Calculus

1. use the definition mtan=(f(x)-f(x))/(x-a) to find the SLOPE of the line tangent to the graph of f at P.

2. Determine an equation of the tangent line at P.

3. Given 1 & 2, how would I plot the graph of f and the tangent line at P if : f(x)=x^2 +4, P(4,20)

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  1. first of all, if you are trying to define the definition of the derivative, it should be
    mtan=Lim (f(x+a)-f(x))/(a) , as a --->∞

    Furthermore, I will assume that f(x) = x^2 + 4, you only state that at the very end

    f(x+a) = (x+a)^2 + 4
    then (f(x+a)-f(x))/(a) = (x^2 + 2ax + a^2 + 4) - x^2 - 4)/a)

    slope of tangent = lim (x^2 + 2ax + a^2 + 4 - x^2 - 4)/a as a --->∞
    = lim (2ax + a^2)/a , as a --->∞
    = lim a(2x + a)/a , as a --->∞
    = lim 2x + a , as a --->∞
    = 2x

    now at the point (4,20)
    the slope of the tangent is 2(4) or 8
    and the tangent equation is
    y-20 = 8(x-4)
    y = 8x - 12

    check with Wolfram:
    http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+%2B+4,+y+%3D+8x+-+12

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  2. a --->∞ should probably be a --->0

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  3. Scott, of course you are right, my silly error.
    Fortunately, my calculations match the a --->0 conditions

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