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#### zhen

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##### Zhen's Mathematical Methods Exam 1 Guide
« on: November 16, 2017, 06:44:26 pm »
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Zhen's Mathematical Methods Exam 1 Guide
I'm going to start this post off by saying that I'm not one of those genius methods students that will end up getting 50 raw. In fact, I probably won't even get a 40 raw in methods. But, I've realised that you don't have to be a maths genius to give people advice. In this guide I'll give some tips on how I improved my exam 1 scores, which will hopefully help out the weaker methods students.

What should I do during reading time?

What should I do during writing time?
I don't have any really amazing tips for during writing time. I personally worked from start to finish in the order the exam was in, so that I wouldn't accidentally forget a question I skipped. But, I feel like if you're spending way too much time on a certain question, you should definitely skip it and go back to it. Also, the last question of the 2017 methods exam 1 was really controversial, as VCAA made us find the gradient at an endpoint. Also, the graph was slightly misleading. My only tip after experience from going through a couple of dodgy practice exams is that you should always put an answer down no matter how dodgy you think it is. Also, if this dodgy answer is the only possible way to solve the question then just put it down. It's better to get some method marks then lose all your marks because you put nothing down.

Ok, I've finished all the questions and I have 20 minutes of exam time left. What do I do now?
During my trial exam at school, all I would do after I finished the exam was stare blankly at my answers and pray that they were correct. (I was doing heaps of praying after all my actual methods exam 2 finished  ) This was the biggest waste of time ever. Instead you want to be properly checking your answers and eliminating those silly mistakes. Personally, after I finished my actual exam, I'd go back to the first question and literally redo the entire exam again. I'd cover my working (so I'm not influenced by the prior answer) and redo it and compare my two answers and see which one was correct. Also, doing the question again with a different method also helps to avoid making the same mistake twice. In the actual exam, doing this literally saved me like 10 marks as I found a ton of mistakes I made. So I definitely recommend this method of rechecking your answers.

Time to cover some common exam questions
Here I'll go through different common VCAA questions
Simple Differentiation
VCAA usually have a simple differentiation question in every exam. Just be careful that you answer the entire question. If the question provides f(x) and finds f'(1), make sure you evaluate the derivative at x=1. Here's my made up example of a question VCAA could ask.
$Let\ y=\frac{sin^2(x)}{x+3}\\ Find\ \frac{dy}{dx}$
Ok, so this type of question is really common in methods exam 1. In reading time the first thing I'd notice is that it'll require the use of the quotient rule and chain rule. In the first try I'd use the quotient rule, which gives
$\frac{dy}{dx}=\frac{2sin(x)cos(x)(x+3)-sin^2(x)}{(x+3)^2}$
Ok, so when I'm checking my answers, I may redo the question using the same method and cover my previous answer, or I'd end up using another method, such as the product rule for this question.
$Rewrite\ it \ as \ y=sin^2(x)(x+3)^{-1}$
Using the product rule, I would get
$\frac{dy}{dx}=2sin(x)cos(x)(x+3)^{-1}-sin^2(x)(x+3)^{-2}$
From just looking at it, you know that these two things are exactly the same and should be the correct answer. Ok I'll give you another example.
$Let\ f(x)=x^4e^{7x}$
$Evaluate \ f'(2)$
$f'(x)=4x^3e^{7x}+x^4\times7e^{7x}$
$f'(x)=4x^3e^{7x}+7x^4e^{7x}$
$f'(x)=(4x^3+7x^4)e^{7x}$
$f'(2)=(4\times2^3+7\times2^4)e^{14}$
$f'(2)=144e^{14}$
Average rate of change
To find the rate of change, normally you differentiate the function. However, to find the average rate of change, you don't differentiate. Instead you're asked to find the rate of change across two points.
$Let\ f:[-π,π]\rightarrow R, \ where \ f(x)=5cos(2x)+2$
$Calculate \ the \ average \ rate \ of \ change \ of \ f \ between \ x=-\frac{π}{4} \ and \ x=\frac{π}{3}$
$Average \ rate \ of \ change =\frac{5cos(\frac{2π}{3})+2-(5cos(-\frac{π}{2})+2)}{\frac{7π}{12}}$
$Average \ rate \ of \ change =\frac{-\frac{5}{2}}{\frac{7π}{12}}$
$Average \ rate \ of \ change =\frac{-30}{7π}$
Average value
Here's an example finding the average value of the above function.
$Let\ f:[-π,π]\rightarrow R, \ where \ f(x)=5cos(2x)+2$
$Calculate \ the \ average \ value \ of \ f \ over \ the \ interval \ -\frac{π}{4}\leq x\leq \frac{π}{3}$
$average \ value=\frac{1}{\frac{7π}{12}}\int_{-\frac{π}{4}}^{\frac{π}{3}}5cos(2x)+2 \ dx$
$average \ value=\frac{14π+15(\sqrt{3}+2)}{7π} \ (I \ used \ the \ CAS \ for \ this \ cause \ I \ was \ lazy)$
Integration by recognition
Here's an example I'm making up on integration by recognition
$Let \ y=3xlog_e(2x)$
$Find \ \frac{dy}{dx}$
$\frac{dy}{dx}=3log_e(2x)+3 \ (product \ rule)$
$Hence \ calculate \ \int3log_e(2x) \ dx$
$3log_e(2x)=\frac{dy}{dx}-3$
$\int 3log_e(2x) \ dx=y-3x+c$

