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Math Help - CAL 1 Project 2

  1. #1
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    Red face CAL 1 Project 2

    1. Let f be a differentiable function. Let g be a function defined by:
    g(x)= the lim. as h approaches 0 of [f(x+h) - f(x-h)]/h
    show that g(x)=2f'(x)

    2. Show that the product of even functions is even
    3. show that the product of odd functions is even
    4. show that the product of even and odd functions is odd.
    Can anyone please help me with this, i dont' even know where to start.
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  2. #2
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    The trick in #1 is to realize that \lim _{h \to 0} \frac{{f(x + h) - f(x)}}{h} = \lim _{h \to 0} \frac{{f(x) - f(x - h)}}{h}\;{\color{red}[1]}.
    To see that realize that t =  - h \Rightarrow \quad h \to 0 \equiv t \to 0 so that <br />
\frac{{f(x) - f(x - h)}}{h} = \frac{{f(x) - f(x + t)}}{{ - t}} = \frac{{f(x + t) - f(x)}}{t}.
    That proves {\color{red}[1]}.
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