Methods - Paul's Exam Advice

[paulsterio]

Methods Exam Checklist!

Both Exams:
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

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

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 = ...

Thanks to Daliu
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 (actually meant to be "0 more than or equal to p more than or equal to 1", couldn't type it though...)
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

Thanks to BoredSaint
'Define the Variable in Probability' - as in' - "Let X be the number of...."

Thanks to jane1234
Re: Methods - Paul's Exam Advice

[paulsterio]

Advice for using a CAS in Extended Response Questions (Most applies to all CAS)

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

- More to come, break time!

[paulsterio]

Bump - New CAS stuff added! (2nd Post)

+ This is reserved for CAS - Multiple Choice

[thatisanote]

If you have any more to add I'd love to read it!

[Wezanator123]

Paul's definitely gonna get a 50 in methods

[RossiJ]

for inverse do you have to write 'for inverse swap x and y'
I just swap them usually without writing it? :O

[thatisanote]

for inverse do you have to write 'for inverse swap x and y'
I just swap them usually without writing it? :O

Make sure you write it!

[paulsterio]

I've been thinking about what else to add, but I'm out of ideas, I've stopped doing practice exams now, so I'm probably not going to have any more ideas. Most of these were just where I'd lose marks whilst doing practice exams

But I might add some CAS tricks later, I'll think about what other little nifty tricks I use often

[DisaFear]

Pro tier advice, thank you

[nacho]

do we lose marks for not stating the use of product/quotient rule?
is it needed for chain rule?

[paulsterio]

do we lose marks for not stating the use of product/quotient rule?
is it needed for chain rule?

For the product/quotient rule, usually you get a mark for "indicating the rule and attempting to use it" but that's open to interpretation, if you write the rule, however, you'll definitely get that mark, furthermore, if it's a 2 mark question, the rule might get you 1/2 marks if you differentiate incorrectly.

I wouldn't say that it's needed for the chain rule, because the chain rule is quite simpler

[jane1234]

Great tips Paul!

Mind if I add some extra advice here? I had a "mistake book" last year which might pick up on anything that you missed... I'll dig it up after I'm done with this stupid spesh exam.

[paulsterio]

Great tips Paul!

Mind if I add some extra advice here? I had a "mistake book" last year which might pick up on anything that you missed... I'll dig it up after I'm done with this stupid spesh exam.

You're on ATARNotes whilst doing a practice exam? :O haha, it's funny cause I'm doing a spesh exam as well *hi-5*

[jane1234]

Okay, I'll try not to repeat Paul... but I probably will...

GENERAL:

• Do ALL working written out. Do as little in your head as possible. This includes adding, subtracting, whatever. Trust me, this is one of the easiest ways to minimize errors, especially in exam 1. Write every step down (makes checking over your work easier too).
• Don't expand/simplify expressions when you don't have to. This again will cut down on the silly mistakes. Worst thing is when you have the right answer and then try and simplify it and get it wrong...
• ALWAYS SUBSTITUTE BACK! This means deriving antiderivatives, subbing in answer to an equation, etc. I did this and picked up about 3 silly mistakes in exam 1 which would have cost me the 50. Always do the 'opposite', essentially, of what the question asks to check.
• More for exam 2, but with worded questions you sometimes need to restrict domains accordingly. You have to THINK, will this answer work in real life? For example, you can't have a negative length, negative time etc.
• Make sure you read the right numbers/signs off your calculator. It takes half a second to double check, but so easy to skip a negative sign or read a number wrong.
• Check that the answer works. This may seem like common sense, but so easy to lose marks on. If you are asked for the height of a building and you get 0.01 cm, then clearly that answer is wrong.
• Be wary of long, complicated working out and answers. I can tell you this from experience of the many practice exams I did, but if you get something like 10234/2278438484 (especially for exam 1 when you have to add by hand) then 9 times out of 10 that answer is probably wrong. For exam 1 they are not testing your adding skills, so they are not likely to make you do a long, complicated sum. Honestly, most answers will be simple (though that doesn't exclude surds, some fractions and pi). So just double check if you're answer is a weird, long number. This doesn't mean it definitely ISN'T the answer, but it's not likely to be.
• READ THE QUESTION! I cannot stress this enough. READ THE QUESTION! If it asks for factors of a polynomial DO NOT GIVE SOLUTIONS!
• Also, READ THE QUESTION! If it asks for x values of intercepts, don't give them co-ordinates and vice versa.
• Use correct pronumerals. I nearly lost a mark on exam 1 for this. If it gives you an equation h = 2a then DON'T use x and y on the graph or say 'dy/dx' when it should be 'dh/da'.
• Check working AS YOU GO, especially with MC. You might not have as much time as you think to go over the paper, ESPECIALLY if there is a difficult question.
• Watch modelling questions. If Day 1 is at x=0, then Day 6 will be at x=5 NOT x=6.

GRAPHS:

• Most people don't worry about this, but it is very easy to lost marks for having a dodgy graph shape. In exam 2 always plot the graph on your calculator BEFORE sketching, and make sure the scale is the same on the page as it is on your calculator.
• With addition of ordinates, sub as many points as possible. Really easy to get the shape wrong for some of these.
• Don't neglect asymptotes. Your calculator will not show these.
• Do graphs in pencil, and then go over them with pen/highlighter as you wish.
• Make sure your stationary points are FLAT at that particular point. This is especially important for stationary points of inflection as people often miss this.

CALCULUS:

• When calculating areas, do yourself a favour and draw the graph. Do not assume it is all above the x-axis
• 'Use calculus' means USE CALCULUS. Especially for exam 2, you must show your working out by hand (though you can do the 'steps' on the calculator).
• Watch max/min problems. The maximum is NOT ALWAYS the turning point. It may infact be an endpoint on a restricted domain. Always, always plot the graph of everything where possible to ensure you don't do silly things like this.
• When differentiating/antidifferentiating don't forget to change cos to sin and sin to cos. Very easy to miss, especially if it is 221sin(23x-1183) or something like that.

FUNCTIONS:

• f(x/2) is wider than f(x). f(2x) is narrower than f(x), even though you might assume as 2x > x, the graph must be wider. This is WRONG.
• Been said before, but be careful with domains and ranges of composite functions. Remember the range of the inner function must be a subset of the domain of the outer function.

That's pretty much most of the stuff I had. Sorry about the lack of probability, I've forgotten how to do most of it... I might add to this later if I think of anything else...

Anyway, good luck guys! Just remember to be really careful when checking over your work, as you don't want to be losing unnecessary marks. Don't panic when you see a hard question, just remind yourself that if it's on the exam, it's in the study design and therefore you KNOW how to do it. I wish you guys all the very best for Tuesday & Wednesday, and I know you'll all ace it!

[daliu]

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

Hahaha Specialist Mathematics ftw!

Here's some more ideas for consideration:

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 (actually meant to be "0 more than or equal to p more than or equal to 1", couldn't type it though...)
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.