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pi

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Specialist Mathematics Resources
« on: November 21, 2011, 03:31:39 pm »
+10
AN Specialist Mathematics Resources
This thread is designed to keep an index of useful threads contributed by our users. If you think a link or thread should be added to this list, or if you would like to contribute resources that you hold copyright to, or are legally available for free distribution, please PM a moderator. We do not condone illegal distribution of copyright materials. If you are seeking practice exams, have a read of the advice in this thread: Practice Exams - Where to get them for free?

Please thank the original authors in their thread. Any additional questions can be posted here.

Guides
Guide to Using the TI-Nspire for SPECIALIST - b^3
All you need to know about inequalities! - TrueTears
General solutions to circular functions - TrueTears
Trinon's Guide to Sketching Trig Graphs - Trinon
Trinon's Guide to Anti-derivatives through derivatives - Trinon
Techniques for Sketching Nice-Looking Graphs - pi
Vector Proofs - ClimbTooHigh
Volumes of Solids of Revolutions: How-To - EulerFan101

Tips
Compilation of Tricky Points + Nifty Stuff Part 1 - kyzoo
Compilation of Tricky Points + Nifty Stuff Part 2 - kyzoo
Specialist Exam 1 - Tips and Predictions - trinon
Calculator Tricks - abes22
5 Simple Tips for Success in VCE Mathematics - Zealous
101 Days Before VCE Maths Exams (Methods/Specialist) [Guide] - Sine


Trial Exams
Puffy 2011 - Specialist Maths - ATARNotes Trial Examination - Paulsterio and Luffy
Re: Specialist Mathematics Resources - Practice exams error list - |ll|lll| (barcode)
VCE PAST PAPER BOOKLETS! - EEEEEEP

Notes
ATAR Notes Specialist Maths Notes
Mao's Bound Reference - Mao
pi's bound reference - pi

Challenging Problems
Recreational Problems (SM Level) - Ahmad
Super-fun Happy Maths Time - dcc
Fun Questions - TrueTears
Resources/Guides: For advanced students - TrueTears

Other
The Matrix Cookbook
Maths Links - Ahmad
GMA Resources - pi
How to choose a CAS calculator? - pi
Generalised Textbook Summaries - pi
Difficult Questions from past VCAA papers- Spesh - insanipi
Euler's Method Program for TI-Nspire - zsteve
« Last Edit: July 22, 2017, 05:26:50 pm by MightyBeh »

pi

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Re: Specialist Mathematics Resources
« Reply #1 on: November 21, 2011, 03:33:33 pm »
+4
Maths Links
Written by Ahmad with contributions by /0 and pi
Original thread

Upon request I will share some useful maths links, which do not necessarily relate to specialist maths, but which are nonetheless useful and interesting.

Our own Mao's bound spesh/muep notes!

Purplemath covers many methods/specialist maths topics

Integrating using Euler's formula

Project Euler

Mathscentre

Math course notes from MIT!

The Art of Problem Solving Forum

Calculus Videos

Nick's Mathematical Puzzles

James Stewart's Calculus Challenging Problems

Complex Numbers and Trigonometry

Algebra through Problem Solving

Heaps of ebooks!
More ebooks

Generating functions, and others

Project Euler, programming
Maths Challenge, (same owner as Project Euler, but no programming)

Euclidean Geometry Notes (PDF)

Trigonometric Delights!

Combinatorics Notes

Terence Tao's Blog

Visual Complex Analysis - NOTE: DJVU FORMAT

Paul's online maths notes

Calculus Lecture Notes

William Chen's Lecture Notes


Some links by /0


Free Calculus Videos
http://midnighttutor.com/math_tutor_online.php

More Calculus Videos (With QuickTime you can open them in a new window and download them [sweet, collecting math videos is so awesome!])
http://online.math.uh.edu/Math1431/
http://online.math.uh.edu/HoustonACT/videocalculus/index.html (Some videos here might overlap with the previous link)

A lot of you probably know these sites, but whatever
http://itute.com/
http://www.wolframalpha.com/

Some Integration Exercises (great fun)
http://archives.math.utk.edu/visual.calculus/4/integrals.2/index.html

Interesting sin(nx) and cos(nx) equations
http://www.trans4mind.com/personal_development/mathematics/trigonometry/deMoivre.htm

Youtube:
MathTV's playlists include Derivatives, Integrals and Sequences as well as other videos. Examples given are easy but explanations are brilliant
http://www.youtube.com/user/MathTV
I found this video very useful for graphing trig: http://www.youtube.com/watch?v=s_NI50p-pcg

Khanacademy makes lots of videos about math and physics! The presentation may feel a bit sloppy but explanations are also great.
http://www.youtube.com/user/khanacademy

Donylee makes math and physics vids, a lot of it is advanced stuff, but some calculus is still relevant and it might be more useful for uni math'ers
http://www.youtube.com/user/donylee

Music Videos:
http://www.youtube.com/watch?v=BjNFzlTdWgo - Displays trig formulae
http://www.youtube.com/watch?v=APmW3iwgbFE - Another trig formulae vid
http://www.youtube.com/watch?v=DIRYQTXrBko - Displays calculus formulae
« Last Edit: April 27, 2014, 08:23:11 pm by pi »

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Re: Specialist Mathematics Resources
« Reply #2 on: November 21, 2011, 03:35:14 pm »
+8
Compilation of Tricky Points + Nifty Stuff Part 1
Written by kyzoo
Original Thread

EXAM TECHNIQUE

●  For Exam 2, ALWAYS ALWAYS ALWAYS USE THE CALCULATOR IF YOU CAN. Try to never do stuff by hand even if you; even simple stuff like “3+2”

●  Always use the maximum possible amount of working out steps you can. DON'T SKIP STEPS, NEVER DO MENTAL ARITHMETIC.

