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justinfung

  • 2 years ago

solving for g'(x) when g(x)=integral from sqrt(x) to x^2, (sin t)(sqrt t) dt

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  1. satellite73
    • 2 years ago
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    derivative of the integral is the integrand, plus chain rule replace \(t\) by \(x^2\) and multiply by the derivative of \(x^2\) get \[\sin(x^2)\sqrt{x^2}\times 2x\] for the top part bottom part is the same, but make it negative

  2. justinfung
    • 2 years ago
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    I think I am suppose to break it up. integral from sqrt x to 0 plus 0 to x^2. But i have no idea why this is the way to do it

  3. satellite73
    • 2 years ago
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    yes but you can break up anyway you like

  4. satellite73
    • 2 years ago
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    the top part gives what i wrote above, the bottom part is analogous, just make it negative

  5. justinfung
    • 2 years ago
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    the bottom can be any number??

  6. justinfung
    • 2 years ago
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    I know I'm suppose to use the second fundamental theorem

  7. satellite73
    • 2 years ago
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    yes \[\int_{\sqrt{x}}^{x^2}f(t)dt=\int_{\sqrt{x}}^af(t)dt+\int_a^{x^2}f(t)dt\]

  8. justinfung
    • 2 years ago
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    My teacher chose 0 though. Is there a reason for that? should I just choose 0 everytime?

  9. satellite73
    • 2 years ago
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    take the derivative of the second part, get \(f(x^2)\times 2x\)

  10. satellite73
    • 2 years ago
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    the derivative of the first part is \(-f(\sqrt{x})\times \frac{1}{2\sqrt{x}}\)

  11. satellite73
    • 2 years ago
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    if you like to pick zero, go ahead a pick zero makes no difference

  12. justinfung
    • 2 years ago
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    Ok I understand it now. Thanks!!!! :)

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