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Calculus

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I have the function f(x)=e^x*sinNx on the interval [0,1] where N is a positive integer. What does it mean describe the graph of the function when N={whatever integer}? And what happens to the graph and to the value of the integral as N approaches infinity? Does the graph confirm the limiting behavior of the integral's value?

  • Calculus - ,

    well, e^0 is 1
    and e^.5 = 1.64
    and e^1 is 2.72
    so it is a sine wave with increasing amplitude as you approach 1 and frequency increasing with N
    The integral of e^ax sin bx dx is
    [e^ax/(a^2+b^2)] [a sin bx -b cos bx}here a = 1 and b = N
    so
    [e^x/(1+N^2)] [sin Nx - N cos Nx]
    as N gets big
    this looks like
    e^x (-N cos Nx)/N^2
    or
    (-e^x/N)(cos Nx)
    e^x is that small constant and cos Nx ranges between -1 and + 1 so as N gets big this goes to zero like 1/N

  • Calculus - ,

    Now how would you describe the graph of this function when say N=5, N=10, and N=100?

    And what does it mean does the graph confirm the limiting behavior of the integral's value?

  • Calculus - ,

    But how did you arrive at "as N gets big this looks like e^x(-NcosNx)/N^2?????

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