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


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


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

    Thanks for that
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