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Thread: Slope by limit process

  1. #1
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    Slope by limit process

    Greetings,

    I need to use the limit definition of :

    $\displaystyle m= \lim_{\Delta x\rightarrow 0} f(x) = (x+\Delta x)- f(x) / \Delta x$

    ( / means all divided by $\displaystyle \Delta x$)

    I have to use the limit definition to find the slope of the tangent line to the graph of $\displaystyle f$ given at these equations:

    $\displaystyle f(x) = 4-x^2$ at point $\displaystyle (2,0)$


    If someone could show me how to solve this step by step that would be cool.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    $\displaystyle f(x)=4-x^2$
    $\displaystyle f'(x)=-2x$

    $\displaystyle m=f'(x)=-4$
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  3. #3
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    I think what he needs is to solve this using First Principle.

    $\displaystyle f(x) = 4-x^2$ at point $\displaystyle (2,0)$, so $\displaystyle x=2$

    Now as you said:

    $\displaystyle m= \lim_{\Delta x\rightarrow 0} \frac{f(x+\Delta x)- f(x)}{\Delta x}$
    Sub $\displaystyle x=2$
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} \frac{f(2+\Delta x)- f(2)}{\Delta x}$
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} \frac{4-(2+\Delta x)^2- (4-(2)^2)}{\Delta x}$
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} \frac{4-(2^2+4\Delta x + (\Delta x)^2)- (4-(2)^2)}{\Delta x}$
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} \frac{4-4-4\Delta x - (\Delta x)^2- 4+4}{\Delta x}$

    We can do some cancelling
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} \frac{-4\Delta x - (\Delta x)^2}{\Delta x}$

    We can do some more cancelling which leaves us with
    $\displaystyle m=\lim_{\Delta x\rightarrow 0} -4 - \Delta x$

    The limit is as $\displaystyle \Delta x \rightarrow 0$, so sub it in, which gives
    $\displaystyle m= -4-0
    =-4$

    So, at the point $\displaystyle (2,0)$, $\displaystyle m=-4$


    You can also verify this by differentiating $\displaystyle f(x)$ as alexmahone did:
    $\displaystyle f' (x)=-2x$
    $\displaystyle f'(2)=-2(2) = -4$

    Which is the same as what we obtained before, although this doesn't directly use the first principles or limits.
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