Just doing the same thing as I did for Methods =X
http://vcenotes.com/forum/index.php/topic,20778.0.html 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 THEORYFUNCTIONS● In an inequation, when you reciprocal the whole thing you must reverse the signs
●
VCAA 2010 Spesh Exam 2 Question 4 c.)
●
(x-b)^2)
has 3 real solutions
)
. But 2 distinct real solutions
)
●
\neq(f(x))^{-1}.)
)
is the inverse function.
)^{-1}=\frac{1}{f(x)})
LOGARITHMS●
 + ln|x| + c = ln|x| + c)
, 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.
=sin k^o)
 = sin\left ( \frac{k\pi}{180} \right )^c)
●
)
is not equivalent to
)
. The former is a relation, the latter is a function. Thus the range of
 is R)
● 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.
^2 )
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
 > 0.5\pi} )
is the entire Argand diagram except for the origin
● Always label complex regions
● For
 = \theta}. )
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.
^{-1} )
● 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.
●
) = \frac{d}{dx}(ln|f(x)| ) )
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●
\,dx )
. 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
 = 10, u(8) = 65 )
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
, v(0))
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 
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