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Chapter 5 addressed the problem of creating an accumulation function in open form when we have its rate of change at every moment as a function in closed form. In the next chapter we will start from where we ended in Chapter 5. In this chapter, we addressed the first foundational problem of calculus:
Start with an exact rate of change function $r_f$ in closed form. Its values gives the rate of change at every moment of an (unknown) exact accumulation function f.
Use the meaning of rate of change at a moment to assume that $r_f$ is essentially constant over small intervals of size $\Delta x$ that contained each moment.
Let $dx$ and $dy$ vary over each $\Delta x$-sized interval. Approximations to the exact accumulation over intervals are then continuous. Each value of $dx$ produces a value of $dy=m dx$ as $dx$ varies, thus defining the approximate net accumulation function A as a piecewise linear function that approximates the exact net accumulation function $A_f$. The meaning of $A_f(x)$ is that any value of $A_f(x)$ is the net accumulation (variation) in the value of f from an initial value a to any value of x.
When $\Delta x$ is so small that making it smaller has no appreciable effect on our estimates of the net accumulation from a to x, we said $A_f$ was the exact net accumulation function for $r_f$. We represented $A_f$ in open form as $A_f(x) = \int_a^x r(t)dt$.
The relationship between $A_f$ and f is that for any given value of a, $A_f(x)=f(x)-f(a)$. Said another way, every value of $A_f(x)$ gives us the net variation in f from a to x, and the net variation in f from a to x is $f(x)-f(a)$. Therefore $f(x) = f(a) + A_f(x)$, which says that $f(x)$—the accumulation in f up to a value of x—is made by the accumulation in f up to a, plus the accumulation in f from a to x.
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