●  Always draw diagrams first before you start solving the question
E.g. for tank inflow outflow problem, always convert the information into a diagram before you start answering the questions

●  Always draw large diagrams

GENERAL

●  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. If it doesn't ask for decimal places, assume it is asking for exact solution

● 
=>

~ Whenever there is an even denominator present when "powering" the equation, the sign is necessary

●  Put all your solutions into the last line for clarity

●  Line up your "= signs" to make it easier for examiners to read your solution

●  For proof question, always state what you have proven at the end

●  Always simplify answers at the end. Always rationalize any surds.

●  Do not confuse SIGNIFICANT FIGURES with DECIMAL PALCES




●  Always check to see if the domain is restricted when they give you a function

DRAWING GRAPHS

●  Always label curves with its corresponding equation ...

●  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).

~ From Derrick Ha book: If you need to draw an or asymptote, draw them directly beside the axis, rather than on

●  When drawing graph lines, put arrows on the end of the lines to indicate that they continue on

●  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

●  Dr. He: A dotted line alone is not an asymptote. You need to indicate that it is an asymptote with the label

●  Always label axis-intercepts with co-ordinates rather than a single number. Label y-intercept as and x-intercept as

●  For reciprocal functions, watch out for the horizontal asymptote. Really easy to miss.

METHODS THEORY

FUNCTIONS

●  In an inequation, when you reciprocal the whole thing you must reverse the signs





● 
VCAA 2010 Spesh Exam 2 Question 4 c.)

●  has 3 real solutions . But 2 distinct real solutions

●  is the inverse function.

LOGARITHMS

●  , because c can be any number.

● 


or alternatively





ELLIPSE AND HYPERBOLAS

●  Major axis of an ellipse is the DIAMETER, not the radius. The semi-major axis is the radius

TRIGNOMETRY

●  Derivatives and antiderivatives for sinusoidal functions only work if they are in radians measurements. Thus if it is in degrees, you must convert to radians.




●  is not equivalent to . The former is a relation, the latter is a function. Thus the range of

●  Make a habit of explicitly stating the restricted domain of inverse sinusoidal functions

VECTORS

●  The zero vector is Not

●  Dr He: It is wrong to say that or That is geometry notation, not vector notation

●  Dot product of and is S NOT or even Not having the dot is a big mistake

●  The angle between two vectors is when they are placed tail-to-tail or head-to-head

●  You cannot square a vector. is invalid notation

COMPLEX NUMBERS

●  VCAA 2010 Exam 1 Question 1: Differentiate between roots/solutions and factors of a polynomial equations

●  Must differentiate between and ; complex region of is the entire Argand diagram except for the origin

●  Always label complex regions

●  For The origin is always an open endpoint.

DIFFERENTIATION

●  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

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

●  When stuff is leaking out that is a negative rate of change

●  You are in trouble if This can be any type of stationary point and you need to use a gradient sign test to ascertain it. Do not automatically assume that it is an inflection point. Example:

●  Gradient sign test (Derrick Ha)
Need to give actual values of gradient immediately to left and to the right of the point, rather saying they are >0 and <0




\ _ /
Local minimum at (0,3)

●  In implicit differentiation where you have a relation like , BE VERY CAREFUL TO DIFFERENTIATE THE “8” TO BECOME “0”

●  Although follows fraction laws
(  is not a fraction), does not obey fraction laws.
    
●  Difference between and
is always a function of x.

is not necessarily a function, it is just the gradient of the tangent at a point.

● 

ANTI-DIFFERENTIATION

●  When anti-differentiating an indefinite integral, take care to include the “+ c” part.

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

●  , unless the domain specified otherwise
NB: 2010 Exam 1 Question 7 DID have the domain specified. So you had to shed the modulus and replace them with brackets. You must replace modulus with brackets when the domain is specified

●  . You must have the expression enclosed within a bracket. is two expressions where is an undefined expression

●  Remember to change limits for substitution method

●  Dr He: For substitution, do not change the limits until the integration variable is du













AREA UNDER THE GRAPH

●  Always draw the graph first before finding the area under the curve

●  Avoid integration across asymptotes

●  For solid of revolution, by careful to put in front of the integral term when finding the volume of a solid of revolution.

