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:
You know how fast a quantity is
varying at every moment; you want to know how much of that quantity
there is at every moment.
Figure 5.4.1 summarizes the work we accomplished in Chapter 5. Study the animation in Figure 5.4.1 before reading its narrative.
Figure 5.4.1. Visual summary of
Chapter 5.
Figure 5.4.1 is a reminder of the method we developed in this chapter.
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.
We knew when values of a function $r_f$ give the rate of change at every moment of a function f, at each value in f’s domain there existed an interval of width ∆x containing that value that was small enough for which $r_f(x)$ was essentially constant.
Not knowing how wide these intervals were, we chose values of ∆x small enough so that our estimates were as accurate as we needed them to be.
We used these ∆x-intervals in $r_f$’s domain so that as $dx$ varied through each ∆x-interval, $r_f(\mathrm{left}(x))dx$ approximated the actual variation in f at every value of x in that interval, thus getting our approximate accumulation function A.
We then introduced integral notation to represent the net accumulation $A_f(x)$ in the exact accumulation function from an initial value to any value of x.
In the next chapter we will address the second foundational problem
of calculus, namely:
You know how much of a quantity
there is at every moment; you want to know how fast it is varying at
every moment.