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    Hi, I've been having some trouble with these implicit differentiation questions. If someone could lend a hand, that would be very much appreciated!

    For Q8, I'm not sure how my answer simplifies to the one that is given in the answers. I think my working out is correct.

    For Q15, how are you supposed to work out the maximum and minimum sizes of the population?



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    (Original post by Bazinga?)
    Hi, I've been having some trouble with these implicit differentiation questions. If someone could lend a hand, that would be very much appreciated!

    ...

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    Notice:

    1 = ln e = (1/3) (ln (e^3))
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    (Original post by 2710)
    Notice:

    1 = ln e = (1/3) (ln (e^3))
    Ah yes, thanks!
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    (Original post by Bazinga?)
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    Hi, I've been having some trouble with these implicit differentiation questions. If someone could lend a hand, that would be very much appreciated!

    For Q8, I'm not sure how my answer simplifies to the one that is given in the answers. I think my working out is correct.

    For Q15, how are you supposed to work out the maximum and minimum sizes of the population?



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    "2710" has given useful advice for the first part, I'm sure what they have said will suffice.

    For the second part, the convention is the write your answer isolated your dependant variable on one side of the equation
    Spoiler:
    Show
    P=P_0 e^{\sin{kt}}


    For what value of t does P take its maximal and minimal values? Granted, it is possible to use calculus but, as you are not looking for the value of t itself, it may be good idea to, as t can take on a sufficiently large range of values, consider the maximum and minimum values of \sin{kt}.

    With this knowledge, is there any obvious property of the exponential function that completes the logic (this is not needed if you do the "brute force" calculus method).

    Hope this helps
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    (Original post by Jkn)
    "2710" has given useful advice for the first part, I'm sure what they have said will suffice.

    For the second part, the convention is the write your answer isolated your dependant variable on one side of the equation
    Spoiler:
    Show
    P=P_0 e^{\sin{kt}}


    For what value of t does P take its maximal and minimal values? Granted, it is possible to use calculus but, as you are not looking for the value of t itself, it may be good idea to, as t can take on a sufficiently large range of values, consider the maximum and minimum values of \sin{kt}.

    With this knowledge, is there any obvious property of the exponential function that completes the logic (this is not needed if you do the "brute force" calculus method).

    Hope this helps
    Yes, I now understand how to answer part 1.

    So I guess I would have to draw the y=sinx graph, which tells me that the maximum point is at y=1 (x=90o) and minimum is at y=-1 (x=270o)

    Wouldn't I need to work out the value of k first? Or the value of Po?

    And what do you mean by the "brute force" calculus method?
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    (Original post by Bazinga?)
    Yes, I now understand how to answer part 1.

    So I guess I would have to draw the y=sinx graph, which tells me that the maximum point is at y=1 (x=90o) and minimum is at y=-1 (x=270o)
    Perfect!

    But you do not need a graph. Simply note that -1 \le sinA \le 1, \ \forall A.
    Wouldn't I need to work out the value of k first? Or the value of Po?

    And what do you mean by the "brute force" calculus method?
    No! That's a beauty of it! There is no way to calculate P_0 without extra information. It represents "initial population". Note, however, that, with the ratio, this quantity cancels nicely

    Finding t such that \frac{dP}{dt}=0 and substituting this in to the original equation
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    (Original post by Jkn)
    Perfect!

    But you do not need a graph. Simply note that -1 \le sinA \le 1, \ \forall A.

    No! That's a beauty of it! There is no way to calculate P_0 without extra information. It represents "initial population". Note, however, that, with the ratio, this quantity cancels nicely

    Finding t such that \frac{dP}{dt}=0 and substituting this in to the original equation
    ...I've never heard anyone mention beauty in the same sentence as maths, but each to their own I guess!

    So would the maximum population be P=P_oe and the minimum is P=P_oe^-1? So P_oe:P_oe^-1. So e : e^-1? This doesn't look quite right
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    (Original post by Bazinga?)
    ...I've never heard anyone mention beauty in the same sentence as maths, but each to their own I guess!
    I meant that as in the colloquial expression rather than the maths actually being beautiful (it's not!)

    But I find this a tragedy! Mathematics is full of beauty! Everything you learn up until the end of a-level isn't real maths it's "mathematical methods" (olympiads being the only exception). If you were in a Spanish class learning how to say "my name is..." and "how much is a...", can you really say that you understand spanish literature? This is the best analogy I can draw between school maths and university/"real" maths!

