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Thread: Calculus & Analysis Marathon & Questions

  1. #101
    Senior Member integral95's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Yeah basically, but part a) could've been smacked straight away with the first FTC and the chain rule

    Hiccup on the final answer for d) though
    wait what?
    I don't know if it's legit lol
    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
    ― Brandon Mull

  2. #102
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by integral95 View Post
    wait what?
    I don't know if it's legit lol
    f'(1) = 3√17 not 3√5

  3. #103
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon


  4. #104
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    I'm taking a massively blank stab at this one for my procrastination from accounting, so the quality of this answer will be again, poor lol.










  5. #105
    Taking a break! dan964's Avatar
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    Re: First Year Uni Calculus Marathon

    Wouldn't you have to also check/assume the individual requirements of L'Hs are satisfied, namely the limits have to all exist?

  6. #106
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Change variables.

    P(ex)/Q(x)

    Without Loss of Generality, we can assume P is of degree 1.

    This expression obviously tends to infinity.

    For those who do not see it, apply L'Hôpital's Rule (with respect to x) n times where n is the degree of Q.

    If P is larger, then the contribution to the explosion is greater, and the expression still diverges.

    [QED]
    Last edited by Paradoxica; 13 Jun 2016 at 1:43 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  7. #107
    Exalted Member BlueGas's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    Change variables.

    P(ex)/Q(x)

    Without Loss of Generality, we can assume P is of degree 1.

    This expression obviously tends to infinity.

    For those who do not see it, apply L'Hôpital's Rule (with respect to x) n times where n is the degree of Q.

    If P is larger, then the contribution to the explosion is greater, and the expression still diverges.

    [QED]
    How are you able to do these uni questions when you're not even at uni?

  8. #108
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by BlueGas View Post
    How are you able to do these uni questions when you're not even at uni?
    These things are fairly simple to understand, you don't have to be taking a high level mathematics course to be able to do them. I'm sure I could find people in year 10 who can do these questions.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  9. #109
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by dan964 View Post
    Wouldn't you have to also check/assume the individual requirements of L'Hs are satisfied, namely the limits have to all exist?
    That's a statement I forgot to (explicitly) make.

    But I obviously assumed so because the final limit is supposed to exist and tend to infinity

  10. #110
    Exalted Member BlueGas's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    These things are fairly simple to understand, you don't have to be taking a high level mathematics course to be able to do them. I'm sure I could find people in year 10 who can do these questions.
    That's abit of an exaggeration isn't it? Would you find people in Year 10 (unless if they accelerate) that understand L'Hôpital's Rule?

  11. #111
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by BlueGas View Post
    That's abit of an exaggeration isn't it? Would you find people in Year 10 (unless if they accelerate) that understand L'Hôpital's Rule?
    Yes. I would, and I can, and I will. I am quasi-mentoring (mainly because he is very self capable) a year 10 student who has self-taught all the basics of Extension 2 Integration, and I will be teaching him competitive integration. Then I will drag him to the MATHSOC Integration Bee next year.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  12. #112
    Taking a break! dan964's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    That's a statement I forgot to (explicitly) make.

    But I obviously assumed so because the final limit is supposed to exist and tend to infinity
    Limits that tend to infinity, do they actually exist in the reals?

  13. #113
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    Re: First Year Uni Calculus Marathon

    Going to infinity counts as "exist" for the purposes of using L'Hôpital's rule. You can't have something that oscillates forever though (like (2+cos(x))/(2-cos(x)) or something), that'd fail to satisfy the requirement for L'Hôpital's rule.
    kawaiipotato likes this.

  14. #114
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Going to infinity counts as "exist" for the purposes of using L'Hôpital's rule. You can't have something that oscillates forever though (like (2+cos(x))/(2-cos(x)) or something), that'd fail to satisfy the requirement for L'Hôpital's rule.
    Identically equal to making a manipulation/substitution so that you get 0/0 instead of ∞/∞
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  15. #115
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon



    Prove that f is differentiable at the origin.

    There is a real positive number θ which satisfies tan-1θ = sinθ.

    Prove that f is continuous at θ.

    Hence prove f is continuous at -θ.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  16. #116
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    Yes. I would, and I can, and I will. I am quasi-mentoring (mainly because he is very self capable) a year 10 student who has self-taught all the basics of Extension 2 Integration, and I will be teaching him competitive integration. Then I will drag him to the MATHSOC Integration Bee next year.
    ...just saw (realised) this

  17. #117
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    ...just saw (realised) this
    cool
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  18. #118
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon


  19. #119
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Yep. MVT implies (f(x)-f(a))/(x-a)=f'(c(x)) for some c(x) strictly between a and x. As x->a, c(x)->a, which implies that the RHS converges to a limit by the question's assumptions. So the LHS converges to a limit as x->a, which is precisely the definition of differentiability.

    A related followup:
    Let f:R->R be a differentiable function. Must f' be continuous?
    Last edited by seanieg89; 2 Jul 2016 at 10:28 AM.
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  20. #120
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    Yep. MVT implies (f(x)-f(a))/(x-a)=f'(c(x)) for some c(x) strictly between a and x. As x->a, c(x)->a, which implies that the RHS converges to a limit by the question's assumptions. So the LHS converges to a limit as x->a, which is precisely the definition of differentiability.

    A related followup:
    Let f:R->R be a differentiable function. Must f' be continuous?
    Going by memory isn't the function f(x)=x. sin(1/x) for x \neq 0 ; 0 for x=0 a case where f is differentiable everywhere but f' is not continuous at 0?

  21. #121
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Going by memory isn't the function f(x)=x. sin(1/x) for x \neq 0 ; 0 for x=0 a case where f is differentiable everywhere but f' is not continuous at 0?
    You should never go just by memory!

    Do you believe that this is a counterexample? And if so, why?

  22. #122
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    And another related Q.

    Last edited by InteGrand; 2 Jul 2016 at 3:22 PM.

  23. #123
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    You should never go just by memory!

    Do you believe that this is a counterexample? And if so, why?
    Ouch! Maybe I meant x2 sin(1/x). Anyway here's an elementary proof





    I think because f follows the same rule from the left and the right there is no need to take 1-sided limits here.






  24. #124
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Bingo. Haha I figured you would realise that the previous exponent wasn't good enough when you tried to work out the details.
    leehuan likes this.

  25. #125
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    And another related Q.

    Doesn't my proof of existence show that f'(a)=L, the assumed limit of f'(x) as x->a? We showed the function was differentiable at a by finding it's derivative at a, which was L.

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