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Thread: Calculate derivative using formal definition

  1. #1
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    Question Calculate derivative using formal definition

    I'm really hoping someone can help me calculate the derivative (using the formal definition) for the following function:
    Calculate derivative using formal definition-screen-shot-2019-06-15-7.42.00-pm.png


    Here is my attempted solutions but I know it is wrong:
    Calculate derivative using formal definition-img_20190615_194600.jpg


    Can someone tell me what I am doing incorrectly?
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    Re: Calculate derivative using formal definition

    Quote Originally Posted by otownsend View Post
    I'm really hoping someone can help me calculate the derivative (using the formal definition) for the following function:
    Click image for larger version. 

Name:	Screen Shot 2019-06-15 at 7.42.00 PM.png 
Views:	8 
Size:	25.1 KB 
ID:	39423


    Here is my attempted solutions but I know it is wrong:
    Click image for larger version. 

Name:	IMG_20190615_194600.jpg 
Views:	12 
Size:	551.3 KB 
ID:	39422


    Can someone tell me what I am doing incorrectly?
    We can't tell you where you went wrong if you don't post the whole problem... What's h(t)?

    -Dan
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    Re: Calculate derivative using formal definition

    h(t) is in the first image amended to my post. Do you not see it?
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    Re: Calculate derivative using formal definition

    Quote Originally Posted by otownsend View Post
    I'm really hoping someone can help me calculate the derivative (using the formal definition) for the following function:
    Click image for larger version. 

Name:	Screen Shot 2019-06-15 at 7.42.00 PM.png 
Views:	8 
Size:	25.1 KB 
ID:	39423
    Here is my attempted solutions but I know it is wrong:
    Click image for larger version. 

Name:	IMG_20190615_194600.jpg 
Views:	12 
Size:	551.3 KB 
ID:	39422Can someone tell me what I am doing incorrectly?
    Let's start from $\dfrac{k^2\sin\left(\tfrac{1}{k}\right)}{k}=k \cdot \sin\left(\tfrac{1}{k}\right)$

    At this point we need to look at a graph, SEE HERE
    Looking at that graph, what does the limit as $k\to 0$ look like?

    How do you prove it? If you need hints here they are:
    1) $-1\le\sin\left(\tfrac{1}{k}\right)\le 1$
    2) If $k>0$ then $-k\le k\sin\left(\tfrac{1}{k}\right)\le k$ and we know $k\to 0^+$
    This is the tricky one, what about $k\to 0^-~?$
    3) Well $\sin$ is an odd function so $\sin\left(\tfrac{1}{-k}\right)=-\sin\left(\tfrac{1}{|k|}\right)$ as can be seen on the graph.
    In other words: the limit as $k\to 0^+$ is equal to the limit as $k\to 0^-$
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    Re: Calculate derivative using formal definition

    Are you saying that sin(1/-k) is the function from the left of 0 ?
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    Re: Calculate derivative using formal definition

    Quote Originally Posted by otownsend View Post
    Are you saying that sin(1/-k) is the function from the left of 0 ?
    To be fair to myself, I did say that this was tricky.
    Odd functions are just that.
    $\sin(-t)=-\sin(t)$ or $\sin(t)=-\sin(-t)$

    THUS $\displaystyle \large{\mathop {\lim }\limits_{t \to {0^ - }} \sin (t) = \mathop {\lim }\limits_{t \to {0^ - }} - \sin ( - t) = \mathop {\lim }\limits_{t \to {0^ + }} - \sin (t)}$
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