●  Solids of revolution
~ Be careful you rotate around the right axis
~ Area you rotate must be adjacent to the axis

DIFERENTIAL EQUATIONS

●  Make sure you have an appropriate number of arbitrary constants

●  The slope field curve does not equal

VECTOR FUNCTIONS

●  The domain of the Cartesian equation WILL ALMOST ALWAYS be restricted. Sketch and graphs in order to ascertain the domain and range of the Cartesian equation

●  Put whenever antidifferentiate vector functions

●  When sketching the path of a particle you have to
~ Indicate initial position at 
~ Indicate direction
~ Indicate restricted domain/range and Cartesian equation

●  If they give you and do not exist as the graphs of is not differentiable at

●  Terminal velocity = asymptote velocity, not maximum velocity

DYNAMICS

●  Equation motion is
e.g. -

●  Reaction force DOES NOT EQUAL Normal force. Normal force is a part of the reaction force. Reaction force can act at any angle to the slope

●  F does not equal unless it is on the brink of moving or it is moving

●  Constant velocity means

●  Remember that weight force is , not

●  Easy to forget component of weight force down the slope especially when there is a force towing the object up the slope.

●  When the question states a “push”, this is almost never included as a force in the force diagram, as the force acts for only a moment.
VCAA 2007 Exam 2 Q20

●  Always use force diagram in working out for dynamics question

●  CONVEYOR BELT QUESTIONS EXPLAINED
~ Friction used to pull object. So in this special case, friction is in the direction of motion.

~ can have a value bigger than 1

~ Belt can accelerate faster than object on belt. When this happens the object slips down the belt because of its negative relative acceleration, yet it still has a positive acceleration relative to the ground.

~ Object has maximum acceleration determined by coefficient of friction of belt. Cannot exceed this acceleration, and if belt exceed this, object still remain at maximum acceleration


« Last Edit: May 06, 2012, 02:13:11 pm by VegemitePi »

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Re: Specialist Mathematics Resources
« Reply #3 on: November 21, 2011, 03:35:35 pm »
+4
Compilation of Tricky Points + Nifty Stuff Part 2
Written by kyzoo
Original Thread

GENERAL

●  Unlike in Methods, speed is essential in Spesh. Keep asking yourself – what is the fastest method I can use?

●  When stuck recite a list of concepts

RANDOM GEOMETRY

●  If for a triangle, than angle C is bigger than


METHODS THEORY

●  To find the maximal domain for a question with many parts to it, find the implied domain of each part, then find the union of each part.

(VCAA 2006 Sample Exam 2)

Consider the function f with rule

State the largest domain for which f is defined.

Easy way to do this. Take all the individuals functions of x, figure out the maximal domain for each, then find the intersection of each maximal domain. That’s how you do these types of questions.

For , the maximal domain is
For , maximal domain is
For maximal domain is

Hence the maximal domain for the entire function is

ELLIPSES AND HYPERBOLAS

● 

Prove this by using Pythagoras theorem and ellipse loci along x-axis

●  The asymptotes to the hyperbola are the solutions to 

●  Area of ellipse = .
Not in syllabus, can only be used to check answer

● 
--> circle
a same sign as b --> ellipse
a different sign from b --> hyperbola


TRIGNOMETRY

●  Inverse Circular Function Properties








VECTORS

●  You can multiply both sides of an equation by a dot product

is equivalent to




●  For 




This set of vectors is linearly dependent if determinant of following matrix = 0


●  Often easier to use geometrical methods than vector algebra methods

●  If , this is the acute angle between two vectors. If , this is the obtuse angle between two vectors


COMPLEX NUMBERS

●  In , get rid of the – sign by converting into

●  is an ellipse

● 
Specifies circle with radius “r” and centre “w”

●  is a circle

●  Solving Complex Equations Methods

~ Equating real and imaginary components
~ Completing the square/Quadratic formula
~ Factorising (Includes “fake” factorising with made-up constants)
~ De Moivre’s Theorem
~ Factor Theorem Substitution
~ Difference of two squares
~ +0
~ Conjugate Factor Theorem


●  Complex Conjugate Formulas
~
~
~

DIFFERENTIATION

●  Some derivatives
~
~
~
~

INTEGRATION

●  When you have to find the area adjacent to the y-axis rather than the x-axis, swap the x- and y- axis, then sketch the inverse function. This makes it easier to see the required integral term

●  Integration by parts
Extension of product rule
Let and be functions of




●  Some antiderivatives

~
~
~
~


●  Antiderivative Properties
~
~ If odd function, then
~ If even function, then
~ : prove this with change of variable method


DIFFERENTIAL EQUATIONS

●  Easier way to ensure your dashes in slope fields have correct slope
Say you have a dash at (2, 0) that has gradient 2. This should also pass through the point (3, 2). So use ruler to connect (2, 0) and (3, 2)

VECTOR FUNCTIONS

●  Distance travelled from to is found by evaluating
Distance = Speed x Time = Area under Speed-Time graph

●  Two ways to find intersection of two particle’s paths
~ Simultaneous equation between two Cartesian equations
~ Use and for the two vector equations, then equate coefficients of unit vectors
DYNAMICS

●  When there are two connected objects on different planes, there is only one force in the direction of motion. Every other force is against the direction of motion

CALCULATOR

●  Always change the domain/range for each graph, and zoom in as well. It's so easy to lose marks from not noticing features that you can't observe from afar

●  For partial fractions, use
and

●  Inputting into calculator


●  Using calculator for partial fractions
To go from single fraction to partial fraction form, use Expand
To go from partial fraction to single fraction, use comDenom

« Last Edit: January 04, 2012, 04:05:18 pm by Rohitpi »

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Re: Specialist Mathematics Resources
« Reply #4 on: November 21, 2011, 03:38:42 pm »
+2
Specialist Exam 1 - Tips and Predictions
Written by trinon
Original Thread

Here are my tips:

First and foremost, take a look at practice exams. From what I've done, I've noticed quite a few trends.

