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Math Help - differentiation rules #2

  1. #1
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    differentiation rules #2

    I need help on these problems. Thanks in advance.
    1. Use differentials to estimate the amount of paint needed to apply a coat of paint .05 cm thick to a hemispherical dome with diameter 50 m.
    2. Use a linear approximation or differentials to estimate the given number. L(x) = f(a)+f '(a)(x-a) I have tried to use the formula and pluged in the numbers but I still couldn't figure out the correct answer.
    a) e^-.015
    b) tan 44 degrees
    Last edited by mr fantastic; October 5th 2009 at 03:34 AM. Reason: Re-titled post
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  2. #2
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    1. Use differentials to estimate the amount of paint needed to apply a coat of paint .05 cm thick to a hemispherical dome with diameter 50 m.

    dV = 2\pi r^2 \, dr

    dV = 2\pi (25 \, m)^2 \, (.0005 \, m)

    dV will be in cubic meters

    2. Use a linear approximation or differentials to estimate the given number. L(x) = f(a)+f '(a)(x-a) I have tried to use the formula and pluged in the numbers but I still couldn't figure out the correct answer.

    a) e^{-.015}

    use the line tangent to y = e^x at (0,1)

    m = f'(0) = e^0

    y - e^0 = e^0(x - 0)

    y - 1 = x

    y = x + 1

    e^{-.015} \approx -.015 + 1 = .985


    b) tan 44 degrees

    note that derivatives for trig functions are only valid for angles in radians.

    f(x) = \tan{x}

    f'(x) = \sec^2{x}

    use line tangent to the point \left(\frac{\pi}{4},1\right)

    m = \sec^2\left(\frac{\pi}{4}\right) = 2

    y - 1 = 2\left(x - \frac{\pi}{4}\right)

    y = 2\left(x - \frac{\pi}{4}\right) + 1

    \tan\left(\frac{44\pi}{180}\right) \approx 2\left(-\frac{\pi}{180}\right) + 1 = 1 - \frac{\pi}{90}
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