Derivative problem

1. Dec 14, 2014

wintermute++

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. Dec 14, 2014

pasmith

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

Perhaps the definition of an "amount function" imposes conditions on $a$ which you haven't told us about, for example that $a(0) = 1$.

3. Dec 14, 2014

Stephen Tashi

$a'(t) = a'(0)$ implies $a'(t)$ is a constant function. You know how to find an antiderivative of a constant function.

4. Dec 14, 2014

LCKurtz

The conclusion is false. Try $a(t) = mt$ for any nonzero constant $m$. It satisfies the hypotheses but not the conclusion.

5. Dec 14, 2014

wintermute++

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. Dec 14, 2014

LCKurtz

No. Since $a'(t) = a'(0)$ then $a(t) = ta'(0) + C$, and you aren't given $a(0)=1$.

7. Dec 14, 2014

wintermute++

My bad. $a(0) = 1$ for accumulation functions.

8. Dec 14, 2014

LCKurtz

Accumulation functions? Who said anything about accumulation functions, whatever they are? Not good to keep secrets when stating a problem...

9. Dec 14, 2014

wintermute++

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. Dec 14, 2014

vela

Staff Emeritus
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 $K$.