There will be a Complex Numbers question in which there's a conversion between Cartesian and Polar form. This will also often include De Moivre's theorem. The rest of complex numbers is a wild guess. There could be a find the factors part in which you may need to use the remainder law or long division. I'm fairly sure there will be an arg/Arg question. Remember that . Take into consideration the positive/negative signs on the x and y values. This is an indication of the quadrant, and therefore the angle that you solve. Don't forget the conjugate factor theorem when working out factors. There will only be a pair of conjugate complex factors if the original equation has no complex components (so no i).

There should be an implicit differentiation question where you may need to find the normal or tangent to the curve. Be careful though, they might ask for the gradient of the tangent and not the equation. In this case you are just finding and subbing in the given value.

I've seen a few trigonometric proofs which can be troublesome. A good knowledge of the various trig equations will be handy.

There will be an integration question (or a find the volume question). This can be coupled (and has been in the past) with trigonometric simplification such as using the double angle formula . I haven't seen many, but they could also trick you with an integration question where you have to find the area and the curve is something like . In this case you'll need to find the inverse function and solve with respect to y instead of x. Don't forget the terminals! As soon as you do anything to an integration equation, remember to fix the terminals! Reading Exam Reports, you'll see that the Chief Examiner (One Doctor Swedosh) comments every year about students forgetting to swap the terminals. It's easy and stupid for you to lose 1 or 2 marks over something so easy and trivial.

There will be a guaranteed dynamics question. This will probably include resolving the i and j components and finding the coefficient of friction, or the tension in the rope, or the acceleration. A few things to look out for. Always check if the plane is smooth or rough. This considerably changes the equation. Also look out for solving the acceleration. They might ask you to solve the acceleration down the plane in which case if you have the i component going up the plane, you'll get a negative acceleration. Simply remove the negative and give indication why you removed the negative and you'll be fine. I personally indicate on the graph which way I'll be resolving the i and j components in, and then at the end I'll say therefore, the acceleration is down the graph. Another handy trick that could gain you a mark is to draw on the given graph the forces. If the question is worth 3 or 4 marks, it is usually expected.

Slope fields is also a guaranteed question. It's fairly new to the course so they will want to test students knowledge. Slope fields are probably the easiest part of the course. You'll be given a equation and the question will tell you which points to draw in the slope. A common mistake is to draw the slope in between the points, or to do them in the correct place but include too many. If the question only asks for the slope at x = -2, -1, 0, 1, 2 and the same for y, only do that. Don't do extra because you will lose marks!. Another key component of slope fields is to draw up a table. Do it whether you like it or not because it will gain you a mark and at the least it won't lose you any marks. It's simple enough, you need two rows, one called y (or x, depending on the given values) and the other called slope or dy/dx. Just fill in the boxes by working out the derivative equation in your head. If it's a multi-part question than they are bound to say at the end, solve the derivative equation from part a and then draw the graph on the axis above (where you drew the slope field). First thing to remember is not to panic. Take a minute to work out 4 or 5 rough points on the graph and then draw a fairly good line. Once done, check that the x and y intercepts are in the right place. This can be a killer for that question.

Euler's Method is a peculiar one. It was on the exam last year but apart from that I've only seen it in a few practice exams. It will be given to you on exam 2 for sure, so it's worth covering now anyway. The equation is given to you on the formula sheet. The most common question is that it will give you dy/dx, h, the first x and y value and then tell you to find y when x equals a certain value. This is also the most screwed up question. Remember what I'm about to say! As an example, I have and I want to know what y is equal to when . The answer to this is when you use in the equation! Don't go overboard and find the next y value because it's wrong.

I think there is a good chance that they will give us a "graph the equation" question. It'll either be something like a reciprocal polynomial or a sec/cosec/cot graph. Pray for the first because it is far easier. To earn the marks in this graph, you need to show axis intercepts, asymptote equations and endpoints. Don't forget the horizontal asymptote equation(s). Most likely it will be at , but depending on the translation, or whether it's a cot graph or not it might not be specifically at that point. All coordinates you list will have to be in exact value. None of this crap. Take care to read over the set domain. If it shows then be sure to indicate the open and closed circles.

Another question that could be on the exam (although I think very unlikely) is one about the rate at which liquid is leaving a tank, or the temperature of an object. These formulas aren't provided on the formula sheet so you need to have them memorized. An easy way to remember the rate at which a liquid leaves a tank is the formula where conc stands for concentration of salt (or some other substance) and stands for the original volume in the tank. For temperature its simply memorising Newton's Law of Cooling which is where represents the surrounding temperature. If anything, they are likely to describe the situation such as "The rate at which the temperature is decreasing is proportional to the reciprocal of the square of the current temperature". (Note, that is an example and not the actual law of cooling).

