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Math Help - integration and differenciation

  1. #1
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    integration and differenciation

    hello! I had my class test today, but there are 2 questions on which I am doubtful of my answer, could you please check?

    1) integral from -1 to x of (3t^4 + 15 t^2)e^(t^2)
    a) is never null
    b) has a point of local minimum at -1
    c) is decreasing
    d) none of the preceiding

    I said b

    2) the mean value of 2xsin(x^2) in (0; root of pi) (the boundaries are included but I don't know how to write that brachets) is
    a) 2
    b)1/root of pi
    c) 0
    d) none of the preceding

    I said d

    Thank you so much!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    hello! I had my class test today, but there are 2 questions on which I am doubtful of my answer, could you please check?

    1) integral from -1 to x of (3t^4 + 15 t^2)e^(t^2)
    a) is never null
    b) has a point of local minimum at -1
    c) is decreasing
    d) none of the preceiding

    I said b
    The integrand is always positive, and so the integral is positive for x>-1,
    zero when x=-1, and negative for x<-1, also it is increasing.

    So we can rule out a), b) and c), so it looks like d) is the required answer.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    2) the mean value of 2xsin(x^2) in (0; root of pi) (the boundaries are included but I don't know how to write that brachets) is
    a) 2
    b)1/root of pi
    c) 0
    d) none of the preceding

    I said d
    you want:

    \left. \left[ \int_0^{\sqrt{\pi}}2x\,\sin(x^2) dx\right] \right/ \sqrt{\pi},

    which looks to equal 2/\sqrt{\pi}, but check this as I'm in a rush and can't do so myself just now.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    The integrand is always positive, and so the integral is positive for x>-1,
    zero when x=-1, and negative for x<-1, also it is increasing.

    So we can rule out a), b) and c), so it looks like d) is the required answer.

    RonL


    (You should add to the "emoticons" the face that cries and despairs...)

    Anyway, I thought that we were considering the interval (-1; x) so in that interval at -1 we could have had a local minimum. where do I mistake? I mean, it seems to me that I haven't understood the meaning of the boundaries of the integral then.
    When yesterday we were plotting that graph that you and topsquark made equal(and I mistook) you started the graph from the lower boundary.
    ...Too much confusion...
    please help me please please please help me understand
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    When yesterday we were plotting that graph that you and topsquark made equal(and I mistook) you started the graph from the lower boundary.
    What we were plotting (if we are thinking of the same problem) was the
    integrand, not an integral with a variable upper limit of integration. The
    integral in question was a definite integral from 0 to infinity and the question
    was does this converge.

    The graph was to show that the integrand went to zero for large x, even
    though the integral diverged.

    (I would not be surprised if you got partial credit for answering "local
    minimum" though)

    RonL
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  6. #6
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    sorry for insisting or annoying you CaptainBlank, but I need to truly understand (the best my teacher can do when you ask smth is looking at you as if you were the most stupid person in the world )... yes the problem I was referring to was exactly that; ok, but now, in the case of the class test, if we end up writing a formula with the "x" and so on, i mean we do not end up with a number(the area), what is that -1 there for???
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by 0123 View Post
    sorry for insisting or annoying you CaptainBlank, but I need to truly understand (the best my teacher can do when you ask smth is looking at you as if you were the most stupid person in the world )... yes the problem I was referring to was exactly that; ok, but now, in the case of the class test, if we end up writing a formula with the "x" and so on, i mean we do not end up with a number(the area), what is that -1 there for???
    It is normaly taken that:

    \int_1^x f(u) \, du=-\int_x^{1}f(u)du

    To illustrate this the attachment shows a computer algebra system output
    for examples of the integral in the problem

    RonL
    Attached Thumbnails Attached Thumbnails integration and differenciation-gash.jpg  
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