VCE Methods Question Thread!

[kinslayer]

My class skipped over Linear Approximation when we were doing differentiation in class. I've tried going back over it now, but i don't think I really understand it. Or at least i really have no idea how to do any questions. I don't understand the steps taken in my textbook for this example:

Given that f(x) = x^4 - x^3, find in terms of p the approximate increase in f(x) as x increases from 2 to 2 + p, where p is small.

For the percentage change questions, do you use the percentage change formula from the start, or is it something you use after you find the linear approximation? Is linear approximation ever actually show up in any exams? Is it more likely to show up in exam 1 or exam 2?

Thanks

Linear approximation is using the first derivative. The idea is that the closer you get to the point x = a, the more your function looks like a line, and if you can get arbitrarily close to x = a then your function looks like the tangent line, so that is what you use to approximate the function.

In this case f'(x) = 4x^3 - 3x^2, and the slope of the tangent line is 4*8 - 3*4 = 20. f(2) = 16 - 8 = 8.

So you take f(2), then you go p units to the right, up the tangent line. The slope of the tangent line is 20 at x = 2, so the approximation is:



You can check with small values of p:



Compare this with the actual value of

So that's a fairly good approximation.

[keltingmeith]

My class skipped over Linear Approximation when we were doing differentiation in class. I've tried going back over it now, but i don't think I really understand it. Or at least i really have no idea how to do any questions. I don't understand the steps taken in my textbook for this example:

If it's any consolation, since you do specialist, you'll cover Euler's method in there, and they're pretty much the same thing. (and unlike linear approximation, Euler's method comes up a bit more regularly)

Given that f(x) = x^4 - x^3, find in terms of p the approximate increase in f(x) as x increases from 2 to 2 + p, where p is small.

Now, the general way to do these questions is identify what you want:

a) How much you have increased by,
b) What you have increased to.

Once you've figured out which of these you are, you can apply the correct formula - the first being , and the second being . However, note that , and , so the second formula reduces to

What does this mean? Welp, if you have no idea what to do, you can actually apply the first formula all the time and you'll be fine - you just need to know when you need to add it to f(x) (scenario b) and when you don't add it (scenario a). So, doing it "this way" instead of kinslayer's (read: they're actually the same method, even if they don't look it):



Now, we add it to f(x), to get:



Which matches kinslayer's answer.

For the percentage change questions, do you use the percentage change formula from the start, or is it something you use after you find the linear approximation?

See above. This is certainly a bit of a "real-life" question focusing on what you're actually doing it for, so instead focus solely on the question. What is it asking for? Percentage increase? Did it ask for an approximation of the value? No? Then just give percentage increase.

Is linear approximation ever actually show up in any exams? Is it more likely to show up in exam 1 or exam 2?

If I'm going to be honest with you, I honestly cannot remember the last time it showed up... To memory, I think it was actually cut from the study design, it never showed up. But, it's more likely to come up in exam 1 - in exam 2, you have a calculator that can do approximations, why do a computational method by hand?

[knightrider]

would you be expected to graph the graph of in methods?

[keltingmeith]

would you be expected to graph the graph of in methods?

Certainly - it's just addition of ordinates/calculator work. You shouldn't be asked to sketch all features, though - specifically, the oblique asymptote at y=x.

[Sundal]

If a graph had undergone the following transformations (in this order):
1.) Dilation by a factor of 1/2 from the y-axis
2.) Reflected in the x-axis
3.) Translated 3 units up

and you needed to show the point (x,y) being mapped onto a new set of points, as a result of these above transformations, would this point be mapped onto (x/2, -y +3) or (x/2, -y-3) ?

Cheers.

[knightrider]

How would you guys go about sketching trig graphs with translations in it.

In particular horizontal translations for trig functions.

i find these very confusing when graphs have both horizontal and vertical translations.

How would you guys go about drawing these.

Any tips.

