**Paul's Mathematical Methods - Pre-Exam 2 Advice for 2012***Paulsterio***Preamble**In order to do well on Exam 2, we need to recognise what skills it is testing. Whilst Exam 1 is a test of our mechanical computational skills, Exam 2 is a test of our ability to mathematically reason, applying what we have learnt in Methods to the cases and scenarios presented. It requires almost next to no ability to do things by hand, given that it is CAS-assisted, even questions which seemingly ask to be solved by hand can usually be done with a CAS.

Thus, in order to excel in Exam 2, we need to be familiar with a CAS and we need to really be able to understand and mathematically interpret the scenarios at hand.

**Multiple Choice Questions**Multiple Choice questions usually test a wide variety of elements within the course and the only real way to be prepared for them is to have a sound THEORETICAL knowledge of the whole Mathematical Methods course. Statistically speaking, you will find that of the 22 MC questions, you will generally be able to group them into 4 categories:

**1) Purely CAS questions**These questions can purely be solved using a CAS and don't really require any sort of interpretation. It could, for example, be something like "solve this equation". One thing to watch out for with these sorts of questions is answer forms. What I mean is that there are often more than one way to express an answer - e.g. log9 and 2log3 are the same. So if the answer your CAS gives isn't present, look out for equivalent options. (Tip, if you can't find it, use DECIMALS! It will always reveal the right answer). Usually, there are quite a few of these questions, 30%-ish of MC will be them.

**2) Purely interpretational questions**An example of this sort of question would be "find the amplitude and period of the following trigonometric function" - they are questions which only require you to look at a bit of information (for example an equation for a trig graph) and make a deduction (the amplitude and period) - they require little mathematical calculation and they require no usage of the CAS. You usually get a few of these, not many, but they are easy - especially given that they can be done in a small amount of time - around 20% of your MC will be this sort of question.

**3) CAS + Interpretational Questions** These questions require both the use of a CAS and making mental mathematical deductions. An example of this sort of question would be the majority of MC probability questions, which require you to make a set-up or interpretation first, and then use the CAS to evaluate it. This will probably make up the most of your MCs, around 40%, and they are usually harder than the other two previous types, so beware of these, if you're able to do really well at this type, you have a chance of scoring 20/20 for MC. Other examples of this sort of question might involve graphs (which of the following is the right graph for this equation - so you have to graph it on your CAS and interpret it)...etc.

**4) Curveball Questions**Then you have the questions which are curveballs, sort of different to the rest. An example of this sort of question was on last year's Methods Exam. There was a question where the logaithmic change of base rule had to be used, it wasn't a CAS question, nor was it really an interpretational question, it wasn't really both either, it was just something unexpected, not many people expect the change of base rule to feature. There will always be one or two curveball questions on every exam paper, so beware of them, they're the differentiators between the best and the very best, essentially what sets apart the 45+'s from the 50's.

**General MC Tips**- MC is worth 22 marks, which is 1/4-ish of the paper, this means you should be spending a MAXIMUM of 30 mins on MC.

- Aim for an average of 1 question per minute, that way, you can finish MC in 20 mins, which leaves you more time for Extended Response questions.

- Be careful with MC questions which require interpretation, the answer options aren't chosen randomly, they're out there to trick you!

- You can usually do around half of the MC questions in your head during reading time if you really wish to

- Be proficient with the CAS - it will help you, generally students who are good with the CAS will find that they can complete the MC with more speed and accuracy.

**Extended Response Questions**Extended Response questions are often the most feared, but generally, I tended to like them, for two main reasons:

- They didn't actually include that much maths - most of the maths is done using a CAS

- They actually require you to think and apply your skills, which is rarely seen elsewhere on the course

Thus, you have to approach ER questions with this mindset. You can't approach them in a mechanical manner and expect to do well, you have to interpret the information they have given you and ask yourself how you can build equations, formulae and a mathematical set up from it. Once you have your set-up, it's all just CAS from there-on in.

