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Math Help - Integration and Differentiation Problem

  1. #1
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    Integration and Differentiation Problem

    I took a math exam and want to see if I got these questions right. These are the ones I can remember.

    3.) Integral of (sin x cos x) from x = -pi to x = pi.

    We weren't allowed to ask questions so I didn't know if they wanted net area or all the area counted as positive, even the negative area so I put answer = 0.

    6a.) Find dy/dx for the following equation of an hyperbola x^2 - y^2 = 9.

    I got the answer dy/dx = -x/-y


    6b.) What is the equation of the tangent line to the hyperbola at (3,0)

    I put no such tangent line exists.
    Last edited by Ackbeet; December 16th 2011 at 03:04 AM.
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    re: Integration and Differentiation Problem

    Quote Originally Posted by Rstewart View Post
    3.) Integral of (sin x cos x) from x = -pi to x = pi.

    We weren't allowed to ask questions so I didn't know if they wanted net area or all the area counted as positive, even the negative area so I put answer = 0.
    \displaystyle \begin{align*} f(x) &= \sin{(x)}\cos{(x)} \\ f(-x) &= \sin{(-x)}\cos{(-x)} \\ f(-x) &= -\sin{(x)}\cos{(x)} \\ f(-x) &= -f(x) \end{align*}

    This is an odd function. You should know that \displaystyle \begin{align*} \int_{-a}^a{f(x)\,dx} = 0 \end{align*} if \displaystyle \begin{align*} f(x) \end{align*} is an odd function. So you are correct.

    You could also have solved the integral using Brute Force.
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    re: Integration and Differentiation Problem

    Quote Originally Posted by Rstewart View Post
    6a.) Find dy/dx for the following equation of an hyperbola x^2 - y^2 = 9.

    I got the answer dy/dx = -x/-y


    6b.) What is the equation of the tangent line to the hyperbola at (3,0)

    I put no such tangent line exists.
    I agree with the calculation of your derivative, though I would simplify it to x/y.

    Anyway, for part b, what do you know about the gradients of vertical lines?
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    re: Integration and Differentiation Problem

    Quote Originally Posted by Prove It View Post
    I agree with the calculation of your derivative, though I would simplify it to x/y.

    Anyway, for part b, what do you know about the gradients of vertical lines?
    >.< So that means it would be a vertical tangent and I got that one wrong?
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    re: Integration and Differentiation Problem

    Quote Originally Posted by Rstewart View Post
    >.< So that means it would be a vertical tangent and I got that one wrong?
    Yes, the equation of the tangent would have been \displaystyle \begin{align*} x = 3 \end{align*} .
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