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Math Help - exam problems (calculus 1)

  1. #1
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    exam problems (calculus 1)

    hey everyone, i would appreciate any help with these problems ..
    Attached Thumbnails Attached Thumbnails exam problems (calculus 1)-exam-problems.png  
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  2. #2
    MHF Contributor chisigma's Avatar
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    18. Is...

    \displaystyle \frac{1}{4} \frac{x-2}{x+2} = \frac{1}{4} - \frac{1}{x+2} (1)

    ... so that...

    \displaystyle f^{(n)} (x) = (-1)^{n-1} \frac{n!}{(x+2)^{n}} (2)



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  3. #3
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    12) Rewrite it as \ln\left[F(x)\right] = \ln\frac{\left[f(x) \right]^\pi}{\left[3+f(x)\right]^e}=\pi \ln\left[f(x) \right]-e \ln\left[3+f(x)\right]
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  4. #4
    MHF Contributor chisigma's Avatar
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    20. Is [l'Hopital's rule...] ...

    \displaystyle \lim_{x \rightarrow 0} \frac{1-\cos 4 x}{x^{2}} =  \lim_{x \rightarrow 0} \frac{4\ \sin 4x}{2 x} =  \lim_{x \rightarrow 0} \frac{16\ \cos 4 x}{2} = 8 (1)

    ... so that...

    \displaystyle \frac{8}{3}\ \alpha = \frac{4}{\beta} \implies \alpha\ \beta= \frac{3}{2} (2)



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  5. #5
    MHF Contributor chisigma's Avatar
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    12. Is...

    \displaystyle \frac{d}{dx} \ln F(x)= \frac{F^{'}(x)}{F(x)} (1)

    ... so that, writing \displaystyle F(x)=  \frac{f(x)}{g(x)}, is also...

    \displaystyle F^{'} (x) = F(x)\  \frac{d}{dx} \ln F(x) = F(x)\ \frac{d}{dx} \{\ln f(x) - \ln g(x) \}=

    \displaystyle = F(x)\ \frac{d}{dx} \{\frac{f^{'}(x)}{f(x)} - \frac{g^{'}(x)}{g(x)} \} (2)

    ... and now You have all the elements to compute F^{'} (0) ...



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  6. #6
    Grand Panjandrum
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    Don't just post a list of questions like this.

    1. There are too many questions for one thread, the limit is 2 or 3. I have closed this thread, repost the remaining questions in new threads with one or two questions per thread.

    2. Please show some working or explain where you are having difficulties.

    We are not here to do your work (I will assme this is not a take home exam) for you but to help you understand.

    CB
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