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Math Help - Geographic Meaning of Differentiation

  1. #1
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    Geographic Meaning of Differentiation

    Hello All,

    The following is a paragraph from my book...

    "Find the slope of the tangent to the curve y=\frac{1}{2x}+3 at the point where x=-1. Find the angle at which this tangent makes with the curve y=2x^2+2. The slope of the tangent is the slope of the curve at the point where they touch one another; that is, it is the \frac{dy}{dx} of the curve for that point. Here \frac{dy}{dx}= -\frac{1}{2x^2} and for x=-1, \frac{dy}{dx}=\frac{-1}{2}, which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation y=ax+b, and its slope is \frac{dy}{dx}=a, hence a=\frac{-1}{2}. Also, if x=-1, y=\frac{1}{(2)(-1)}+3=2\frac{1}{2}, and as the tangent passes by this point, the coordinates of that point must satisfy the equation of the tangent, namely: y=-\frac{1}{2}x+b so that 2\frac{1}{2}=\frac{-1}{2}(-1)+b and b=2; the equation of the tangent is therefore y=\frac{-1}{2}x+2 "


    So here are my questions regarding this paragraph:
    1.) When I take the tangent of the first curve, I get \frac{dy}{dx}=\frac{1}{2} ...how does he come up with \frac{dy}{dx}=\frac{1}{2x^2} ?


    (depending on the answer, I might have more questions to follow) Thanks.
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  2. #2
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    Question

    How did you get dy/dx = 1/2 ?
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  3. #3
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    Quote Originally Posted by wonderboy1953 View Post
    How did you get dy/dx = 1/2 ?
    I was looking at it as \frac{1}{2}x instead of \frac{1}{2x}

    Thanks for that
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