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Thread: Fundamental Theorem of Calculus help!

  1. #1
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    Fundamental Theorem of Calculus help!

    So, my problem is that I know we have to replace the variable in the function with the upper limit(x), and then multiply by the derivative of x. However, the problem I have here has the lower limit of cosx, and an upper limit of sinx. I know that in order to replace our variable, the upper limit has to be unknown (an x). How would I do this?
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    Re: Fundamental Theorem of Calculus help!

    Quote Originally Posted by scorks View Post
    So, my problem is that I know we have to replace the variable in the function with the upper limit(x), and then multiply by the derivative of x. However, the problem I have here has the lower limit of cosx, and an upper limit of sinx. I know that in order to replace our variable, the upper limit has to be unknown (an x). How would I do this?
    If each of f~\&~g is a differentiable function and H(x) = \int_{g(x)}^{f(x)} {h(t)dt} then
    H'(x) = \left[ {h \circ f(x)\,} \right]f'(x) - \left[ {h \circ g(x)} \right]\,g'(x)
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    Re: Fundamental Theorem of Calculus help!

    Quote Originally Posted by Plato View Post
    If each of f~\&~g is a differentiable function and H(x) = \int_{g(x)}^{f(x)} {h(t)dt} then
    H'(x) = \left[ {h \circ f(x)\,} \right]f'(x) - \left[ {h \circ g(x)} \right]\,g'(x)
    Ah, okay. So my question is:

    Fundamental Theorem of Calculus help!-codecogseqn.gif

    I filled in what you told me, but I'm not sure where to go from there...
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    Re: Fundamental Theorem of Calculus help!

    Quote Originally Posted by scorks View Post
    Ah, okay. So my question is:

    Click image for larger version. 

Name:	CodeCogsEqn.gif 
Views:	0 
Size:	1.1 KB 
ID:	29384

    I filled in what you told me, but I'm not sure where to go from there...
    There is nothing more to do. What did you get?
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    Re: Fundamental Theorem of Calculus help!

    For my final answer, I simplified it to:

    Fundamental Theorem of Calculus help!-codecogseqn.gif
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    Re: Fundamental Theorem of Calculus help!

    Quote Originally Posted by scorks View Post
    For my final answer, I simplified it to:
    Click image for larger version. 

Name:	CodeCogsEqn.gif 
Views:	0 
Size:	828 Bytes 
ID:	29385
    I have absolutely no idea what you did. The answer is:
    \cos(x)[\sin^2(x)+2\sin(x)]+\sin(x)[\cos^2(x)+2\cos(x)]
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    Re: Fundamental Theorem of Calculus help!

    Quote Originally Posted by Plato View Post
    I have absolutely no idea what you did. The answer is:
    \cos(x)[\sin^2(x)+2\sin(x)]+\sin(x)[\cos^2(x)+2\cos(x)]
    I multiplied cos(x) by the sin^2(x) and the 2sin(x), then the same type of thing for my sin(x). Then I took out the common factor. Also, shouldn't that be a -cos(x), since the integral of sin(x) is -cos(x)?
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  8. #8
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    Re: Fundamental Theorem of Calculus help!

    Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule). And,



    is the FTC: Fundamental theorem of calculus - Wikipedia, the free encyclopedia


    Full size


    __________________________________________________ __________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; Oct 6th 2013 at 11:17 AM.
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  9. #9
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    Re: Fundamental Theorem of Calculus help!

    Gotcha. Thanks! I'll try again, I think is was accidentally multiplying by the integral rather than the derivative.
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