This guide is going to cover the extra parts of calculus in the Extension 1 course. These are usually some of the hardest parts of an exam, so read carefully, think about the examples, and of course, ask questions!
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. And finally, be sure to check out the 2 unit guide on differentiation
So there are a few, fairly distinct areas to cover here, so we'll go one by one. First, rate of change questions
. These have the potential to be quite nasty, but really, all you need to know is this basic idea:
The rest of the work comes with tricky proofs, usually linking volume, radius, and height of a container. The best way to prepare for these is practice, so let's look at one from 2013
EG:A spherical raindrop of radius r metres loses water through evaporation at a rate that depends on its surface area. The rate of change of the volume V of the raindrop is given by
where t is time in seconds and A is the surface area of the raindrop. The surface area and the volume of the raindrop are given by
(i) Show that
(ii) How long does it take for a raindrop of volume
to completely evaporate?
This example is fairly straightforward, but a good way to introduce the concept. We require the change in r with respect to time. We have the change in volume with respect to time. So we can use the formula above to get what we need as shown:
which is a constant.
For part (b), the easiest way to consider this is when the radius is zero. We first need to know the initial radius. In the interest of brevity, I'll skip the algebra:
From part (a), we know that the radius decreases at the constant rate found. This makes the next part easy:
So the answer is 62 seconds. The hardest part of these questions is definitely knowing what to do with the data. Do some derivatives, write down everything you know, and look for the magic pairing. It will be there.
Next is projectile motion. These are usually one of the last questions in the exam, and there is no other way to say it... They are brutal
. These are the questions which will earn you a Band E4, if you can do them. Now there is an almost infinite number of things they can ask, but one thing that pops up a lot of deriving the standard formulae you learn. In fact, it's normally worth a good 4-5 marks! Let's make sure you can do it:
Remember that we can find velocity by integrating with respect to time, and position by integrating with respect to velocity. The hardest part is considering the constants of integration. Make sure you memorise this process. First the y (vertical) equations:
The x equations are much easier in comparison:
Remember that V is the initial speed, and it is separated into it's xy components with some simple trigonometry. These questions are varied in what they ask, and the best way to prepare is to practice as many past questions as possible. However, some tips for these questions:1-
Remember that maximum range is achieved when the angle of projection is 45 degrees!2-
Examine the equation you need to derive. The trig functions are a big clue. How do you get to tan from sin and cos? Never just start manipulating, think about where you are heading and how you might get there.3-
Remember that, all other things constant, two different launch angles give the same landing position. This factors into lots of questions.4-
Draw a diagram! I cannot recommend this enough, it will give you best idea of what's going on, and make your logic clearer for the marker.
The final big area is simple harmonic motion. These are easier than projectile questions, but they can still be tricky! Be sure to remember the few key formulas:
Simple harmonic motion occurs when the particle oscillates about an equilibrium position. It is a consequence of the properties of the acceleration; the first formula shows this, and is what you must prove to prove SHM is occurring. A is the amplitude of the motion, it indicates how far the particle moves from the equilibrium position, indicated by the phase constant. If the formula is a sine formula, the particle starts at this equilibrium position, if it is cos, it starts at it's max/min position.
Knowing these formulas and their consequences makes most questions fairly easy. Take this question I did in my HSC:
EG: A particle is moving in simple harmonic motion about the origin, with displacement x metres. The displacement is given by x = 2 sin 3t, where t is time in seconds. The motion starts when t = 0.
(i) What is the total distance travelled by the particle when it first returns to the origin?
(ii) What is the acceleration of the particle when it is first at rest?
For part (i), we can immediately read the amplitude of 2 from the equation, so the answer is 4. BE CAREFUL HERE GUYS
. It travels to the end point, then back, so the answer is 4, not 2.
For part (ii), if you know your SHM, we know that the particle is first at rest when it reaches it's maxima (2 units from the origin). We use the standard formulae for a simple 2 marks:
The actual worked solution provided by BOSTES for this question is 15 lines long, 15! Just by knowing a little bit more about SHM, these questions become easy, so remember these two other key facts:
1- Velocity is maximum at the equilibrium, and zero at the endpoints
2- Acceleration is maximum at the endpoints, and zero at the equilibrium
Of course, these are not the be all and end all of potential questions. Miscellaneous questions on velocity and acceleration pop up a fair bit, and be prepared for more complex applications of the chain, product and quotient rules. And of course, you have a few extra derivatives to remember as well. The trick is knowing your formulas like the back of your hand... If you have this, you can focus on interpretation and getting logically developed solutions. In terms of approaching the really tricky projectile questions, practice makes perfect! And of course, if you get a specific question you want examined or explained, post it here and I will post a worked solution for everyone to benefit from!
That's about it for this guide. I hope these are proving beneficial, feel free to post some feedback below, check out the notes, and talk to each other! You have an amazing community to capitalise on