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Math Help - Fundamental theorem problem

  1. #1
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    Fundamental theorem problem



    Ok, so you split up into two intervals I did from 2x to 0 and 0 to 4x.

    Then you reverse the 2x to 0 and the integral becomes negative.

    Then, replace 2x and 4x with u and chain rule

    answer: (-2(x)2-5/(2x)2+5)*2 + (4(x)2-5/(4x)2+5)*4


    Is my work correct? I was marked wrong and it said we didnt have to simplify.
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  2. #2
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    Re: Fundamental theorem problem

    Quote Originally Posted by Steelers72 View Post
    What the heck are you doing? What are the instructions?
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    Re: Fundamental theorem problem

    Quote Originally Posted by Steelers72 View Post


    Ok, so you split up into two intervals I did from 2x to 0 and 0 to 4x.

    Then you reverse the 2x to 0 and the integral becomes negative.

    Then, replace 2x and 4x with u and chain rule

    answer: (-2(x)2-5/(2x)2+5)*2 + (4(x)2-5/(4x)2+5)*4

    Did you intend this as an integral?

    Is my work correct? I was marked wrong and it said we didnt have to simplify.
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  4. #4
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    Re: Fundamental theorem problem

    Fundamental theorem, it says to find the derivative. So what I did was split up into two integrals because we have an x for both b and a of the integral. I split into intervals 2x to 0 and 0 to 4x. Since 2x is still on the bottom of integral, you move it by law of integrals to the top and it becomes a negative integral from 0 to 2x. Then I plugged in 2x and 4x in for each u value and did chain rule of 2x and 4x and got the answer from above. Sorry I don't know how to make integrals on the computer
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    Re: Fundamental theorem problem

    Quote Originally Posted by Steelers72 View Post
    Fundamental theorem, it says to find the derivative. So what I did was split up into two integrals because we have an x for both b and a of the integral. I split into intervals 2x to 0 and 0 to 4x. Since 2x is still on the bottom of integral, you move it by law of integrals to the top and it becomes a negative integral from 0 to 2x. Then I plugged in 2x and 4x in for each u value and did chain rule of 2x and 4x and got the answer from above. Sorry I don't know how to make integrals on the computer
    There is absolutely no reason to 'split' anything.

    Suppose that each of g~\&~h is a differentiable function then if
    F(x) = \int_{h(x)}^{g(x)} {\Phi (t)dt} then F'(x) = g'(x)\Phi (g(x)) - h'(x)\Phi (h(x)).
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  6. #6
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    Re: Fundamental theorem problem

    My professor told us to split that's why I did it because he said when you have an x for both the top and bottom intervals you should split into two separate integrals. I guess you are saying some sort of rule that explains this without splitting it up?
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    Re: Fundamental theorem problem

    Quote Originally Posted by Steelers72 View Post
    My professor told us to split that's why I did it because he said when you have an x for both the top and bottom intervals you should split into two separate integrals. I guess you are saying some sort of rule that explains this without splitting it up?
    It is just a simple application of the chain rule.

    Now it is true that \int_{h(x)}^{g(x)} {\Phi (t)dt}  = \int_0^{g(x)} {\Phi (t)dt}  - \int_0^{h(x)} {\Phi (t)dt} .

    But why waste time?
    Thanks from Steelers72
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  8. #8
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    Re: Fundamental theorem problem

    You are right but I think he didn't want to confuse us but he'll probably do it next class haha
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