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Math Help - What do these calculus questions actually mean?

  1. #1
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    What do these calculus questions actually mean?

    Hiya. I'm doing calculus and I thought I understood it all, but I'm getting some questions and I don't understand what the questions actually mean. Here they are:

    "Evaluate the limits by interpreting each as a derivative," followed by some limits of functions.

    "Express the derivatives of the given functions in terms of the derivatives f' and g' of the differentiable functions f and g," followed by some functions. Could anyone please explain what it is they are asking?
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  2. #2
    Bar0n janvdl's Avatar
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    Quote Originally Posted by mari View Post
    Hiya. I'm doing calculus and I thought I understood it all, but I'm getting some questions and I don't understand what the questions actually mean. Here they are:

    "Evaluate the limits by interpreting each as a derivative," followed by some limits of functions.

    "Express the derivatives of the given functions in terms of the derivatives f' and g' of the differentiable functions f and g," followed by some functions. Could anyone please explain what it is they are asking?
    This sounds like L'Hospital's rule... Is that what you are currently busy with in your course?

    Here is something for you to read up on: L'H├┤pital's rule - Wikipedia, the free encyclopedia
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by mari View Post
    "Evaluate the limits by interpreting each as a derivative,"
    They must be talking of the definition of the derivative which involves limits.
    For example, the derivative of the sine is the cosine hence, \lim_{x\to 0} \frac{ \sin x }{x}=\lim_{x\to 0} \frac{ \sin x -\sin 0}{x-0}=\cos 0 =1. You can evaluate a couple of limits this way.
    "Express the derivatives of the given functions in terms of the derivatives f' and g' of the differentiable functions f and g,"
    I don't know. It might have to do with the chain rule or with the derivative of a product. Can you show us an example of the functions which are given ?
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  4. #4
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    Interesting thing about that rule...L'Hopital's Rule or L'Hospital's Rule?

    Neither is actually wrong. It is named after French mathematician, Guillaume Franšois Antoine, Marquis de l'H˘pital.

    l'H˘pital is commonly spelled as both "l'Hospital" and "l'H˘pital." Marquis spelled his name with an 's'; however, the French language has since dropped the 's' (it was silent anyway) and added a circumflex to the preceding vowel.
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  5. #5
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    Thank you for the replies! L'Hopital's Rule isn't until the next chapter, actually. I'm on Chapter 2 of Adams' "Calculus: A Complete Course" if that makes any difference. Here are some examples.

    "Evaluate the limits by interpreting each as a derivative"
    \lim_{h\to 0} \frac{(x+h)^(20)-x^(20)}{h}
    Sorry, but I can't figure out how to do double digit powers correctly. I hope you can see what I mean, that they are to the power of 20.

    "Express the derivatives of the given functions in terms of the derivatives f' and g' of the differentiable functions f and g"
    f(3-x^2)
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by mari View Post
    Sorry, but I can't figure out how to do double digit powers correctly. I hope you can see what I mean, that they are to the power of 20.
    Write x^{20} instead of x^(20).
    Quote Originally Posted by mari View Post
    "Evaluate the limits by interpreting each as a derivative"
    \lim_{h\to 0} \frac{(x+h)^{20}-x^{20}}{h}
    Denote f(x)=x^{20}. The limit you're looking for is \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} which is the definition of the derivative of f taken at x. As you can evaluate f', you know the value of this limit.

    Quote Originally Posted by mari View Post
    "Express the derivatives of the given functions in terms of the derivatives f' and g' of the differentiable functions f and g"
    f(3-x^2)
    You have to use the chain rule : (f\circ g)'=g'\times f'\circ g. Here, g(x)=3-x^2 and as you don't know f, f' will appear in the final result.
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  7. #7
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    Oh right. I was thinking it was something like f(x) = 3-x^2 and I didn't know what I was supposed to do with it. I just didn't understand how to read the mathematical notation, apparently. I understand it now.


    Thank you, flying squirrel!
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