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Homework Help: ? I don't understand this answer

  1. Nov 21, 2008 #1
    ?? I don't understand this answer

    1. The problem statement, all variables and given/known data

    The pdf attached is the answer to a question from a Calculus quiz. (the right hand side is a bit cut off, but the formula is repeated further down)

    Edit: the "f(x) = ..., and a=0" part is the solution, not part of the the question.

    2. Relevant equations



    3. The attempt at a solution

    ok. I have no idea what she's doing here. when I do the derivative of the f(x) she gives as the answer, I get

    1/2 (1+3cosx)^1/2 is

    1/4 ((1+3cosx)^-1/2)(-3sinx) = -3sinx / 4 sqrt(1+3cosx)

    which is not even close to the original f'(x) she posted...

    can anyone show me how she got that answer? thanks.
     

    Attached Files:

    Last edited: Nov 21, 2008
  2. jcsd
  3. Nov 21, 2008 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Re: ?? I don't understand this answer

    The derivative df/dx of f(x) at x= a is, by definition
    [tex]\lim_{h\righrarrow 0}\frac{f(a+h)- f(a)}{h}[/tex]

    You are asked to find a function f and number a such that
    [tex]\lim_{x\rightarrow 0}\frac{\sqrt{1+ 3cos(x)}- 2}{2x}= f'(a)[/itex]

    Compare that to the definition of the derivative. If you replace the x with h then you have
    [tex]\lim_{h\rightarrow 0}\frac{\sqrt{1+ 3cos(h)}- 2}{2h}= f'(a)[/itex]
    or
    [tex]\lim_{h\rightarrow 0}\frac{(1/2)\sqrt{1+ 3cos(h)}- 1}{h}= f'(a)[/itex]

    Since [itex](1/2)\sqrt{1+ 3cos(0)}= (1/2)\sqrt{4}= 1[/itex], it should be clear that you take f(x) to be [itex]f(x)= (1/2)\sqrt{1+ 3cos(x)}[/itex] and a= 0.
     
  4. Nov 21, 2008 #3
    Re: ?? I don't understand this answer

    thanks. wow. I can't believe I didn't see that. gah. I did so poorly on that quiz. I don't know what happened. I was getting 80's and 90's, and on this quiz this morning I failed... and miserably. the whole thing was just a staring contest between me and the page. It's like I woke up 50% dumber today. I felt like writing an apology at the beginning to whoever marks it. This quiz is going to bring my overall mark down by 5%. bah. :cry:
     
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