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Math Help - Revision Questions - Help!

  1. #1
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    Revision Questions - Help!

    Hi, I am doing some revision questions to keep up with some course work, and I am afraid that the course is starting to get away from me slightly.

    I would be very much obliged if someone would be kind enough to give me a brief explanation about the following questions...


    1. Suppose the output of a firm is given by Y = KL where K denotes the number of units of capital, L denotes the number of units of labour, and Y denotes the number of units of output
      What is the first derivative of Y with respect to K?
      Answer:
      Zero
      None of these
      L
      K
      KL

    I think the answer is 1 (i.e. none of these) as the units are to the power of 1, but am not sure.
    Question 2


    1. Suppose the output of a firm is given by Y = KL where K denotes the number of units of capital, L denotes the number of units of labour, and Y denotes the number of units of output
      What is the second derivative of Y with respect to K? How would you interpret this?

      Answer
      Second derivative of Y with respect to K is KL.
      This means that the marginal product of output with respect to capital increases as capital increases.


      Second derivative of Y with respect to K is K.
      This means that the marginal product of output with respect to capital increases as capital increases.

      Second derivative of Y with respect to K is L.
      This means that the marginal product of output with respect to capital increases as capital increases.


      Second derivative of Y with respect to K is 0.
      This means that the marginal product of output with respect to capital does not vary as capital increases.


    Based on my previous answer, I have to go with none, but doubting I understand this part either...

    Uggggh, exams coming up soon, and to be honest, on this part of the course, I am clueless, would really really appreciate some help.

    Thanks so much.

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  2. #2
    MHF Contributor
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    Re: Revision Questions - Help!

    You need to know how to find derivatives of basic functions. In particular, if c is a constant, then (cx)'=\frac{d(cx)}{dx}=c and c'=\frac{dc}{dx}=0.

    When a function has two or more variables and you take a partial derivative with respect to one of them, the other variables are considered constant. For example, \frac{\partial (xy)}{\partial x}=y and \frac{\partial y}{\partial x}=0. Since the derivative is taken with respect to x, the variable y is considered to be a constant.

    A joke about partial derivatives:
    A constant function and ex are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e^x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says ex, "he won't bother ME, I'm e to the x!" and he walks on. Of course he meets the differential operator after a short distance.

    ex: "Hi, I'm e^x"

    diff.op.: "Hi, I'm d/dy"
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