Composite functions
$Let \ f(x)=x^2 \ and \ g(x)=sin(x)$
$State \ the \ domain \ and \ range \ of \ f(g(x))$
Ok, so assuming that f(g(x)) exists, the domain of the composite function is the domain of the inside function. The domain of g(x)=R, so the domain of f(g(x))=R. The range is a bit trickier. First evaluate the range of g(x), the inside function. Range of g(x) is [-1,1]. Ok, how I find the range of f(g(x)) is I treat the range of g(x) like the domain of f(x). I find the range f(x) given that -1≤x≤1. From a quick sketch of f(x) over this domain, or a bit of intuition, you can see that the range of f(g(x))=[0,1]

Exponentials and Logarithms
Solving exponentials and logarithms generally pop up now and then, so I'll do a bit on it. This example will be a hidden quadratics one that's a bit tricky and one that VCAA does a bit.
$Solve \ e^{x}+2=15e^{-x} \ for \ x$
$(e^x)^2+2e^x-15=0$
$Let \ z=e^x$
$z^2+2z-15=0$
$(z+5)(z-3)=0$
$z=3 \ and \ z=-5$
$e^x=3 \ and \ e^x \neq -5, \ as \ e^x>0$
$So, \ x=ln(3)$

Normal distribution
Sometimes they include some simple questions involving the normal distribution. Here is an example.
$Let \ the \ random \ variable \ X \ be \ normally \ distributed \ with \ mean \ 3.0 \ and \ standard \ deviation \ 0.1 \\ Let \ Z \ be \ the \ standard \ normal \ random \ variable, \ such \ that \ Z\sim N(0,1)$
$Find \ b \ such \ that \ Pr(X>3.3)=Pr(Z< b)$
$z=\frac{x-\mu}{\sigma}$
$z=\frac{3.3-3.0}{0.1}$
$z=3$
$So, Pr(z>3)=Pr(X>3.3) \\ As \ the \ standard \ normal \ distribution \ is \ symmetrical \ Pr(z<-3)=Pr(z>3) \\ So, \ b=-3$

I'm really tired right now and honestly too lazy to do anymore today, especially since I don’t know latex. I'm going to post it now and I might add to it later if anyone has any more suggestions of stuff I should add on. I've added on some things after I initially posted it and I think I'm done for now. If anyone has any things they want me to add on, please leave a post below or PM me. Also, can someone notify me if I've made any mistakes with my maths or if you disagree with the stuff said especially since I’m not the best maths student.
« Last Edit: November 06, 2018, 09:03:34 pm by Sine »

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #1 on: November 16, 2017, 07:21:45 pm »
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Oh my goodness, this is amazing!!! This will definitely be of huge help for future 1/2 and 3/4 Methods students
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#### zhen

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #2 on: November 16, 2017, 07:37:40 pm »
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Oh my goodness, this is amazing!!! This will definitely be of huge help for future 1/2 and 3/4 Methods students
Thanks. It’s honestly no where near as detailed or as good as your amazing methods guide that you posted at the start if the year, but I still hope it’ll end up helping the future generations.

#### Eric11267

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #3 on: November 16, 2017, 08:25:35 pm »
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Wow this looks great! Can't wait to see the Exam 2 Guide
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#### exit

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #4 on: November 16, 2017, 08:38:03 pm »
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#### Joseph41

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #5 on: November 23, 2017, 09:15:54 am »
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Great work, zhen.

#### Legolas

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #6 on: November 06, 2018, 06:09:18 pm »
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Wow impressive work!
Just a friendly correction here, for the Log example you gave above, shouldn't it be e^-x instead of e^-1 ?

#### ImTryingIGuess

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #7 on: November 06, 2018, 08:27:53 pm »
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Wow impressive work!
Just a friendly correction here, for the Log example you gave above, shouldn't it be e^-x instead of e^-1 ?

I was curious about this aswell, but it turns out, as 15e^-1 can bee seen as 15/e  you could multiply both sides by e and get the equation above
I hope this cleared things up

Edit: I thought about this and it doesn't really make sense haha, I guess I wanted to reason with it sorry about that!
« Last Edit: November 06, 2018, 09:50:25 pm by ImTryingIGuess »
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#### Sine

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##### Re: Zhen's Mathematical Methods Exam 1 Guide
« Reply #8 on: November 06, 2018, 09:02:20 pm »
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Wow impressive work!
Just a friendly correction here, for the Log example you gave above, shouldn't it be e^-x instead of e^-1 ?

I was curious about this aswell, but it turns out, as 15e^-1 can bee seen as 15/e  you could multiply both sides by e and get the equation above
I hope this cleared things up
I belive it is a typing error. Even if you were to multiply e across to the LHS "e" is still just a constant (2.71...) and wouldn't yield that equation involving e^2x.

I will edit Zhen's post so that no one gets confused in the future.