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
 = artan(y/x))
. 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
 = 2cos^2(x) - 1 = 1-2sin^2(x))
. 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. For those that don't know what a slope table is:
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
 = 1, h = 0.1)
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
 = 1.73)
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
 - \frac{Q * flow out }{V_0 + (flow in - flow out)t })
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
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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