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Year 13 Maths Help Thread

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Original post by Zacken
You even wrote "differentiation" on top! :tongue:


haha. :facepalm:. Ohhh I am so not ready for C3
To work out a range do you make the equation 0 then solve it. Then the use the value you get. E.g. if it was 3/2 then the range would be f(x) > or equal to 3/2 ?
Original post by kiiten
To work out a range do you make the equation 0 then solve it. Then the use the value you get. E.g. if it was 3/2 then the range would be f(x) > or equal to 3/2 ?


No specific criterion, sketching a graph usually helps.
Original post by Zacken
No specific criterion, sketching a graph usually helps.


How would you go about finding the range without a graph (its just possible y values, right?) e.g. if the equation is something like 3x^4 + 2x^3 etc.
Original post by kiiten
How would you go about finding the range without a graph (its just possible y values, right?) e.g. if the equation is something like 3x^4 + 2x^3 etc.


Differentiate and find minimum/maximum points. It's a positive quartic so the range will be >= (min value of quartic), if it was a cubic, the range would be all of R, etc..
Original post by kiiten
How would you go about finding the range without a graph (its just possible y values, right?) e.g. if the equation is something like 3x^4 + 2x^3 etc.


Find the minimum point and it's everything above it/equal to it.
Original post by Zacken
Differentiate and find minimum/maximum points. It's a positive quartic so the range will be >= (min value of quartic), if it was a cubic, the range would be all of R, etc..


Original post by RDKGames
Find the minimum point and it's everything above it/equal to it.


Thanks :smile: - so if you find the min point will this work for any equation?
Original post by kiiten
Thanks :smile: - so if you find the min point will this work for any equation?


No because by that logic you would get the range to f(x)=sin(x)f(x)=sin(x) wrong. Yes with typical polynomials but you must have knowledge of the curves you're working with because differentiation helps find the turning points and with higher degree polynomials there will be multiple of those; up to you to find which one gives the minimum point. Min points are only indications and potential candidates to the boundary of the range.
(edited 7 years ago)
Original post by kiiten
Thanks :smile: - so if you find the min point will this work for any equation?


No, lots of functions aren't continuous.
Original post by RDKGames
No because by that logic you would get the range to f(x)=sin(x)f(x)=sin(x) wrong. Yes with typical polynomials but you must have knowledge of the curves you're working with because differentiation helps find the turning points and with higher degree polynomials there will be multiple of those; up to you to find which one gives the minimum point. Min points are only indications and potential candidates to the boundary of the range.


Now im confused. What are the general rules for finding the range? :s-smilie:
Original post by kiiten
Now im confused. What are the general rules for finding the range? :s-smilie:


For any function you deal with in C3 it's just the matter of finding the stationary points I believe (dont remember any questions from the module that suggest otherwise). For more complex functions there's different things to consider such as asymptotes and limits.
(edited 7 years ago)
Original post by RDKGames
For any function you deal with in C3 it's just the matter of finding the stationary points I believe (dont remember any questions from the module that suggest otherwise). For more complex functions there's different things to consider such as asymptotes and limits.


The stationary points are the range?
Original post by kiiten
The stationary points are the range?


No. Stationary points just indicate min/max points where the gradient is 0.
Original post by kiiten
:argh: what is the range and how do i find it?


For what?

There is no single rule for finding the range of ANY function. Give me an example and I'll tell you what you can consider.
Original post by kiiten
:argh: what is the range and how do i find it?


Edit: Ignore my post - I should have seen the 'thread'.
(edited 7 years ago)
Original post by kiiten
Now im confused. What are the general rules for finding the range? :s-smilie:


There aren't any (for the second time!). You need to just look at the function and use some thinking skills. Drawing a graph in the beginning when learning about this sorta stuff helps a lot.
Original post by kiiten
Its fine i know its a stupid question. I already looked it up but im so confused about finding the range without a graph so ill just leave it for now.


You need knowledge of how various graphs look like before you can move onto finding the range without graphs because then you know what methods you can use; such as finding the minimum/maximum points.

If in doubt, sketch the graph yourself.
I'll volunteer to be a helper as I really like to help people and hopefully in September I'll be studying maths at university.
Original post by kiiten
Its fine i know its a stupid question. I already looked it up but im so confused about finding the range without a graph so ill just leave it for now.


I take it back, because I don't know what textbook you're using (if any) and how the range is defined may be quite unclear - I was just jumping to an incorrect conclusion.