Generally you will get either 4 or 5 Extended Response Questions in Methods, 1 of them will probably be an algebraic one which involves solving a few equations and drawing a graph, all things which can be done on a CAS. One of them will definitely be a probability one, so that is mostly CAS as well, but be familiar with probability, you need to know that much to set it up. It will probably involve the normal distribution somewhere within the question, so be familiar with those commands on the CAS as well. Then you always have the difficult last question, which usually involves calculus and some sort of minimisation/maximisation sort of question. These are difficult because students often have run out of time or they no longer have the energy and mental stamina to solve them. One way to test this out is to do the following, take a past VCAA exam and just do the last question, sure it'll be hard, but it won't be THAT hard, because when you're fresh and thinking straight, you'll find it much easier.

**General Tips for ER Questions**- Take your time and think the questions through, if there is a really easy question, you might not have fully grasped what it's asking.

- Always use correct notation, be mindful of how many marks are allocated and use that to guide your working out

- Whenever unsure, always put MORE working out than you think is necessary - you can never have too much - don't be lazy

- Always keep the instinct of using the CAS on your mind, you want to use it as much as possible, but know its limitations

- Never use CAS notation, always use the correct mathematical forms

**CAS Tips for ER Questions**How to use a CAS to evaluate areas, showing full working

- Write down the integral statement for the area, for example, the integral of x^2 with respect to x from 0 to 5

- Type the function into the CAS, without the bounds, and get the antiderivative

- Now write the antiderivative and put in the correct bounds, using the square brackets

- Now, by hand, substitute the numbers into the anti-dervative, so F(a) - F(b), but don't evaluate it

- Go back to the CAS, and enter in the integral, this time with bounds, then copy the answer across to your paper

- So you've just worked out an area, supposedly showing "full working" and "using calculus" but you're assured of a right answer

How to use a CAS to find derivatives, showing full working

- Say we want to find the derivative of a complicated function, but it's worth 3 marks, this is what I'd do

- Determine the rule to be used. Say it's a quotient

- First, let u=... and v=...

- Now write down the rule dy/dx = (v.du/dx - u.dv/dx)/v^2

- Go to your CAS, and find du/dx and dv/dx

- Substitute all into the rule, but leave unsimplified - dy/dx = ( (......) x (........) - (.........) x (........))/(.......)

- Now use the CAS to find the derivative, dy/dx

- Copy it down, and voila, 100% correct derivative

Finding f(x) given f'(x) - a shortcut

- If we know a derivative and a point on the curve f(x), there is a shortcut to solving it

- It's using a command on the CAS called dSolve - for the ClassPad

- Go interactive, advanced, dSolve

- In the first column, type y'=...(derivative)...

- Independent Variable - x

- Dependent Variable - y

- Initial condition, type, for example if we had the point (1, 5) - "x=1,y=5"

Finding f(x) given f'(x) - a shortcut using definite integrals

- Similar to above but for people on TI Calcs - may be a fast way

- Type the integral sign with bounds, but instead of using x, use another letter, for example t

- So type in the integral sign, and then the derivative using t instead of x

- Now look at your initial conditions, say you have the point (0,5)

- Put the lower bound as your x-co-ordinate "0"

- Put the upper bound as the variable "x"

- Now after the integral (i.e. after the dt) put + the y-co-ordinate so here you would put +5

- Remember it's "dt" not "dx"

- Hit enter, and you should get your function of x

Finding a,b,c...etc in equations knowing the points

- You can use the regression function to check that your values are correct

Extra Resource -

Re: Mathematical Methods Guides and Tips (b^3's TI nSpire Guide)

**Final Tips for the 2012 Exam 2**General:

Remember to read the question again, once you have finished and ensure you have answered it

Check that decimals are used where required and that they are to the correct accuracy (decimal places)

Check to see that you have transcribed all information given correctly, don't make mistakes copying equations

Check especially for adding, multiplying, subtracting and division errors

Functions:

When finding the inverse of a function, remember to write: "for inverse swap x and y"

