preamble

In order to do well on Exam 2, we need to recognise what skills it is testing. Whilst Exam 1 is a test of our mechanical computational skills, Exam 2 is a test of our ability to mathematically reason, applying what we have learnt in Methods to the cases and scenarios presented. It requires almost next to no ability to do things by hand, given that it is CAS-assisted, even questions which seemingly ask to be solved by hand can usually be done with a CAS.

Thus, in order to excel in Exam 2, we need to be familiar with a CAS and we need to really be able to understand and mathematically interpret the scenarios at hand.

Multiple Choice Questions

Multiple Choice questions usually test a wide variety of elements within the course and the only real way to be prepared for them is to have a sound THEORETICAL knowledge of the whole Mathematical Methods course. Statistically speaking, you will find that of the 22 MC questions, you will generally be able to group them into 4 categories:

1) Purely CAS questions

These questions can purely be solved using a CAS and don't really require any sort of interpretation. It could, for example, be something like "solve this equation". One thing to watch out for with these sorts of questions is answer forms. What I mean is that there are often more than one way to express an answer - e.g. log9 and 2log3 are the same. So if the answer your CAS gives isn't present, look out for equivalent options. (Tip, if you can't find it, use DECIMALS! It will always reveal the right answer). Usually, there are quite a few of these questions, 30%-ish of MC will be them.

2) Purely interpretational questions

An example of this sort of question would be "find the amplitude and period of the following trigonometric function" - they are questions which only require you to look at a bit of information (for example an equation for a trig graph) and make a deduction (the amplitude and period) - they require little mathematical calculation and they require no usage of the CAS. You usually get a few of these, not many, but they are easy - especially given that they can be done in a small amount of time - around 20% of your MC will be this sort of question.

3) CAS + Interpretational Questions

These questions require both the use of a CAS and making mental mathematical deductions. An example of this sort of question would be the majority of MC probability questions, which require you to make a set-up or interpretation first, and then use the CAS to evaluate it. This will probably make up the most of your MCs, around 40%, and they are usually harder than the other two previous types, so beware of these, if you're able to do really well at this type, you have a chance of scoring 20/20 for MC. Other examples of this sort of question might involve graphs (which of the following is the right graph for this equation - so you have to graph it on your CAS and interpret it)...etc.

4) Curveball Questions

Then you have the questions which are curveballs, sort of different to the rest. An example of this sort of question was on last year's Methods Exam. There was a question where the logaithmic change of base rule had to be used, it wasn't a CAS question, nor was it really an interpretational question, it wasn't really both either, it was just something unexpected, not many people expect the change of base rule to feature. There will always be one or two curveball questions on every exam paper, so beware of them, they're the differentiators between the best and the very best, essentially what sets apart the 45+'s from the 50's.

General MC Tips

- MC is worth 22 marks, which is 1/4-ish of the paper, this means you should be spending a MAXIMUM of 30 mins on MC.
- Aim for an average of 1 question per minute, that way, you can finish MC in 20 mins, which leaves you more time for Extended Response questions.
- Be careful with MC questions which require interpretation, the answer options aren't chosen randomly, they're out there to trick you!
- You can usually do around half of the MC questions in your head during reading time if you really wish to
- Be proficient with the CAS - it will help you, generally students who are good with the CAS will find that they can complete the MC with more speed and accuracy.

Extended Response Questions

Extended Response questions are often the most feared, but generally, I tended to like them, for two main reasons:
- They didn't actually include that much maths - most of the maths is done using a CAS
- They actually require you to think and apply your skills, which is rarely seen elsewhere on the course

Thus, you have to approach ER questions with this mindset. You can't approach them in a mechanical manner and expect to do well, you have to interpret the information they have given you and ask yourself how you can build equations, formulae and a mathematical set up from it. Once you have your set-up, it's all just CAS from there-on in.

Generally you will get either 4 or 5 Extended Response Questions in Methods, 1 of them will probably be an algebraic one which involves solving a few equations and drawing a graph, all things which can be done on a CAS. One of them will definitely be a probability one, so that is mostly CAS as well, but be familiar with probability, you need to know that much to set it up. It will probably involve the normal distribution somewhere within the question, so be familiar with those commands on the CAS as well. Then you always have the difficult last question, which usually involves calculus and some sort of minimisation/maximisation sort of question. These are difficult because students often have run out of time or they no longer have the energy and mental stamina to solve them. One way to test this out is to do the following, take a past VCAA exam and just do the last question, sure it'll be hard, but it won't be THAT hard, because when you're fresh and thinking straight, you'll find it much easier.

