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Thread: questions about differentiation of trigo

  1. #1
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    questions about differentiation of trigo

    1.
    Find the equation of the tangent and normal to the curve y= 1 + sin 2x at the point x=$\displaystyle \frac{\pi}{12}$.


    2.
    The volume of a cone V $\displaystyle cm^3$ is given by the formula V= $\displaystyle kr^3 cot \theta$ , where k is constant, r is the radius of the cone and $\displaystyle \theta$ is a semi vertical angle. Given that r is fixed and $\displaystyle \theta$ is increasing at a rate of 0.2 radian per second, calculate the rate of change of volume with respect to time when $\displaystyle \theta$ = $\displaystyle \frac{\pi}{6}$.


    ================
    i think i know how to do, but the problem is my answer is not the same as the book's answer. Can someone please tell me where i went wrong ??

    1.
    $\displaystyle \frac{dy}{dx} = -2 cos 2x$
    When x= $\displaystyle \frac{\pi}{12}$, $\displaystyle \frac{dy}{dx} = -\sqrt3$
    When x= $\displaystyle \frac{\pi}{12}$, y= $\displaystyle \frac{3}{2}$

    equation of tangent :
    $\displaystyle y - \frac{3}{2} = - \sqrt3 (x - \frac{\pi}{12} )$
    $\displaystyle 2y -3 = -2 \sqrt3 x + \frac{2 \sqrt3 \pi}{12}$
    $\displaystyle 2y= -2 \sqrt3 x + 3 + \frac{\sqrt3 \pi}{6}$

    the book's answer is $\displaystyle 2y= 2 \sqrt3 x +3 - \frac{\sqrt3 \pi}{6}$
    so my answer is different from the book's answer in terms of the positive sign and negative sign .....

    gradient of normal= $\displaystyle \frac{1}{\sqrt3}$

    equation of normal:
    $\displaystyle y- \frac{3}{2} = \frac{1}{\sqrt3} (x - \frac{\pi}{12})$
    $\displaystyle 2y-3 = \frac{2}{\sqrt3} x - \frac{2\pi}{12 \sqrt3}$
    $\displaystyle 2y= \frac{2}{\sqrt3} x + 3 - \frac{\pi}{6 \sqrt3}$
    $\displaystyle 2 \sqrt3 y - 2x = 3 \sqrt3 - \frac{\pi}{6}$

    the book's answer is $\displaystyle 2 \sqrt3 y + 2x = 3 \sqrt3 + \frac{\pi}{6}$

    ======================

    2.
    V = $\displaystyle kr^3 cot \theta$
    = $\displaystyle kr^3 (tan \theta)^{-1}$

    $\displaystyle \frac{dv}{d \theta} $
    =$\displaystyle - kr^3 (tan \theta)^{-2} sec^2 \theta$
    =$\displaystyle - kr^3 sec^4 \theta$

    $\displaystyle \frac{d \theta}{dt} = 0.2$

    Find $\displaystyle \frac{dV}{dt}$ when $\displaystyle \theta = \frac{\pi}{6}$ .

    $\displaystyle \frac{dV}{dt} = \frac{d \theta}{dt} * \frac{dV}{d \theta}$
    = 0.2 * ( $\displaystyle - kr^3 sec^4 \frac{\pi}{6}$)
    =$\displaystyle ( - \frac{kr^3}{5})(\frac{1}{tan \frac{\pi}{6}}^4$
    =$\displaystyle ( - \frac{kr^3}{5})(\frac{3}{\sqrt3} )^4$
    =( - $\displaystyle \frac{kr^3}{5}$)($\displaystyle \frac{81}{9}$)
    =$\displaystyle - \frac{9kr^3}{5} $

    the book's answer is $\displaystyle - \frac{4kr^3}{5}$.....

    Can someone please tell me where i went wrong ?? Thanks.
    Last edited by wintersoltice; Mar 11th 2009 at 05:13 AM.
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  2. #2
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    Quote Originally Posted by wintersoltice View Post
    1.
    Find the equation of the tangent and normal to the curve y= 1 + sin 2x at the point x=$\displaystyle \frac{\pi}{12}$.