Vector functions is also a good possibility. They could spin this in many directions. They could get you to solve for the distance/velocity using the different acceleration formulas. They could get you to find the Cartesian equation and this could result in an ellipse or a hyperbola (very likely that an ellipse/hyperbola question will show up). Remember when graphing these to be mindful of the domain of t. It is very easy to screw this up. I recommend working out 2 or 3 points of the vector equation so you get the general look of the curve, and the direction the object is traveling in. When they ask for the speed, don't forget that this is the modulus of the velocity function.

Vector resolutes are a possibility. I would say they are fairly easy, with only a few things to remember. When they ask to find the vector resolute of a perpendicular to b, that means they are asking for . Other than that it's just remembering the formula.

Vector proofs. I'm sorry to say that there is a good chance a vector proof made it onto the exam. Don't worry though, it can't be too hard because it has to be done in the set time limit. Best hopes are a "prove that this shape is a rhombus" question or something similar. Just remember the standard properties of vectors and you'll be fine.

This is a taste of what might be on the exam. I'll stop here for fear of carpal tunnel and probably add a bit more later. Feedback is welcome and please tell me if I wrote anything wrong.. this was spur of the moment :P

Edit:
Fixed spelling of carpal tunnel. Cheers Polky.
Fixed Euler's.
Added graphing equations and related rates.
Added complex number stuff.
Fixed rate of change.
Added Vector resolutes/proofs/functions.
Thanks to Ben for various help.
<3 to Mao
Shoutout to Larry
« Last Edit: May 28, 2012, 03:09:00 pm by VegemitePi »

pi

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Re: Specialist Mathematics Resources
« Reply #5 on: November 21, 2011, 03:40:11 pm »
+4
Guide to Using the TI-Nspire for SPECIALIST
Written by b^3
Original Thread

Version 2.00
Ok guys and girls, this is a guide/reference for using the Ti-nspire for Specialist Maths. It will cover the simplest of things to a few tricks. This guide has been written for 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 METHODS - The simple and the overcomplicated: http://www.atarnotes.com/forum/index.php?topic=125386.msg466347#msg466347
Printer Friendly PDF version: http://www.atarnotes.com/?p=notes&a=feedback&id=661


NOTE: There is a mistake in the printable version. 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. Remember this should help speed through those Multiple Choice and to double check your answers for Extended Respons quickly.

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)


Vectors
These way the Ti-nspire handles vectors is to set them up like a 1 X 3 matrix. E.g. The vector 2i+2j+1k would be represented by the matrix You can enter a matrix by pressing [ctrl] + ["x"], then select the 3 X 3 matrix and enter in the appropriate dimensions.
It’s easier to work with the vectors if you define them. E.g. [Menu] [1] [1] a =
The functions that can be applied to the vectors are:
Unit Vector: [Menu] [7] [C] [1] - unitV()
Dot Product: [Menu] [7] [C] [3] – dotP(,
Magnitude: type "norm()" – norm()
E.g. a=2i+2j+k, b=6i+2j-16k, Find the Unit vector of a and a.b

E.g. a and b are perpendicular

Graphing Vectors Equations
Normally expresses as a function of t. Graphed as parametric equations. Select the graph entry bar, [ctrl] + [Menu] [2:Graph Type] [2:Parametric]
Enter in the i coefficient as x1(t) and the j coefficient as x2(t)
e.g. Graph



Complex Numbers
There are two important functions related to complex numbers. They work the same as the original functions, but will give complex solutions aswell.
cSolve: [Menu] [3] [C] [1]
cFactor: [Menu] [3] [C] [1]
E.g. Solve for z and factorise


Quicker Cis(θ) Evaluations
1. Define ([Menu] [1] [1]) cis(θ)=\cos(θ)+i\sin(θ)
2. Simply plug in the value of theta


Finding Arguments
1. Use the angle function (i.e. find it in the catalogue of type “angle(*)”
E.g. Find the Argument of


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 for
Enter 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 when and
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 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.
[Menu] [3] [5] - (function,variable)
e.g. Find the turning point of

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 has a maxmimum and a minimum for

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 when .


Visualisation of Addition of Ordinates
Graph f(x) and g(x), then graph f(x)+g(x)
E.g. Graph
Then


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 and solving for x.
e.g. Find the vertical asymptotes for

So for there is a vertical asymptotes at and
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 , which of the following is true?
A.
B.
C.
D.
E.
 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).
Derivative:  [Menu] [4] [1]
E.g. Find the derivative of
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)

Implicit Differentiation
[Menu] [4] [E] impDif(equation, variable 1, variable 2)
E.g. Differentiate with respect to 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 cuts the x-axis at (-2,0) and (2,0). It cuts the y-axis at (0,1) and has a local maximum when . Find the values of a, b, c & d.
1. Define (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 and and the equation of the curve is

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. are in radians NOT degrees.
RADIAN MODE                                                          DEGREES MODE


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.
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
.

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.


Differntial Equation Solver
[Menu] [4] [D] – DeSolve(equation, variable on bottom, variable on top)


Integrals
[Menu] [4] [3]
E.g. If find if and y=0 when x=0


Plotting Differential Equations + Slope Fields
Firstly you will need to open a graphing screen.
Then you need to setup up the mode for differential equations. This can be done in two ways:
A. Select the graph entry bar and press [Ctrl] [Menu] then select [2] (Graph Type) [6] (Differential Equation)
or
B. [Menu] [3] [6]
Now the interface comes up.