General Tips for ER Questions

- Take your time and think the questions through, if there is a really easy question, you might not have fully grasped what it's asking.
- Always use correct notation, be mindful of how many marks are allocated and use that to guide your working out
- Whenever unsure, always put MORE working out than you think is necessary - you can never have too much - don't be lazy
- Always keep the instinct of using the CAS on your mind, you want to use it as much as possible, but know its limitations
- Never use CAS notation, always use the correct mathematical forms

CAS Tips for ER Questions

How to use a CAS to evaluate areas, showing full working
- Write down the integral statement for the area, for example, the integral of x^2 with respect to x from 0 to 5
- Type the function into the CAS, without the bounds, and get the antiderivative
- Now write the antiderivative and put in the correct bounds, using the square brackets
- Now, by hand, substitute the numbers into the anti-dervative, so F(a) - F(b), but don't evaluate it
- Go back to the CAS, and enter in the integral, this time with bounds, then copy the answer across to your paper
- So you've just worked out an area, supposedly showing "full working" and "using calculus" but you're assured of a right answer

How to use a CAS to find derivatives, showing full working
- Say we want to find the derivative of a complicated function, but it's worth 3 marks, this is what I'd do
- Determine the rule to be used. Say it's a quotient
- First, let u=... and v=...
- Now write down the rule dy/dx = (v.du/dx - u.dv/dx)/v^2
- Go to your CAS, and find du/dx and dv/dx
- Substitute all into the rule, but leave unsimplified - dy/dx = ( (......) x (........) - (.........) x (........))/(.......)
- Now use the CAS to find the derivative, dy/dx
- Copy it down, and voila, 100% correct derivative

Finding f(x) given f'(x) - a shortcut
- If we know a derivative and a point on the curve f(x), there is a shortcut to solving it
- It's using a command on the CAS called dSolve - for the ClassPad
- In the first column, type y'=...(derivative)...
- Independent Variable - x
- Dependent Variable - y
- Initial condition, type, for example if we had the point (1, 5) - "x=1,y=5"

Finding f(x) given f'(x) - a shortcut using definite integrals
- Similar to above but for people on TI Calcs - may be a fast way
- Type the integral sign with bounds, but instead of using x, use another letter, for example t
- So type in the integral sign, and then the derivative using t instead of x
- Now look at your initial conditions, say you have the point (0,5)
- Put the lower bound as your x-co-ordinate "0"
- Put the upper bound as the variable "x"
- Now after the integral (i.e. after the dt) put + the y-co-ordinate so here you would put +5
- Remember it's "dt" not "dx"
- Hit enter, and you should get your function of x

Finding a,b,c...etc in equations knowing the points
- You can use the regression function to check that your values are correct

Extra Resource - Re: Mathematical Methods Guides and Tips (b^3's TI nSpire Guide)

Final Tips for the 2012 Exam 2

General:
Remember to read the question again, once you have finished and ensure you have answered it
Check that decimals are used where required and that they are to the correct accuracy (decimal places)
Check to see that you have transcribed all information given correctly, don't make mistakes copying equations
Check especially for adding, multiplying, subtracting and division errors

Functions:
When finding the inverse of a function, remember to write: "for inverse swap x and y"
If you are asked for find a composite function, remember to check if it exists
The composite function of some function and it's inverse will be y = x
It may be faster, when trying to find the intersection points between two inverses to find the intersection with y = x for one of them
If evaluating g(x) = a and you have the inverse, just find g^-1(a)
When giving the general solution to trigonometric equations, remember to write that "k" or whatever variable you use is an integer
For transformations where they give you a matrix, it is safer to multiply the matrix out and then substitute into the equation, but it may be faster to use recognition
Remember that similar triangles may be on the exam
When you define a variable, you should make it clear "Let $x$ be"
Always solve equations using the CAS - set up the equation, then use the CAS to solve it
Be very familiar with the graphing screen, including the different types of Zooms and what can be found on the graphing screen

Calculus:
When using the product of quotient rule, remember to state the rule
The derivative of f(x) is f'(x), the derivative of y=... is dy/dx = ...
Sometimes you're asked to find the derivative at a certain point, remember to do so, not just find the derivative function.
Remember to put the "+c" in antiderivatives, unless "an antiderivative is asked for
Remember the "dx" at the end of the integral
When a question says "use calculus" - you must show the derivative or antiderivative, HOWEVER, I would suggest that you always show the derivative or antiderivative (where possible)
Always do derivatives and integrals using the CAS, never do them by hand
Know how to find both local and global minima and maxima using the CAS
Know how to do a linear approximation using the CAS

Probability:
When solving questions to do with conditional probability, remember to include the rule
Remember to include the statements X~Bi(n,p) and X~N(m, var) when dealing with Binomial and Normal
No Calculator syntax - No invNorm
To express normcdf in the correct way, write, for example, X~N(1, 0). Pr(X>10) = ...
To express inverse normal in the correct way, write, for example, Given that X~N(1,0) and Pr(X>a) = 0.5, a = ...

