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Specialist Mathematics Resources
pi:
AN Specialist Mathematics ResourcesThis 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
pi:
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
pi:
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
pi:
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
pi:
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
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