NOTE 1: When entering y in the bar, you will have to enter y1.
NOTE 2: If you want to plot a second differential equation that is not related to the first, you will need to either, open a new document (not just a graphing screen, for some reason the original equation that you plotted will be shown again) or clear out all the differential equations in the graph entry bar (i.e. y1, y2...) or open a new problem in the current document by pressing [Ctrl] [Home] [4] [1] [2]
e.g. Sketch the slope field

e.g. Sketch the slope field of  for
NOTE: Make sure you use y1

You will only need to draw the lines in the red box since if you draw the unrequited lines you may lose marks
e.g. Sketch the slope field for with initial conditions x=1 when y=0

Don’t forget a slope field should have a table of values with it.

Graphing Circles, Elipses, Hyperbolas in 1.5 easy steps
This allows you to plot equations in their zero form easily without having to rearrange for y and forming two (or more) equations.
Step 0: Firstly what you have to is rearrange the equation so that it equals 0.
e.g. becomes
becomes
becomes
Now remove the part
Step 1: Enter in the graph bar zeros(equation, dependent variable)



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 vgardiy for the real easy sketching of equations in their zero form.

Remember you can always do other funs things like 3-D graphs. Enjoy. Yey 800th post.


« Last Edit: July 28, 2012, 02:06:50 pm by VegemitePi »

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Re: Specialist Mathematics Resources
« Reply #6 on: November 21, 2011, 03:41:37 pm »
+3
Techniques for Sketching Nice-Looking Graphs
Written by pi
Original thread

So, throughout my time being a spesh student, I saw some pretty horrendous looking graphs. Personally, I think that drawing nice looking graphs using solid techniques and a pencil+ruler (this is important!) can do any of the following:
1) Speed up the time you spend on a question
2) Makes it easier for not only the examiner to find things, but also yourself
3) Impress an examiner (this is important!)

For me, I found that visualising points was equally as important as calculating them, and in fact, the simple visualisation of certain points can make a complicated graph very easy to sketch. Just remember, "sketch" does not mean messy!

So here's my quick guide on how I like to go about things :)

Implied domains of certain scenarios

#1

Take and solve for
Domain = \{}

#2

Take and solve (use a "quick-sketch" if needed)
Domain = solution of inequation

#3

Take and solve (use a "quick-sketch if needed)
Domain = solution of inequation

#4

Take and solve
Domain = solution



Sketching a nice looking graph (addition of ordinates)
1. Recognise the case: ,   
   (i) Straight line and hyperbola
   (ii) Parabola and hyperbola
   (iii) Straight line and truncus
   (iv) Parabola and truncus

2. Find Domain and Range,

3. Find any vertical asymptotes:
     ⇒ for is undefined

4. Find any oblique or curved asymptotes:
     ⇒ Resolve

5. Find critical points:
   (i) -int, let
   (ii) -int, let
   (iii) Stationary points, let and solve for and
   (iv) Any crossing of the horizontal asymptote, let and solve equation for
   (v) Endpoints (if any)

6. Do a light dotted sketch of both and .

7. Find key points (used to aid graphing):
   (i) Zeroes of and . The -co-ordinate is on the other curve
   (ii) Cancelling points, the is -int (don’t solve for these points, do this visually), this should match your above calculation
   (iii) Visually use -ints of and to find the -int of , this should match your calculation
   (iv) Intersections of and . The -co-ordinate is double of this.

8. Look left/right of each key-point, realising the behaviour of the curve

9. Sketch, rub-out any unnecessary dotted line graphs

10. Label the graph, axes, all asymptotes with their equations (as Asym or Asym ) and all critical points in co-ordinate form



Sketching a nice looking graph (reciprocation)*
1. Recognise curve as (or manipulate mentally to see this)

2. Draw a light dotted sketch of

3. Horizontal asymptote is

4. Draw vertical asymptotes through -ints of

5. Find key points:
   (i) , these points will also be on
   (ii) Stationary points, let and solve for and
   (iii) Endpoints (if any)

6. If , , if ,

7. Sketch, rub-out any unnecessary dotted line graphs

8. Label the graph, axes, all asymptotes with their equations (as Asym or Asym ) and all critical points in co-ordinate form


*If has been translated vertically, then there may be -ints and the horizontal asymptote will also change. These need to found if this is the case.



What examiners like to look for
- General shape
- Appropriate and realistic scaling used
- Correct -ints if they exist in co-ordinate form
- Correct -ints if they exist in co-ordinate form
- Correct local max/mins if they exist in co-ordinate form
- Correct end-points if they exist in co-ordinate form
- Correct and labelled asymptotes if they exist
What they like in addition to above:
- Labelled axes
- Labelled graph
- Domain and range given
- Straight lines done with a ruler
- No deviations away from an asymptote
- Clear labels, no smudging, good presentation



Example (for partial fractions - a twist on 'reciprocations')
From my notes, so it's not to the standard I would have done in a SAC or exam, but its alright and shows the working too :) Apologies for my crappy hand-writing too :D

Notes:
- I have left the "dotted-lines" of the various "parts" to the full graph, this is because this was done for notes purposes. You should erase these in SACs/exams.
- I have jumped to the second line of working, I used a CAS to save time and get the problem on one page :P
- Against my tips, I have labelled my asymptotes as only "" instead of "Asym "



N.B.
- For any asymptotes that may appear to lie on an axis, draw it in a coloured pen (preferably blue) or just above the axis.
- You may or may not need to prove a stationary point. If you do, I'd suggest either the use of second derivatives or a gradient-sign table. Check the question to make sure.
- After some practise, you may feel comfortable with skipping some of the steps listed above, but I find that the above list is a very good start for "beginners".