If you are asked for find a composite function, remember to check if it exists

The composite function of some function and it's inverse will be y = x

It may be faster, when trying to find the intersection points between two inverses to find the intersection with y = x for one of them

If evaluating g(x) = a and you have the inverse, just find g^-1(a)

When giving the general solution to trigonometric equations, remember to write that "k" or whatever variable you use is an integer

For transformations where they give you a matrix, it is safer to multiply the matrix out and then substitute into the equation, but it may be faster to use recognition

Remember that similar triangles may be on the exam

When you define a variable, you should make it clear "Let

be"

Always solve equations using the CAS - set up the equation, then use the CAS to solve it

Be very familiar with the graphing screen, including the different types of Zooms and what can be found on the graphing screen

Calculus:

When using the product of quotient rule, remember to state the rule

The derivative of f(x) is f'(x), the derivative of y=... is dy/dx = ...

Sometimes you're asked to find the derivative at a certain point, remember to do so, not just find the derivative function.

Remember to put the "+c" in antiderivatives, unless "an antiderivative is asked for

Remember the "dx" at the end of the integral

When a question says "use calculus" - you must show the derivative or antiderivative, HOWEVER, I would suggest that you always show the derivative or antiderivative (where possible)

Always do derivatives and integrals using the CAS, never do them by hand

Know how to find both local and global minima and maxima using the CAS

Know how to do a linear approximation using the CAS

Probability:

When solving questions to do with conditional probability, remember to include the rule

Remember to include the statements X~Bi(n,p) and X~N(m, var) when dealing with Binomial and Normal

No Calculator syntax - No invNorm

To express normcdf in the correct way, write, for example, X~N(1, 0). Pr(X>10) = ...

To express inverse normal in the correct way, write, for example, Given that X~N(1,0) and Pr(X>a) = 0.5, a = ...

Additions:

1. (x^2)/|x|=|x| [that is, x squared divided by mod x is equal to mod x]

2. Probabilities are always 0≤p≤1

3. If you log something, whatver is inside the log HAS to be above zero (and not including zero). ln(x) where x<0 doesn't exist.

4. If given a probability distribution function, you HAVE to draw the parts of the the function where f(x)=0 as well. Otherwise you get marks taken off.

6. Period of tan(nx) is pi/n, not 2pi/n

**Last Week Before the Exam! Practice Exams?**Yes, now is the time for you (if you haven't started) to really start doing exams under perfect exam conditions. This includes doing them to exact time and making the marks count. I know that most of the time when I did practice exams, I didn't really care, so I just rushed, looked at the answers...etc. Don't do this! Set yourself an aim and reward yourself - "if I get X/80 for this exam, I will get Maccas" - if you're able to really simulate exam conditions, you'll know how well you can work under pressure and under adrenaline.

Now, after you've done all the practice exams you've wanted to do, it's important you go over your mistakes. Redo the questions, make sure you can do them, if not, look at the solutions and keep trying until you can do them. You haven't finished a practice exam and gotten everything out of it until you can do ALL of the questions and get 100% on that exam, if you're not there, you have heaps to improve on, so why are you doing more practice exams when the ones you've been doing aren't perfect? Review them first, then move on.

**Other Resources**Re: Mathematical Methods Guides and TipsRe: Mathematical Methods Guides and TipsRe: Mathematical Methods Guides and TipsRe: Mathematical Methods Guides and TipsRe: Mathematical Methods Guides and TipsRe: Mathematical Methods CAS Resources**Final Advice**There comes a time during our preparation where we become obsessed with marks, the moment where getting a 80/80 on our Exam 2 becomes more important than enjoying the maths we do and enjoying the learning and the applications that mathematics provides. Remember that no matter what happens on the exam, we have spent two years learning Methods, in many ways, what's even more important than doing well on these exams is what we've been able to gain over that period of time.

I wish you all well, and I hope you all perform to your desired potentials come the Mathematical Methods Exam 2.