    When and if you graduate to "real" maths, you will find that the beauty lies in the subjectivity! In realising there were may ways to prove a theorem and your method is different to someone else's. People can even have a sort of personality that comes across int their work in that respect! To take a simple statement like "\pi goes on forever in a non-repeating way" and then, using mathematics to prove it in a matter of 20 minutes is one of the greatest satisfactions/beauties that exist; to know that, despite mankind having speculated over that question for 5000 years and computers checking up to trillions of digits, the proof is right there is front of you; pure truth! And to know that this truth has always been and will always be no matter who questions it. This is why mathematics is so much more than a scientific tool and so much more important than any other subject! :eek:
    So would the maximum population be P=P_oe and the minimum is P=P_oe^-1? So P_oe:P_oe^-1. So e : e^-1? This doesn't look quite right
    Yes or, simply, 1:e^2 :lol:
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    (Original post by Jkn)
    I meant that as in the colloquial expression rather than the maths actually being beautiful (it's not!)

    But I find this a tragedy! Mathematics is full of beauty! Everything you learn up until the end of a-level isn't real maths it's "mathematical methods" (olympiads being the only exception). If you were in a Spanish class learning how to say "my name is..." and "how much is a...", can you really say that you understand spanish literature? This is the best analogy I can draw between school maths and university/"real" maths!

    When and if you graduate to "real" maths, you will find that the beauty lies in the subjectivity! In realising there were may ways to prove a theorem and your method is different to someone else's. People can even have a sort of personality that comes across int their work in that respect! To take a simple statement like "\pi goes on forever in a non-repeating way" and then, using mathematics to prove it in a matter of 20 minutes is one of the greatest satisfactions/beauties that exist; to know that, despite mankind having speculated over that question for 5000 years and computers checking up to trillions of digits, the proof is right there is front of you; pure truth! And to know that this truth has always been and will always be no matter who questions it. This is why mathematics is so much more than a scientific tool and so much more important than any other subject! :eek:
    I can't tell if you're being extremely sarcastic or if you're actually being serious! I just don't see the point of maths (this is probably not wise- seeing as I'm posting this on the maths forum!) when it's more complicated that simple sums, i.e. addition and subtraction The only part of maths which I did enjoy was statistics as that seemed to have some sort of real-life application. Though I do kind of understand where you're coming from, as maths is unique in the sense that it really does give a sense of accomplishment when you complete a tricky question!

    After a quick check of your profile, I now understand why you're so passionate about maths! I might be seeing you at university next year! Though I'll be off to study Medicine Which in my opinion, is superior to maths

    Yes or, simply, 1:e^2 :lol:[/QUOTE]

    If e = e^-1, then surely 1 = e^-2? Does it not matter which way round it is? Or would e=e^-1 generally not be accepted?
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    (Original post by Bazinga?)
    I can't tell if you're being extremely sarcastic or if you're actually being serious! I just don't see the point of maths (this is probably not wise- seeing as I'm posting this on the maths forum!) when it's more complicated that simple sums, i.e. addition and subtraction The only part of maths which I did enjoy was statistics as that seemed to have some sort of real-life application. Though I do kind of understand where you're coming from, as maths is unique in the sense that it really does give a sense of accomplishment when you complete a tricky question!
    Hahaha very few people really get to experience the joys of pure Mathematics (which is maths with no immediate application - that aims to create maths for the subject's sake!) Well I personally hated/hate all of the maths a-levels (especially statistics!)

    Well you get that same sense of accomplishment in the sciences as well, so it's not unique to maths.

    What I particularly like is that, from a philosophical point of view, real truth can only and will only existence in mathematics. Science is approximation-based (empirical proof can never be truly definitive) and everything else is limited by our own humanity. For example, if there were intelligent civilisations on other planets, whose form could be so different that we couldn't even recognise them, they may not have language in the way we can recognise it. If they had science, it might consist of different theories that "work" yet are still radically different, their medicine would certainly be different as they may be of a different form and yet their mathematics, if they have it, can only be the same, never different!
    After a quick check of your profile, I now understand why you're so passionate about maths! I might be seeing you at university next year! Though I'll be off to study Medicine Which in my opinion, is superior to maths

    If e = e^-1, then surely 1 = e^-2? Does it not matter which way round it is? Or would e=e^-1 generally not be accepted?
    Oh congrats! Which college are you going to?

    Pffffffft, how does it feel to be doing the second hardest subject to get into uni for? And please don't make me rank it based on intellectual elitism!

    They are all equivalent so, in theory, it would not matter. However, there is often an expectation that students will leave answers in a sensible form which, in this case, amounts to the one where there are no denominators and one side of the ratio is "1" (which makes the result more easy to interpret than the others).

    The final result can be summarised: "The maximum population size is a factor of e^2 larger than the minimum population size"
 
 
 
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