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Derivative problem

  1. Dec 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose that an amount function ## a(t) ## is differentiable and satisfies the property
    ## a(s + t) = a(s) + a(t) − a(0) ##
    for all non-negative real numbers ## s ## and ## t ##.

    (a) Using the definition of derivative as a limit of a difference quotient, show that ## a'(t) = a'(0) ##.

    (b) Show that ## a(t) = 1 + it ## where ## i = a(1) − a(0) = a(1) − 1 ##.

    2. Relevant equations
    N/A

    3. The attempt at a solution

    I do not understand what part b. expects me to do. If ## a'(t) = a'(0) ##, then I can show that equivalency using the definition of ## i ##. But, does that really show that ## a(t) = 1 + it ##? Perhaps the question is poorly worded, and it should read ## a(t) ## is a possible solution? Or am I looking at this the wrong way?
     
  2. jcsd
  3. Dec 14, 2014 #2

    pasmith

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    Any function of the form [itex]a(t) = Ct + D[/itex] for constants [itex]C[/itex] and [itex]D[/itex] satisfies [itex]a(s + t) = a(s) + a(t) - a(0)[/itex] for all nonnegative [itex]s[/itex] and [itex]t[/itex].

    Perhaps the definition of an "amount function" imposes conditions on [itex]a[/itex] which you haven't told us about, for example that [itex]a(0) = 1[/itex].
     
  4. Dec 14, 2014 #3

    Stephen Tashi

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    [itex] a'(t) = a'(0) [/itex] implies [itex] a'(t) [/itex] is a constant function. You know how to find an antiderivative of a constant function.
     
  5. Dec 14, 2014 #4

    LCKurtz

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    The conclusion is false. Try ##a(t) = mt## for any nonzero constant ##m##. It satisfies the hypotheses but not the conclusion.
     
  6. Dec 14, 2014 #5
    The textbook writes True, True for the solutions, for whatever that's worth.

    My approach was:
    Since ## a'(t) = a'(0) ##, ## a(t) = a(0) = 1 ##. Then ## a'(t) = a(1) - 1 = 0 = a'(0) ##.
     
  7. Dec 14, 2014 #6

    LCKurtz

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    No. Since ## a'(t) = a'(0) ## then ##a(t) = ta'(0) + C##, and you aren't given ##a(0)=1##.
     
  8. Dec 14, 2014 #7
    My bad. ## a(0) = 1 ## for accumulation functions.
     
  9. Dec 14, 2014 #8

    LCKurtz

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    Accumulation functions? Who said anything about accumulation functions, whatever they are? Not good to keep secrets when stating a problem...
     
  10. Dec 14, 2014 #9
    Miswrote, meant amount function as specified in problem. And sorry, I was lazy and assumed too much of whoever was going to help me.
     
  11. Dec 14, 2014 #10

    vela

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    Apparently the terms amount function and accumulation function come from finance. The accumulation function says how $1 would grow over time. In this problem, the accumulation function ##a(t) = 1+it## corresponds to simple interest. The amount function ##A(t)=K a(t)## is the balance at time ##t## if you start with a principal amount ##K##.
     
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