For the function f(x)=x^2-4 construct and simplify the difference quotient of the fraction f(x+h)-f(x)/h?

can anyone help find the difference quotient

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  • sepia
    Lv 7
    3 years ago

    f(x) = x^2 - 4

    [f(x + h) - f(x)] / h

    = [(x + h)^2 - 4 - (x^2 - 4)] / h

    = 2hx + h^2 - 4 + 4) / h

    = h + 2x

  • 3 years ago

    The "difference quotient" is the basis of... differentiation.

    It gives you the rate at which the function varies at some point.

    This will become useful as you progress in mathematics.

    A function is like a recipe.

    f(x) = x^2 - 4

    f is the name of the recipe

    (x) is the "secret ingredient"

    anything after the equal sign is the set of instructions (what to do with the secret ingredient, once you are told what it is).

    f(x) = x^2 - 4

    Once I give you a value for x (the secret ingredient) you simply (and blindly) apply it in the recipe

    f(0) means "use recipe f, with 0 being the secret ingredient)

    f(0) = 0^2 - 4 = 0 - 4 = -4

    f(5) = 5^2 - 4 = 25 - 4 = 21

    f(x+h) = (x+h)^2 - 4 = x^2 - 2hx + h^2 - 4

    f(cat) = (cat)^2 - 4

    OK, that last one does not make much sense, but it does show how you blindly apply the rule: once you are given the identity of the "secret ingredient", you use it. Period. Even if the secret ingredient includes itself... (which would be weird in a real recipe)**.

    f(x+ h) - f(x)

    becomes

    (x+h)^2 - 4 - (x^2 - 4)

    work on it

    x^2 + 2hx + h^2 - 4 - x^2 + 4

    combine and cancel (as appropriate)

    2hx + h^2

    Next, we are asked to divide this whole thing by h

    (2hx + h^2)/h = 2x + h

    That's it.

    -----

    Later, you will learn that you have to find the limit of that expression, as h gets closer and closer to zero (it can never be exactly zero, otherwise you would be dividing by zero - which is not "well defined")

    As h gets closer to zero, the term "h" is closer and closer to zero, until it can be ignored, so that the limit simply becomes 2x

    But that is for later.

    -----

    **There exists a recipe called "friendship bread" where one of the ingredients is... a piece of "friendship bread". It is called friendship bread because once you have successfully made friendship bread, you break off a few pieces (before cooking) and give them to your friends so that they too can make their own bread (and pass more pieces to other friends).

    The idea being that this bread is made from natural yeasts (which do exist in nature, otherwise wine would not exist) that are kept alive by "living" in uncooked pieces of bread dough.

    The analogy, in mathematics, would be a function where you are asked to find "the function of the function"

    f(f(x)) where the "secret ingredient" is the function itself.

    Since we started with

    f(x) = x^2 - 4

    we would get

    f(f(x)) = (f(x))^2 - 4 = (x^2 - 4)^2 - 4

    As I said earlier: you just apply the recipe without asking questions.

  • 3 years ago

    First, the difference quotient should be written with the numerator grouped:

    [f(x+h) - f(x)] / h

    You have:

    f(x+h) = (x+h)² - 4 = x² + 2xh + h² - 4

    f(x) = x² - 4

    Subtract those and cancel terms:

    f(x+h) - f(x) = x² + 2xh + h² - 4 - (x² - 4)

    f(x+h) - f(x) = 2xh + h²

    Now divide that by h.

    [f(x+h) - f(x)] / h = (2xh + h²) / h

    = 2x + h

    Answer:

    2x + h

    P.S. And then if you take the limit h->0, the result is 2x.

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