1. (x^2)/|x|=|x| [that is, x squared divided by mod x is equal to mod x]
2. Probabilities are always 0≤p≤1
3. If you log something, whatver is inside the log HAS to be above zero (and not including zero). ln(x) where x<0 doesn't exist.
4. If given a probability distribution function, you HAVE to draw the parts of the the function where f(x)=0 as well. Otherwise you get marks taken off.
6. Period of tan(nx) is pi/n, not 2pi/n

Last Week Before the Exam! Practice Exams?

Yes, now is the time for you (if you haven't started) to really start doing exams under perfect exam conditions. This includes doing them to exact time and making the marks count. I know that most of the time when I did practice exams, I didn't really care, so I just rushed, looked at the answers...etc. Don't do this! Set yourself an aim and reward yourself - "if I get X/80 for this exam, I will get Maccas" - if you're able to really simulate exam conditions, you'll know how well you can work under pressure and under adrenaline.

Now, after you've done all the practice exams you've wanted to do, it's important you go over your mistakes. Redo the questions, make sure you can do them, if not, look at the solutions and keep trying until you can do them. You haven't finished a practice exam and gotten everything out of it until you can do ALL of the questions and get 100% on that exam, if you're not there, you have heaps to improve on, so why are you doing more practice exams when the ones you've been doing aren't perfect? Review them first, then move on.

Other Resources
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods Guides and Tips
Re: Mathematical Methods CAS Resources

There comes a time during our preparation where we become obsessed with marks, the moment where getting a 80/80 on our Exam 2 becomes more important than enjoying the maths we do and enjoying the learning and the applications that mathematics provides. Remember that no matter what happens on the exam, we have spent two years learning Methods, in many ways, what's even more important than doing well on these exams is what we've been able to gain over that period of time.

I wish you all well, and I hope you all perform to your desired potentials come the Mathematical Methods Exam 2.
« Last Edit: November 17, 2012, 11:25:17 pm by pi »

pi

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Re: Mathematical Methods CAS Resources
« Reply #14 on: May 06, 2012, 05:27:36 pm »
+4
The foolproof guide to transformations
Ancora_Imparo

Transformations can be a very sticky topic to deal with, especially at first. Many schools teach the 'Dash' or 'Mapping' method. Below is another method that may be easier to understand and requires very little working. Examples are also shown below.

The most important thing is that you understand one method really well and you consistently use that method for all problems that you solve. If you are confused with one, don’t use it.

-----------------------------------------------------------------------------------------------------------------------------------------------------------------
The ‘Function’ Method

Dilations
kf(x): Dilation by a factor of k from the x-axis or parallel to the y-axis (up-down stretch)
Ie: Multiply the whole function by a factor 'k'.

f(kx): Dilation by a factor of 1/k from the y-axis or parallel to the x-axis (left-right stretch)
Ie: Put a 'k' in front of every 'x' you see in the function.

Reflections
-f(x): Reflection over/in the x-axis
Ie: Multiply the whole function by -1.

f(-x): Reflection over/in the y-axis
Ie: Put a minus sign in front of every 'x' you see in the function.

Translations
f(x)+k: Translation of k units in the positive direction of the y-axis
Ie: Add 'k' to the end of the function.

f(x-k): Translation of k units in the positive direction of the x-axis
Ie: Put a 'minus k' after every 'x' you see in the function.

When listing transformations, list them in the following order:
Dilations, Reflections, Translations (DRT) or
Reflections, Dilations, Translations (RDT)

If you are trying to transform a complex function into a simple one, working backwards is usually easier (ie: translations first, then dilations and reflections).

Eg: State the transformations that map $y=\sqrt{x}$ to $y=-3\sqrt{2x-4}-1$.
Ans:
1) Dilation by a factor of 3 from the x-axis: $y=3\sqrt{x}$
2) Dilation by a factor of 1/2 from the y-axis: $y=3\sqrt{2x}$
3) Reflection over the x-axis: $y=-3\sqrt{2x}$
4) Translation of 2 units in the positive direction of the x-axis: $y=-3\sqrt{2(x-2)}=-3\sqrt{2x-4}$
5) Translation of 1 unit in the negative direction of the y-axis: $y=-3\sqrt{2x-4}-1$

Eg: State the transformations that map $y=-5(3x+6)^2+2$ to $y=x^2$.
Ans:
1) Translation of 2 units in the negative direction of the y-axis: $y=-5(3x+6)^2+2-2=-5(3x+6)^2=-5(3(x+2))^2$
2) Translation of 2 units in the positive direction of the x-axis: $y=-5(3(x+2-2))^2=-5(3x)^2$
3) Reflection over the x-axis: $y=-(-5(3x)^2)=5(3x)^2$
4) Dilation by a factor of 1/5 from the x-axis: $y=\frac{1}{5}×5(3x)^2=(3x)^2$
5) Dilation by a factor of 3 from the y-axis: $y=(\frac{1}{3}×3x)^2=x^2$
« Last Edit: March 13, 2013, 08:04:49 pm by pi »