Derivative definition problem

  • #1

Homework Statement


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 ##.

Homework Equations


N/A

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?
 

Answers and Replies

  • #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].
 
  • #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.
 
  • #4
LCKurtz
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Homework Statement


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 ##.

Homework Equations


N/A

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?
The conclusion is false. Try ##a(t) = mt## for any nonzero constant ##m##. It satisfies the hypotheses but not the conclusion.
 
  • #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) ##.
 
  • #6
LCKurtz
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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) ##.
No. Since ## a'(t) = a'(0) ## then ##a(t) = ta'(0) + C##, and you aren't given ##a(0)=1##.
 
  • #7
No. Since ## a'(t) = a'(0) ## then ##a(t) = ta'(0) + C##, and you aren't given ##a(0)=1##.
My bad. ## a(0) = 1 ## for accumulation functions.
 
  • #8
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##.
My bad. ## a(0) = 1 ## for accumulation functions.
Accumulation functions? Who said anything about accumulation functions, whatever they are? Not good to keep secrets when stating a problem...
 
  • #9
Accumulation functions? Who said anything about accumulation functions, whatever they are? Not good to keep secrets when stating a problem...
Miswrote, meant amount function as specified in problem. And sorry, I was lazy and assumed too much of whoever was going to help me.
 
  • #10
vela
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Accumulation functions? Who said anything about accumulation functions, whatever they are? Not good to keep secrets when stating a problem...
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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