AP test problems, Derivatives and Tangent lines

  1. 1. The problem statement, all variables and given/known data
    Let f be the function given by f(x) = (2x-5)/(x^2-4).
    a.Find the domain of f.
    b.Write an equation for each vertical and each horizontal asymptote for the graph of f.
    c.Find f'(x).
    d.Write an equation for the line tangent to the graph of f at the point (0,f(0)).

    3. The attempt at a solution
    Note: Just checking answers!
    a.domain: x does not equal 2 or -2 because those values of x make the denominator 0
    b. vertical asymptote
    x = 2, x = -2
    horizontal asymptote
    as x goes to positive or negative infinity
    f(x) goes to 0
    y = 0

    c. f'(x) = (2(x^2 - 4) - (2x - 5)(2x))/(x^2-4)^2
    = (2x^2 - 8 - 4x^2 + 10x)/(x^2-4)^2
    = -2(x^2 - 5x + 4)/(x^2-4)^2
    = -2(x-4)(x-1)/(x^2-4)^2
    f(0) = -5/(-4) = 5/4

    d. y - 5/4 = slope (x - 0)
    slope = f'(0) = -2(-4)(-1)/(-4)^2 = -8/16 = -1/2
    y - 5/4 = (-1/2)x
    y = (-1/2)x + 5/4
     
  2. jcsd
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