    2.
    The volume of a cone V $\displaystyle cm^3$ is given by the formula V= $\displaystyle kr^3 cot \theta$ , where k is constant, r is the radius of the cone and $\displaystyle \theta$ is a semi vertical angle. Given that r is fixed and $\displaystyle \theta$ is increasing at a rate of 0.2 radian per second, calculate the rate of change of volume with respect to time when $\displaystyle \theta$ = $\displaystyle \frac{\pi}{6}$.


    ================
    i think i know how to do, but the problem is my answer is not the same as the book's answer. Can someone please tell me where i went wrong ??

    1.
    $\displaystyle \frac{dy}{dx} = -2 cos 2x$
    When x= $\displaystyle \frac{\pi}{12}$, $\displaystyle \frac{dy}{dx} = -\sqrt3$
    When x= $\displaystyle \frac{\pi}{12}$, y= $\displaystyle \frac{3}{2}$

    equation of tangent :
    $\displaystyle y - \frac{3}{2} = - \sqrt3 (x - \frac{\pi}{12} )$
    $\displaystyle 2y -3 = -2 \sqrt3 x + \frac{2 \sqrt3 \pi}{12}$
    $\displaystyle 2y= -2 \sqrt3 x + 3 + \frac{\sqrt3 \pi}{6}$

    the book's answer is $\displaystyle 2y= 2 \sqrt3 x +3 - \frac{\sqrt3 \pi}{6}$
    so my answer is different from the book's answer in terms of the positive sign and negative sign .....

    gradient of normal= $\displaystyle \frac{1}{\sqrt3}$

    equation of normal:
    $\displaystyle y- \frac{3}{2} = \frac{1}{\sqrt3} (x - \frac{\pi}{12})$
    $\displaystyle 2y-3 = \frac{2}{\sqrt3} x - \frac{2\pi}{12 \sqrt3}$
    $\displaystyle 2y= \frac{2}{\sqrt3} x + 3 - \frac{\pi}{6 \sqrt3}$
    $\displaystyle 2 \sqrt3 y - 2x = 3 \sqrt3 - \frac{\pi}{6}$

    the book's answer is $\displaystyle 2 \sqrt3 y + 2x = 3 \sqrt3 + \frac{\pi}{6}$
    $\displaystyle y= 1 + sin 2x $
    $\displaystyle \therefore \frac{\mathrm{d}y}{\mathrm{d}x} = 2 \cos 2x$

    If you are differentiating $\displaystyle \cos x$ then you get $\displaystyle -\sin x$ (Negative) but if you are differentiating $\displaystyle \sin x $ then you get $\displaystyle \cos x$ (Positive).

    Quote Originally Posted by wintersoltice View Post
    ======================

    2.
    V = $\displaystyle kr^3 cot \theta$
    = $\displaystyle kr^3 (tan \theta)^-1$

    $\displaystyle \frac{dv}{d \theta} $
    =$\displaystyle - kr^3 (tan \theta)^(-2) sec^2 \theta$
    =$\displaystyle - kr^3 sec^4 \theta$

    $\displaystyle \frac{d \theta}{dt} = 0.2$

    Find $\displaystyle \frac{dV}{dt}$ when $\displaystyle \theta = \frac{\pi}{6}$ .

    $\displaystyle \frac{dV}{dt} = \frac{d \theta}{dt} * \frac{dV}{d \theta}$
    = 0.2 * ( $\displaystyle - kr^3 sec^4 \frac{\pi}{6}$)
    =$\displaystyle ( - \frac{kr^3}{5})(\frac{1}{tan \frac{\pi}{6}}^4$
    =$\displaystyle ( - \frac{kr^3}{5})(\frac{3}{\sqrt3} )^4$
    =( - $\displaystyle \frac{kr^3}{5}$)($\displaystyle \frac{81}{9}$)
    =$\displaystyle - \frac{9kr^3}{5} $

    the book's answer is $\displaystyle - \frac{4kr^3}{5}$.....

    Can someone please tell me where i went wrong ?? Thanks.
    You correctly wrote that $\displaystyle \cot x = \frac{1}{\tan x}$ but further down you changed your $\displaystyle \frac{1}{\tan ^2 x}$ to $\displaystyle \sec ^2 x$ which is incorrect.

    $\displaystyle \frac{\mathrm{d}v}{\mathrm{d} \theta} = - kr^3 (\tan ^{-2} \theta)\sec ^2 \theta = -\frac{kr^3}{\tan ^2\theta \cos ^2 \theta} $
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