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Derivative Question

  1. Feb 10, 2005 #1

    DB

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    I'm probably more wrong here then I've ever been, but I'll ask the question.

    I've read that the derivative of a function is a new function showing how fast the original function was changing. So if you had a function of acceleration vs time, would the derivative be a function of jolts vs time?

    Thanks
     
  2. jcsd
  3. Feb 10, 2005 #2

    mathwonk

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    what is a jolt? if it is the rate of change of acceleration, i.e. of force, then yes I guess so.

    i.e. the derivative of acceleration wrt time is approximated AT ANY INSTANT, by the change in aCCELERATION over a small time interval, divided by length of the the time interval.
     
  4. Feb 10, 2005 #3

    DB

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    Yup, a jolt is the rate of change of acceleration. So is this wat you mean,
    Derivative:
    [tex]\frac{\Delta a}{\Delta t}[/tex]
    ???

    P.S I think my equation is wrong, it seems to classical...
     
  5. Feb 10, 2005 #4

    dextercioby

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    Yes,it should be a limit there:

    [tex] \dot{a}=:\lim_{\Delta t \rightarrow 0} \frac{\Delta a}{\Delta t} [/tex]

    Daniel.
     
  6. Feb 10, 2005 #5

    DB

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    Thanks
    I've got 2 questions now, I don't want to bother you guys because I'm not taking calculus, Im just curious, but Im assuming that the dot on top of the a means the derivative of acceleration?

    And should I bother asking what a limit is? Or is it too complicated... It seems to me like a kind of restriction.
     
  7. Feb 10, 2005 #6

    dextercioby

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    Yes,in Newton's notation,a dot over a physical quantity means the derivative wrt time.As for limits (and hence the understanding of derivatives in general),i advise u to take calculus...

    Daniel.
     
  8. Feb 10, 2005 #7

    DB

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    Thanks, Ya its comming up in about 2 years, maybe less. I'm not exactly sure...
     
  9. Feb 10, 2005 #8

    dextercioby

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    In that case,be patient and u'll understand everything... :wink:

    Daniel.
     
  10. Feb 10, 2005 #9

    mathwonk

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    if you have an infinite number of approximations, the number they are approximating to is called their limit.

    so each "differerence quotient": delta(a)/delta(t) is an approximation, and the number they are approximations to, is the derivative, i.e. the derivative is the limit of the differerence quotients
     
  11. Feb 10, 2005 #10

    DB

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    Ok, it took me a while to process that sentence lol, but heres what I'm understanding. A limit is a description of a function when it "reaches" infinity. And the derivative is a way to express the rate of change of a function, but only with a limit, so your approximations don't reach infinity. Is it that right? So when not dealing with linear situations, are derivatives and limits sort of a more eleborate way to use a tangent line in a linear situation?

    And what exactly does [tex]\lim_{\Delta t \rightarrow 0}[/tex] mean? Is delta t--> 0 a way of expressing where the limit "starts and stops"?
     
  12. Feb 10, 2005 #11

    mathwonk

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    the "lim as delta t approaches zero" of some quantity, means you have one approximation of some kind, for each value of delta t. and the approximations are getting better, the smaller delta t gets.

    the number that is being approximated by whatever quantity you have, is called the limit.

    for example, .33333...... is an infinite sequence of approximations, .3, .33, .333, ... and so on.

    they are approximations to the number 1/3. the more decimal places you take, the better the approximation. none of them ever gets to 1/3. but 1/3 is the only number they are getting closer and closer to, as you go further out in the sequence, so it is called the limit.

    the precise definition is this: "given any positive number e, there is a finite decimal in that sequence, such that after that decimal, all the rest of the decimals differ from 1/3 by less than e."

    no other number but 1/3 has that property for every e, so 1/3 is the limit, even though those approximations never reach 1/3.

    (not even after "infinity", which is nonsense terminology.)
     
  13. Feb 10, 2005 #12

    DB

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    Thanks for your help. I get it now :biggrin:
     
  14. Feb 11, 2005 #13

    HallsofIvy

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    1. Strictly speaking, we define the derivative of a function AT A SPECIFIC VALUE OF X.

    We THEN define the derivative FUNCTION as the function that has the derivative value at each x.


    2. I learned the term "jerk" for the derivative of the acceleration rather than "jolt".
     
  15. Feb 11, 2005 #14

    DB

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    jerk, jolt or surge really, I prefer jolts, it sounds better than jerks. :rofl:
     
  16. Feb 11, 2005 #15

    BobG

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    I also learned that the derivative of acceleration was 'jerk'. Or, you could refer to it as the third derivative of position (and use 3 dots over your position).

    The idea of a limit is that you are approcahing something without ever quite reaching it. For example, if I add 1/2 + 1/4 + 1/8 and so on, on and on, I'll never quite reach 1, but I'll get awful darn close, so close you could, for all practical purposes, pretend you really did reach one. (By the way, this is how I progress to work every morning. My coworkers get really annoyed watching me nearly pass through the doorway. This morning, one of the guys from sales smashed his forearm through the middle of my back and screamed at me, calling me a freak. Actually, I was little relieved, even if in agony. I think the warning he got from HR last week made him go a little easier on me this week.)

    What you want for the derivative is the 'instantaneous' rate of change. But, if you let [tex]\Delta t[/tex] equal zero, your equation would be undefined, so we use the concept of a limit to almost divide by zero without actually doing so. Look up the limit definition of a derivative and you'll see what I mean ( Michael Kelley's Chapter 2, Lesson 1, "The difference quotient" ). Newton had no limits (I think Cauchy invented them some time later), so when he tried to explain the concept of a derivative, he took a lot of flack about the concept of 'infinitesimals' and 'instantaneous' rate of change, especially from his arch-enemy, Robert Hooke.
     
    Last edited: Feb 11, 2005
  17. Feb 11, 2005 #16

    DB

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    Wow, thanks, thats a great site!
     
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