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 Dont 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
)
not just
)
. There's a difference between
and
)
. To be safe always include the
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
)
and x-intercept as
)
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
=>
Whenever there is an even denominator present when "powering" the equation, the
sign is neccesary
DRAWING GRAPHS Always label curves with its corresponding equation
...
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
, 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
)
). It is automatically assumed that the domain does not exist for
. 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
)
, the vertical axis is labelled
, and the horizontal axis is labelled
. Furthermore in probability the vertical axis is labelled
)
.
This also applies to the asymptotes. Dont label
... when
... 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 dont mistake minimum for maximum and visa versa.
(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)})
COMPOSITE FUNCTIONS Be able to identify expressions such as
))
immediately as composite functions
Regarding
))
, make sure you go through the functions in the right order. Dont think
)
is the base function that is subbed into
. Carefully look at the expression and interpret, the correct order is
subbed into
.
Regarding
 = g(f(x)))
. The notation that is used to denote
)
in exams is always
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. Dont 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
^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
 = 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
, 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.
 + ln|x| + c = ln|x| + c)
, because c can be any number.
or alternatively
CIRCULAR FUNCTIONS Whenever you have a variables in degrees for a sinusoidal function
=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).
 = 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 dont get the wrong derivative. E.g. when the rate of change is volume is
, the corresponding rate of change is
, not
TANGENT Be careful about whether it is asking for the normal or the tangent
When the gradient of the tangent is
, 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.
The anti-derivative does not include "+c." The antiderivative of
is
not
 + c)
Whenever you have an expression
dx)
always transform it into
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
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 its a < or
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 its 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
^{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
^{3-1})
CONTINUOUS PROBABILITY When writing the hybrid functions of probability density function, you can write the domain for the parts defined by
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
. Take care to indicate where the function is on the x-axis with some coloured line.
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