Hope this helped, post any queries/suggestions/errors in this thread, good luck! :)
« Last Edit: November 11, 2017, 06:30:14 pm by pi »

pi

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Re: Specialist Mathematics Resources
« Reply #7 on: November 21, 2011, 03:41:44 pm »
+2
Calculator Tricks
Written by abes22
Original thread

For all who want the highest score with the least work:
(For the TI-89 Titanimum)

IN THE HOME SCREEN
1) Define:
Press F4, then enter, this gives the word, Define. This can be used to define functions that can then be used in other things.
E.g, enter:
Define f(x)=x^2     then;
Define g(x)=ln(x)

Entering f(g(x)) will give (ln(x))^2, and of course, more complex functions can be used.

Entering Define y1(x)=x^2 will put x^2 in the [ Y= ] window, which can be accessed by pressing diamond F1.

The function can then be used in all other operations, entering f(2) will give 4, and f(y+2) will give (y+2)^2.

2) Differentiating.
Differentiating can be achieved by pressing 2nd 8, typing a function and specifying parameters.
ie, entering d(f(x),x) will differentiate f(x) with respect to x. finding the 2nd derivitive can be done as follows:
d(f(x),x,2).

3) Integrating.
Press 2nd 7, and specify parameters:
integral(f(x),x) will give integral. lower and upper bounds for definite integrals can be placed as follows:
integral(f(x),x,lowerbound,upperbound)

4) Solving.
Easily the most useful tool. You MUST be able to use the "Given that" symbol to specify domain and restrictions, this is achieved by pressing the button with just a line that is 4 buttons up from the left.
ie, entering:
solve(sin(x)=1/2,x) will provide the general solution to the sinusoidal function. However, entering
solve(sin(x)=1/2,x)|0<x<2pi will give you the distinct solutions. Similarly, you can specify other restrictions.
solve(y=3x^4 + 2x^2 + x -11,x)|y=0 will give the x intercept,
solve(y=3x^4 + 2x^2 + x -11,x)|x=0 will give the y intercept.

with functions of several variables, more restrictions can be added.
solve(y+t=3x^4 + 2t^2 + x -11,x)|y=0 and t = 3
will give the value of x when y=0 and t=3. similarly, just typing
3x^2 + 2|x=2 will give the value of that function at x = 2.

solving simultaneous equations:
enter:
solve(y + 2 = x/3 and 3x-4y=5,x) to achieve the solution, (-9/5,-13/5).
If there are 2 functions in terms of 3 variables, you can specify which variable you want to solve for using curly brackets, by pressing 2nd ( or 2nd ).
solve(3x+2a - y/2 = 0 and -2x+y=a+2,{x,y}) will give x = -(3a - 2)/4 and y = -(a-6)/2. alternatively, entering
solve(3x+2a - y/2 = 0 and -2x+y=a+2,{x,a}) will give x and a in terms of y. again, using "given that" and substituting a known value of y will solve for x and a.

you can use previously defined functions within solve.

5) Expand
When given a fraction, you can split it up into partial fractions using F2 expand. Entering:
expand((3x^2+2x)/(x^2 + 5x + 6)) will give the partial fraction form, 33/(x+3) - 12/(x+2) + x - 5.

It can also expand more simple expressions, such as
expand((x+3)*(x+2)) will give x^2 + 5x + 6.

6) ComDenom
this will reverse the partial fractions back to its original form, sometimes useful and can be found in F2.

7) "Hidden" Keys.
Pressing diamond EE will display what each key will do if you press diamond before hand. eg, pressing 2nd 0 will give a less than sign. Pressing diamond 0 will give a less than or equal to sign, helpful in setting domain restrictions.

8 ) Matrices
You can enter matrices by pressing 2nd , and 2nd / to give square brackets. Entries in a row are separated by commas, and each row is separated by a semicolon. eg, the matrix with 1 2 3 4 as the four entries from top left to bottom right is entered:
[1,2;3,4]

you can use define to give matrices a letter and then actually use them in solve etc:
Define a = [1,2;3,4]
Define b = [x;y]
define c = [5;6]
now the equation AB = C
can be solved for x and y by entering solve(a*b = c,x)
Also, a^-1 will give the inverse matrix of a, and det(a) will give you the determinant.

COMPLEX NUMBERS:
The i used in complex numbers is found by pressing 2nd catalog
to solve equations using complex numbers, enter:
csolve(z^2 + 2iz + 6 =0,z) gives z = +/-(root(7)+1)i
simliarly, using cfactor will give the factors of a complex polynomial.

to find the Argument of a complex number, enter:
abs(a+bi) to get the magnitude, and angle(a+bi) to get the Argument.

