VCE Specialist 3/4 Question Thread!

[keltingmeith]

The best way to do euler's method questions is to either, a) write a program on your calculator that does it for you (I did this, it was a time saver), b) set up a little table.

So, going with the table idea for now, let's recall the formula for euler's method is (or whatever variant you want). So, we get:

x        y           dy/dx                    h*dy/dx
1        0          (3-0)/1 = 3            0.5*3=1.5
1.5     1.5       (3-1.5)/1.5 = 1      0.5*1=0.5
2        2          (3-2)/2 = 0.5         0.5*0.5=0.25
2.5     2.25

So, y is approximately equal to 2.25 when x = 2.5       

[hyunah]

So for this question
If dy/dx = tan(x^2) and y = 1 when x = 0, use your CAS to find y when x = 0.4
When I type in intergral tan(x^2) in CAS it gives me intergral tan(x^2), how am I supposed to solve this question?

thank you

[keltingmeith]

The CAS spits that out because that's something we cannot integrate - you see, the derivative of any elementary function is another elementary function, but the integral of an elementary function isn't always an elementary function. So, for cases like these, we have numerical methods and estimations to solve them, such as Euler's method, or drawing rectangles under curves.

The reason the question says "use your CAS" is because your textbook wants you to learn "faster" ways of using Euler's method, like the pseudo-excel spreadsheet.

[Conic]

Using the fundamental theorem of calculus, we have





                     

                     

[hyunah]

thank you so much

If a freezer is at a constant – 6 degrees and a bottle of champage at 24 degrees is place in freezer, its temp drops by 4 degrees in 1 min
a)   Time taken to reach 10 degree?
4degree - - - 1 min
14 degree - - - x min       ans – 3.5 min?
b)   Time when bottle at 4 degrees
4degree - - - 1 min
20 degree - - - x min       ans – 5 min?
c)   Time range between 10 degrees and 4 degrees is
3.5 to 5 mins?

is it just a simple ratio question?

[keltingmeith]

Nope - this is differential equations.

Using Newton's Law of Cooling,



And now that you've made that you've set up this equation, you use that to solve the question.

[hyunah]

nothing about the 4 degrees in one min thing?

I tried using that for a similar question:
If the freezer is at a constant -6 degrees celcius and a tray of water at 12 degrees is placed in the freezer, its temperature drops by 4 degrees in 1 min
a)   Find the time taken for the water to reach freezing point?
180 secs
b)   Temp of the tray at time t
but i got a ridiculous result here.. 18 . 7^t - 6???

[Nato]

With DE's - the inflow/outflow problems, do we always need to rearrange the equation?

say we find the equation for and then antidifferentiate it. it will look like t=(sutff here in terms of Q) + C
if we wanted to find "the amount of salt in the tank after 20 minutes" - do we need to rearrange that equation to make it like Q=(stuff in terms of t)? or can we just substitute the "point" into the one that is in terms of Q?

hope this makes sense.

[nhmn0301]

Find the integral of sin^-1(x) dx

How would we do this? Is it integration by recognition?

If they give you some previous question, you can use hence integration, otherwise I think we need to use integration by parts. But it's definitely not in VCE course and VCAA would not ask you these types of question. Anw, let me know if there is any errors!

[keltingmeith]

With DE's - the inflow/outflow problems, do we always need to rearrange the equation?

say we find the equation for and then antidifferentiate it. it will look like t=(sutff here in terms of Q) + C
if we wanted to find "the amount of salt in the tank after 20 minutes" - do we need to rearrange that equation to make it like Q=(stuff in terms of t)? or can we just substitute the "point" into the one that is in terms of Q?

hope this makes sense.

You can use whichever equation you like - they're not inherently different at all, really. Just that one is rearranged for Q, and one is rearranged for t. If you have to rearrange for Q to get your answer, you'll have to do some more work, sure, but you'll have to have done that work anyway to rearrange for Q in the first place.

Oh  thanks! I definitely don't need to know integration by parts. Interesting though... this question was on an Insight Exam 1 without guidance (i.e.. "hence integrate etc..) so I think they are being dodgy...

Yeah, don't worry, some of the third party papers are a little dodgy like that. I had to integrate cosec^2(x) on a Heffernan exam 1 once, my spec teacher applauded me for setting up my own integration by recognition, and then said there's no other way to do it.

(also, kudos to nhmn for knowing integration by parts! There's one way to keep ahead of the pack, hahah)

[keltingmeith]

I hate to burst your bubble, but it doesn't quite work that way - often, if something has an anti-derivative, that anti-derivative can be found by substitution OR integration by parts, not both. For example, cannot be found with integration by parts, only substitution.

Integration by recognition, which you should know from methods, is the precursor to integration by parts. So, it is worth learning if you like to know things on a deeper level. But, no, it's not worth knowing just to pick up some extra marks in specialist, because they'll never give you an integral that you'll need integration by parts for.

[pi]

So it's not worth learning integration by parts?

You can use it to check your answer but you can't use it in any exam. The exams are written so you don't /need/ to use it either, they're not out to get you.

[lzxnl]

I hate to burst your bubble, but it doesn't quite work that way - often, if something has an anti-derivative, that anti-derivative can be found by substitution OR integration by parts, not both. For example, cannot be found with integration by parts, only substitution.

Integration by recognition, which you should know from methods, is the precursor to integration by parts. So, it is worth learning if you like to know things on a deeper level. But, no, it's not worth knowing just to pick up some extra marks in specialist, because they'll never give you an integral that you'll need integration by parts for.

I hate to burst your bubble too, but I can integrate that by parts.
Let

So integral becomes

So it's not worth learning integration by parts? Like if I am stuck on a substitution question on the VCAA exam, can I use it? Would I be awarded any more marks than the answer mark?
Also, since I haven't looked at integration parts, does it work for any integral? If so, I would definitely invest a few hours into learning it to check answers...

If you EVER don't know how to integrate a function f(x), try differentiating x*f(x) and seeing if that helps. This trick single handedly allows you to integrate logs and all inverse trig functions.

[keltingmeith]

I hate to burst your bubble too, but I can integrate that by parts.
Let

So integral becomes

Welp, that'll teach me for not checking my examples, hahah.

[keltingmeith]

can you give an example/explain how the x * f(x) trick works?

To find the integral of ln(x):



So basically, if they say "find the integral of f(x)", and you can't do it, instead find the derivative of x*f(x), and then see if that helps you find the integral of what you're looking for.