I personally haven't come across a situation (at A-level) where you can't get away with doing the following.

Think of the range as what you can get out from the function, which is defined not only by a formula but by the domain of the formula as well.

Say I had a function y = x + 5, where the domain (the x values the function accepts) is between 2 and 4 inclusive. Then the range of values that you can get out (i.e the range) is between 7 and 9 inclusive. It's a good idea to sketch the graph when doing this, but you can test the 'edges' of the domain (i.e 2 and 4) and notice that the function is increasing inbetween, so 2 and 4 cover the range of values that y can take.

The most difficult bit is getting your head around domain and range. After that, it's about practice. You quickly become familiar with the domains and ranges of standard functions, like e^x, lnx, trig functions in particular.

y = e^x for example lets you put in any x value but has a range of '>0' as you can see by the graph. Changing it so that it's y = e^3x makes no difference to the domain, nor the range for that matter, as you still get the same values out for some value of x. You can draw the two graphs if you are not sure.

lnx does not let you take in negative x values, so when it's of the form ln(f(x)) where f is any function of x, f(x) can not be less than or equal to 0. A good way to remind yourself of this is trying to put in ln0 or ln-5 or into your calculator.

So the domain of lnx is x>0. The range of values that can come out though, as you can see from the graph, is also any value, so it's the domain that is important here, not the range. If we say f(x) = (x-5), then lnf(x) = ln(x-5) and f(x) can not be less than or equal to 0 (otherwise you are putting in a negative value, which ln doesn't like) so x must be >5 for the domain. So with all of these it's just about recognising what you know and how it's different to find the domain/range.

Lastly, trig functions. You know that the range of cosx and sinx is between -1 and 1. No difference to the domain or range if it's 3x or 5x or x^2 + 3x + 5 instead of x. So if you had y = 3 + 5cosx, you take it in steps. cosx has a range of -1 to 1. So 5cosx has a range from -5 to 5. So 3 + 5cosx has a range from -2 to 8. Tan takes all real values as the range but does not like it when you put in 90 degrees or 270 or adding any multiple of 180 to those numbers. (Why? Because sin90/cos90 = 1/0 = :s-smilie:).

So if you had y = tan(x-10) x can not take 100 + 180k where k is any integer, eg if k = 0 then y = undefined. So that limits the domain.
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Avatar for mik1a
mik1a
10
The way I work out the range? I have a mental image of what general curves look like (trig functions, quadratic, cubic, logs etc....), then in my head I identify where on these vague shapes are the minimum and maximum heights.

For example in a quadratic function y = x^2 + x + 2, you know it looks like a "U" shape, so its range is going to be infinite in the upwards direction (because the top of the U never stops going up), but bounded in the negative direction at the very bottom of the "U" shape.

The next mental step is figure out how to find that point. Sometimes you just need to know some properties of each curve (e.g. exp(x) cannot go negative, sin(x) goes between -1 and +1), then work from there.

For a quadratic "U" shape it's got a single minimum point, so you find it: differentiate (2x + 1) set equal to zero (2x + 1 = 0) and solve (x = -1/2), then find the corresponding height value (y = (-1/2)^2 + (-1/2) + 2 = +1/4 - 1/2 + 2 = 7/4), and you know that all other points in the curve must be higher than that. So the range is y >= 7/4.

A sanity check always helps when you've got time. We said the curve reaches a minimum of 7/4 at -1/2. So let's check what it is either side of this. When x = 0, y = 2 which is > 7/4 so that's good. When x = -1, y = 2 again, which is also in the range.

Practice and some experience, that's all it takes.

I might also add one useful thing to know, since you asked about the range of a y = ax^4 + bx^3 type curve. The shape of the curve is kind of dominated by the highest order term. So this particular curve has the shape of a quartic curve, which in general is like a "W" shape. Lots of vague maths going on here but it's served me well through grad school. With a "W", it's almost the same as the "U" of a quadratic, except you've got two minima, and it's not immediately obvious which one is the minimum, so you've got to find both of them and take the lower of the two as your minimum range. As before, there's no upper range because the two side edges of the "W" continue upwards forever.

One more hint: if the curve is linear (x^1) or cubic (x^3) (or any odd highest term), it has an infinite ranges, because when x gets hugely positive x^3 gets hugely positive, and when x gets hugely negative, x^3 gets hugely negative. This doesn't happen with the even ones.
(edited 7 years ago)

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