Foundations — Net change theorem

The parent note fires a lot of notation at you in one breath: $F$, $F^{\prime }$, ${\int }_a^b$, $dx$, $\Delta x$, $\sum$, $\lim$, $v(t)$, $|v|$, the subscripts $x_i$ and $x_i^{\ast }$. If any one of these is a fog, the whole derivation is a fog. This page builds each from nothing, in the order they lean on one another.

1. A quantity and its name: $F(x)$

The picture is a graph: the horizontal axis is the input $x$ (often time), the vertical axis is the output $F(x)$ (the amount). A single dot on the curve says "at this time, this much stuff".

Why does the topic need $F$? Because "net change" is change in $F$ — you cannot talk about the change of something until you can name the something.

2. A small change: $\Delta$ (delta)

Before we can talk about a rate of change, we need a plain word for change itself.

The picture: pick two nearby points on the graph of $F$. Step right by $\Delta x$; the curve rises (or falls) by $\Delta F$. Two little arrows — one horizontal, one vertical — measure exactly these two.

Why the topic needs $\Delta$: everything downstream — rate, slice width, tiny change — is built from "end minus start".

3. Settling-down: the limit $\lim$

The rate we want next is really a $\frac{\Delta F}{\Delta x}$ ratio as the step shrinks to nothing. "Shrinks to nothing" is a limit, so we define that word now.

Picture punching numbers into "something" with $h=0.1$, then $0.01$, then $0.001$. If the answers home in on one value, that value is the limit. It is the honest way to say "at the instant" without ever dividing by zero.

Why the topic needs it: both the derivative (§4) and the integral (§8) are limits — approximations made exact by shrinking.

4. The rate of change: $F^{\prime }(x)$

Picture a tiny straight ruler laid flat against the curve so it just grazes it at one point — the tangent line. Its slope is $F^{\prime }(x)$. Steep uphill = big positive $F^{\prime }$; downhill = negative $F^{\prime }$; flat = zero.

Why the topic needs $F^{\prime }$: the net change theorem starts from a rate and recovers the amount. $F^{\prime }$ is the rate; $F$ is the amount. The prime symbol is the bridge between them.

5. Labelling the slice edges: subscripts $x_0,x_1,\dots ,x_n$

To add up many tiny changes we first cut $[a,b]$ into $n$ equal slices and name every cut.

The star version $x_i^{\ast }$ (read "$x_i$ star") is a sample point — some chosen spot inside slice number $i$ (no star = an edge; star = a point inside). We meet its exact job in §7.

Why the topic needs subscripts: to add up many slices you must be able to talk about "slice $i$" without inventing a new letter each time.

6. Adding many things: the sum symbol $\sum$

Written out, $\sum _{i=1}^3{a}_i=a_1+a_2+a_3$. Nothing mysterious — it is a compact "add these".

Why the topic needs $\sum$: net change is many little changes added up; $\sum$ is the word "added up" in maths.

7. Turning one slice's change into rate × width (Mean Value Theorem)

The telescoping sum of §6 adds up the true little changes $F(x_i)-F(x_{i-1})$. But we want those changes written in terms of the rate $F^{\prime }$, so that a sum of rates appears. The tool that guarantees this is the Mean Value Theorem.

This is the exact meaning of the star point promised in §5: it is the spot where small change = rate there × width, turning each true change into a rate contribution.

Why the topic needs it: it is the hinge that swaps $F(x_i)-F(x_{i-1})$ for $F^{\prime }(x_i^{\ast })\Delta x$, letting a sum of rates replace the sum of raw changes.

8. The definite integral: ${\int }_a^b{F}^{\prime }(x)dx$

Now assemble the pieces. Feed the boxed Mean-Value line into the telescoping sum of §6: $F(b)-F(a)={\sum }_{i=1}^n[F(x_i)-F(x_{i-1})]={\sum }_{i=1}^n{F}^{\prime }(x_i^{\ast })\Delta x.$ The far-right side is a Riemann sum: a pile of (rate × width) strips. Let the slices go infinitely thin using the limit of §3, and this pile becomes the definite integral: $\boxed{{\int }_a^b{F}^{\prime }(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^n{F}^{\prime }(x_i^{\ast })\Delta x=F(b)-F(a)}$

That single line is where every symbol on this page meets. Reading the pieces:

So the whole symbol literally means: the signed area between the rate curve and the axis, from $a$ to $b$. Because each strip is (rate)×(width) = a tiny change, adding the strips gives the total change. That is the Riemann-sum idea, and the punchline of the whole parent topic.

9. Signed area, and the bars: $|v(t)|$

The parent splits displacement ($\int v$, keeps signs — area below the axis counts negative) from distance ($\int |v|$, flips the below-axis area up so everything counts positive). See Displacement vs Distance. The velocity $v(t)$ is just an $F^{\prime }$ whose $F$ is position — same machinery, physics costume.

Why the topic needs it: signed vs unsigned is the single most-tested distinction, and it lives entirely in whether the bars go inside the integral.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Net change theorem — the parent this page equips you for.
• Definite Integral as a Riemann Sum — where $\int$, $\sum$, and $\lim$ all meet.
• Fundamental Theorem of Calculus — links $F^{\prime }$ back to $F$.
• Telescoping Sums — the cancellation of §6.
• Mean Value Theorem — supplies the sample point $x_i^{\ast }$ (§7).
• Displacement vs Distance — signed vs unsigned, the bars of §9.
• Marginal Cost and Revenue — the economics face of $F^{\prime }$.