Pre-Calculus

Determine the derivative at the point (2,−43) on the curve given by f(x)=7−7x−9x^2.

I know that the answer is -43, but I was wondering if it was just a coincidence that the derivative at the point (2,−43) is -43, or is there a reason why -43 is the same as the y-value of the original point.

asked by Ann
  1. suppose you try another point, say (1, -9)
    then
    f ' (x) = -7 - 18x
    f ' (1) = -7 - 9 = -16

    Yup, just a coincidence.



    Question: is there another point where this happens,
    that is,
    solve f(x) = f'(x)
    7 - 7x - 9x^2 = -7 - 18x
    -9x^2 + 11x +14 = 0
    9x^2 - 11x - 14 = 0
    (x - 2)(9x + 7) = 0
    x = 2 or x = -7/9

    we know about the x=2

    when x = -7/9 , f(-7/9) = 7 - 7(-7/9) - 9(49/81)
    = 7
    and f'(-7/9) = -7 - 18(-7/9) = 7

    yup, two cases where it happens.

    posted by Reiny
  2. this is weird...but thanks!

    posted by Ann
  3. noooooooooo

    posted by Eli

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