WITHING THE GRAPHING SCREEN:
in the y= screen, enter:
y1(x) = x^2.
Obviously, this graphs x^2. But if you enter:
y1(x) = x^2|-2<x<2 will only draw the graph over [-2,2]. *Sorry, I cant get a less than or equal to sign on this thing!*

you can also enter things like y1(x) = f(x) if you have previously defined some function f.

obviously, you can set your desired window with diamond F2.

Whilst in the actual graph, pressing F5 enter and then entering a value for x will give the corresponding value for y.
eg, entering (for y1(x) = x^2|-2<x<2) within the graphing screen:
F5
Enter
1.5
Enter

will give you (in the bottom right hand corner) y =  2.25

If you have two graphs drawn, you can toggle between graphs by pressing up or down.

IN THE GRAPHING SCREEN:
F5 Value - this does what i just said above! Enter x = 0 for the y intercept.
F5 Zero - Placing an upper and lower bound will give you the x intercepts
F5 Minimum - Placing a lower and upper bound will give you the coordinates of the local minimum
F5 Maximum - Placing a lower and upper bound will give you the coordinates of the local maximum (you can scroll through quicker by holding 2nd and pressing left and right)
F5 Intersection will give the intersecting point between two graphs. When prompted for "1st Curve" toggle the graphs using up and down arrows until the cursor is on the graph you're working with, then press enter on the second graph, and enter a lower and upper bound for the intersection on this second graph.
F5 Derivitives will give dy/dx at the input point.
F6 Integrate will give the area under the graph you select for the upper and lower bounds you input.
F5 Inflection will look for a point of inflection between the lower and upper bound you input. If no p.o.i is present, the calculator will say "No Solutions found"
F5 Distance will give the distance between two points inputted (these can be on two separate graphs)
F5 Tangent will draw and give the equation of the tangent at a given point


So as you can see, the graphing screen does pretty much everything.

And now, to blow your mind. METHODS ONLY
Probability. Input this EXACTLY, and you can get capital letters by pressing the button with an arrow pointing up (its next to 2nd).

Define noc(l,u,m,s)=TI.Stat.normCdf(l,u,m,s)

Then press enter. Then press 2nd - ( to get to VAR-LINK), then find "noc", select it and press F1 and archive this variable.

Now in the homescreen, if you want to find the probability in a normal distribution that X lies between l and m, with a mean of m and standard deviation of n, then you input:
noc(l,u,m,s)

eg, for a normal distribution with mean 5, standard deviation 2, and you want to find Pr(3<X<6), you input
noc(3,6,5,2) and it will give you the answer of 0.532807

I chose l for lower bound, u for upper bound, m for mean and s for standard deviation. For the rest of these, p will be probability, n will be number of trials, x will be number of successes (for binomial probability)

After each input, Archive the variable to avoid losing it later!

Define ino(p,m,s)=TI.Stat.invNorm(p,m,s)

This is the inverse normal function.

Define bip(n,p,x)=TI.Stat.binomPdf(n,p,x)
This is for binomial probability.

Define bic(n,p,x1,x2)=TI.Stat.binomCdf(n,p,x1,x2)
This is for cumulative binomial probability, ie, if x does not take a single value, but a range of values.

My french exam is tomorrow, so yeah, i'll post worked specialist and methods exams for the 2009 exam to show you all how easy it is with a calculator at a later date (within a week)! if you know what youre doing, these exams can be done in about 40 minutes - thats how long it was taking me when i did methods!
So i'll walk you all through it step by step, and trust me, you dont need to know any maths. you just need to know how to use a calculator!

Happy studying everyone :)
« Last Edit: May 20, 2012, 02:16:13 am by VegemitePi »

pi

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Re: Specialist Mathematics Resources
« Reply #8 on: November 21, 2011, 03:41:52 pm »
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pi

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Re: Specialist Mathematics Resources
« Reply #9 on: November 21, 2011, 03:41:59 pm »
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Bhootnike

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Re: Specialist Mathematics Resources
« Reply #10 on: November 21, 2011, 10:53:48 pm »
+3
Nice stuff.
There's some errors in some images /latex in some of the notes though,  thought I might point that out :) 
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Re: Specialist Mathematics Resources
« Reply #11 on: November 25, 2011, 08:09:58 pm »
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 ;)

pi

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Re: Specialist Mathematics Resources
« Reply #12 on: May 28, 2012, 03:10:49 pm »
0
Nice stuff.
There's some errors in some images /latex in some of the notes though,  thought I might point that out :) 

Should all be fixed now, in this thread as well as in the Methods equivalent :)

Jenny_2108

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Re: Specialist Mathematics Resources
« Reply #13 on: June 17, 2012, 08:57:14 pm »
0
OMG, ITS SO AWESOME!!!
This is what I'm looking for after long time wandering AN!
I wonder how long you spent to type and upload all of those :O

pi

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Re: Specialist Mathematics Resources
« Reply #14 on: June 17, 2012, 10:17:19 pm »
0
OMG, ITS SO AWESOME!!!
This is what I'm looking for after long time wandering AN!
I wonder how long you spent to type and upload all of those :O

Haha, but don't thank me for all of this! The real authors of all these post is written italicised in red under the title of the post :) I think I've only come up with one of these guides (+ the compilation of them all) :)

Hope they come in handy though :)