VCE Methods Question Thread!

[MB_]

"At what time of the day is the temperature in the office decreasing most rapidly?"
How would you approach this?
Could you always find the derivative of the derivative of the graph and set that = 0? It worked in this case but I don't know if it always does. What also kind of worked is adding the stationary points and dividing them by 2.
Are these always true? If so, why?

Thank you in advance!

The steepest ascent/descent occurs at the points where the gradient is maximal/minimal. These points are given by the max's and min's of the derivative.

[tigerclouds]

The steepest ascent/descent occurs at the points where the gradient is maximal/minimal. These points are given by the max's and min's of the derivative.

Oh ok that makes sense, thank you!

[sarah15]

Hi! Could someone please explain MCQ Q4 from 2017 VCAA exam 2. I don't know how to approach the question. Also, for MCQ Q20, why is pi/4 used as the base to find the area of the triangle? Why is it not pi/2?
I don't understand the solutions for q4h and q4i(ii) from section B of that exam.

Thanks

[Matthew_Whelan]

Q4 - f(g(3)) = ?
 G(3)=2 so it’s f(2) = 5

Q20 - I think the answers are a bit vague, if you split the triangle into two triangles it’s easier
So area = 0.5 x Pi/6 x ((3)^1/2)/2 +  0.5 x Pi/3 x ((3)^1/2)/2

[KiNSKi01]

Came across an MCQ with really unfamiliar notation:

Let X and Y be two random variables such that X~N (-1, 3^2) and Y~(5,2^2). The value of c such that Pr(X<c) = Pr(Y>c) is closest to

I could tell that somehow the question involved the normal distribution but I zero clue what the whole "X~N (-1, 3^2) and Y~(5,2^2)" business was 
Can someone please explain what they mean

[ArtyDreams]

Came across an MCQ with really unfamiliar notation:

Let X and Y be two random variables such that X~N (-1, 3^2) and Y~(5,2^2). The value of c such that Pr(X<c) = Pr(Y>c) is closest to

I could tell that somehow the question involved the normal distribution but I zero clue what the whole "X~N (-1, 3^2) and Y~(5,2^2)" business was 
Can someone please explain what they mean

The notation works like this
The N just means that it is a normal distribution. The X is the letter that is being defined. The first number in the bracket gives the mean and the second number is the standard deviation squared. (i.e. the variance)

Like this;
X ~ N(mean, standard deviation^2)

Hope that made sense!

[KiNSKi01]

The notation works like this
The N just means that it is a normal distribution. The X is the letter that is being defined. The first number in the bracket gives the mean and the second number is the standard deviation squared. (i.e. the variance)

Like this;
X ~ N(mean, standard deviation^2)

Hope that made sense!

Yeah cool! Had never seen it written in that form before

[ArtyDreams]

Yeah cool! Had never seen it written in that form before

Just as another note, a similar notation is used for Binomial distributions.

ie. X ~ Bi(number of trials, probability of success)

[KiNSKi01]

Just as another note, a similar notation is used for Binomial distributions.

ie. X ~ Bi(number of trials, probability of success)

Lol yeah I did know that so I probably should've have used some common sense and realised what the equivalent expression for norm distribution was  Reckon the sd squared threw me off

[BitcoinEagle]

Could anyone help me in general with how to approach questions that say: Find the value of ___ such that the inverse and function are tangential and do not cross?

Eg: How would you do the final question of MAV 2019? (The final three parts)

[Matthew_Whelan]

So I haven’t started a summary book because I haven’t really had to all year, but what do most people find useful to have in it? I’m thinking I’ll just put a bunch of difficult past questions that could be emulated this year but if anyone has suggestions that would be great. Thanks in advance.

Could anyone help me in general with how to approach questions that say: Find the value of ___ such that the inverse and function are tangential and do not cross?

Eg: How would you do the final question of MAV 2019? (The final three parts)

Are there answers for these?

A) is b>3 I think
B) tbh this confused me a bit, so unsure
C) y = 6ex + 19e
D) graph and it’s tangent at x=-3 interesect at x =-3, -1.97, 0.56
So area is integral between those points, use the graph to assist, area = 86.83 sq units
E) requires an explanation
F) Switch x and y, use CAS to rearrange for solution
G) switch x and y, rearrange on CAS again for inverse = (ln(x-a) - 2x - 1)^1/2 for x>a
H) and above I understand what they’re asking but I don’t quite understand how to attain the answer.

The first few I’m somewhat confident with but not sure about the last few.

[xxxjss]

Could anyone help me in general with how to approach questions that say: Find the value of ___ such that the inverse and function are tangential and do not cross?

Eg: How would you do the final question of MAV 2019? (The final three parts)

To build from Matthew,
for B) you want the interval at which the graph is one-one, you could possibly differentiate x^3 + bx^2 +3x+1, and do the discriminate for no solutions, as if there are no stationary points(turns) it is most likely one-one . Doing a quick CAS, i think the values should be -3 < b < 3 !!!

For E) this is composite function, and g(h(x)) will only exist if ranh is a subset of domg; prove this and bingo bango

F) swap x and y and rearrange
you would get to the point of loge(x)= y2 + 2y + 1
Complete the square for y --> (y+1)^2 = loge(x) ... can do the rest [remember the domain is positive so accept + not -]

G) hmm my inverse is y= (loge(x-a))^0.5 -1

H) tangential to each other means they have to touch and as they are inverses they would touch on y=x (I think), thereby they must have the same gradient, as well have the same gradient of y=x. I would differentiate f(x)=x and solve to find a value of x, then differentiate f^-1(x)=x, sub in the value of x and then algebra to find a value of a (I'm not too sure if this is correct though...)

i) i actually have no idea how to do this, but my best guess is how a is only a translation (no dilation/reflection), thus if you can translate the inverse end point to be at x=-1 (end point of f(x)), then one of the intersecting points is -1 and a would be the translation to that point
 Could then solve to find the other intersection

j) just area between two curves with the terminals found in i) ?

Sorry if this doesn't help or is wrong XD

[KiNSKi01]

yo for q2b on Methods Exam 2 2013 im kinda confused

It says 'if she is not fit one month' so I assumed this intial month wouldn't come into the calculations but it did
I've seen similar VCAA questions where the initial condition wasn't involved in the calculation

So what would be a dead give away on whether or not we have to take the intial condition into consideration, would it have something like 'given that.."

[antigony]

https://postimg.cc/7bM2cJ0Y

I can narrow it down to C or E, but I don't understand why you have to apply the translation first in order to get the right answer (I thought you had to do dilations before translations?):
(5+5)x3 = 30 (correct)
3x5+5 = 20

By the way, this is MC20 from the 2016 VCE paper. I don't really understand the explanation in the examiner's report.

[Tau]

https://postimg.cc/7bM2cJ0Y

I can narrow it down to C or E, but I don't understand why you have to apply the translation first in order to get the right answer (I thought you had to do dilations before translations?):
(5+5)x3 = 30 (correct)
3x5+5 = 20

By the way, this is MC20 from the 2016 VCE paper. I don't really understand the explanation in the examiner's report.

Hope this helps, it’s a bit messy . I’ve included the algebraic approach and geometric.

First find the transformation, then (1) draw it out graphically using the given integrals as areas of rectangles over regions, then manipulate those regions per the transformations and calculate the new area.

Or (2), using the transformation, write out the integral in terms of the old function and evaluate that.