Thanks 

[cosine]

If a graph had undergone the following transformations (in this order):
1.) Dilation by a factor of 1/2 from the y-axis
2.) Reflected in the x-axis
3.) Translated 3 units up

and you needed to show the point (x,y) being mapped onto a new set of points, as a result of these above transformations, would this point be mapped onto (x/2, -y +3) or (x/2, -y-3) ?

Cheers.

(x, y):

1). (2x, y)
2). (2x, -y)
3). (2x, 3-y)

You should note that dilations from the y-axis go in the form of , where n is the coefficient of x. So if they say dilation by factor of then:




So you place a 2 in front of the x.

[qwerty101]

for the last question , part d

http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2010mmcas2-w.pdf

can someone explain to me why one needs to let the two solutions from c equal each other to get it in terms of b (as in why this means there is one stationary point). I don't get it conceptually, the algebra is not the problem.

Does it have anything to do with b only being present to the power of 1 (or one time) so re arranging will give one solution. I would have never considered letting those two solutions equal each other in the first place?


Also, how do you go about answering part e. How can quartics have a maximum of three stationary points (even though the answer to this part is 2). I can see two, i.e. 2 local min/max etc, but i dont see three. I also dont see why f, has a stationary point of inflection and a max turning point. This should only be to the power of 5? since stationary poi is 3 and maximum point is 2 (in terms of powers)?

Also has me thinking why a cubic can have a max and a min, making it 4....
thanks

[qwerty101]

soz for the questions

[qwerty101]

http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2011mmcas2-w.pdf

for 1a),

when they integrated it, why did they keep the positive?, shouldnt the volume be decreasing as time elapses , so there would be a neg parabola?

[Sine]

for the last question , part d

http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2010mmcas2-w.pdf

can someone explain to me why one needs to let the two solutions from c equal each other to get it in terms of b (as in why this means there is one stationary point). I don't get it conceptually, the algebra is not the problem.

Does it have anything to do with b only being present to the power of 1 (or one time) so re arranging will give one solution. I would have never considered letting those two solutions equal each other in the first place?


Also, how do you go about answering part e. How can quartics have a maximum of three stationary points (even though the answer to this part is 2). I can see two, i.e. 2 local min/max etc, but i dont see three. I also dont see why f, has a stationary point of inflection and a max turning point. This should only be to the power of 5? since stationary poi is 3 and maximum point is 2 (in terms of powers)?

Also has me thinking why a cubic can have a max and a min, making it 4....
thanks

(ab+1)/4a & 1/a Are equations for the S.P, e.g. If a=2 & b=2 there are S.P at x=1/2 and x=5/8 & . We let both equations equal to find which value will be the same for both equations for S.P and therefore have only one distinct S.P
Quartics have a maximum of three S.P because its derivative (a cubic) can have a maximum of 3 x-axis intercepts.

[MTrinita]

Hey guys just wondering does anyone have the worked solutions to Heinemann Maths Methods CAS Units 3/4?

[Sine]

http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2011mmcas2-w.pdf

for 1a),

when they integrated it, why did they keep the positive?, shouldnt the volume be decreasing as time elapses , so there would be a neg parabola?

Volume is decreasing but the rate of loss of fuel with respect to time is increasing. e.g Is the rate of loss of Volume when t is time in mins. when the rate of loss of fuel is and when the rate of loss of fuel is

[qwerty101]

Volume is decreasing but the rate of loss of fuel with respect to time is increasing. e.g Is the rate of loss of Volume when t is time in mins. when the rate of loss of fuel is and when the rate of loss of fuel is

i see, its like a reverse way of thinking.

Say i was setting up related rates of change equation i.e. dh/dt = dh/dv etc.... , would i input the rate as a negative for dv/dt? i.e. dv/dt = -6 ?

[qwerty101]

We let both equations equal to find which value will be the same for both equations for S.P and therefore have only one distinct S.P

could you please explain this part again. I dont fully understand. I kind of understand but want it to be elaborated please.