Interpretation — instantaneous rate of change, slope of tangent: Foundations

This page assumes nothing. Every squiggle, letter, and word the parent note used is unpacked here, in the order you need them. If you can read a graph, you can finish this page.

0. The picture everything lives on: the coordinate plane

Before any symbol, we need the stage.

Figure — Interpretation — instantaneous rate of change, slope of tangent

Figure s01 — a dark chalkboard. A horizontal white line (the $x$ -axis) and a vertical white line (the $y$ -axis) cross at the centre. A yellow dot sits up and to the right; a blue dashed line drops from it to the $x$ -axis (showing "go right $x$ ") and a pink dashed line runs left to the $y$ -axis (showing "go up $y$ "). The dot is labelled "point (x, y)."

Why we need this: the whole topic is about a curve — a set of points — and its steepness. Steepness is an up-vs-right comparison, so we must first agree on what "up" and "right" mean. That agreement is the plane.

1. The symbol $f (x)$ — a machine that turns input into output

Why the topic needs it: the derivative is written $f^{'} (a)$ and uses $f (a + h)$ and $f (a)$ everywhere. If $f ()$ is mysterious, the entire formula is noise.

1a. Domain — the set of legal inputs

Why the topic needs it: we are about to build the fraction $\frac{f ( a + h ) - f ( a )}{h}$ , which uses two input values, $a$ and $a + h$ . Both must be things $f$ can eat. This quietly matters at the edge of a domain (e.g. $x$ at $x = 0$ , where only $h > 0$ keeps us legal — so only a one-sided approach is possible there).

2. The letters $a$ and $h$ — "the spot" and "the tiny step"

Figure s02 — the curve $y = f (x)$ climbs across the board. On the $x$ -axis a yellow dot marks $a$ and a blue dot marks $a + h$ ; a pink arrow between them is labelled "step h." Two dashed vertical lines rise from these inputs up to the curve, meeting it at heights labelled $f (a)$ (yellow) and $f (a + h)$ (blue).

Why the topic needs it: you cannot measure steepness at a single point (nothing to compare against). $h$ manufactures a second point so we have something to compare. Later we shrink $h$ toward $0$ — that is the heart of the whole chapter.

3. Slope — the number that means "steepness"

This is the most important idea to nail before anything else.

Figure s03 — a straight white line slopes upward. Two yellow dots sit on it; a blue horizontal segment between them (labelled "run — sideways") and a pink vertical segment (labelled "rise — up") form a right triangle under the line. Caption text reads "slope = rise / run."

Why the topic needs it: "slope of the tangent" is the answer the derivative gives. Slope is rise/run — this exact fraction reappears as the difference quotient in Section 6.

4. Average rate of change — slope wearing a "speed" costume

Figure s05 — the curve $y = f (x)$ again. A yellow dot at $(a, f (a))$ and a blue dot at $(a + h, f (a + h))$ are joined by a dashed secant line. A blue horizontal segment underneath is labelled "run = h" and a pink vertical segment on the right is labelled "rise = f(a+h) − f(a)." Caption: "avg rate = rise / run."

Why the topic needs it: the derivative is the limit of this quantity. It is the raw material we are about to refine.

5. The secant and tangent lines — two pictures of a line

Figure s04 — the curve with a yellow dot fixed at $a$ . Three faint-to-bright blue secant lines pass through $a$ and second points that creep closer to $a$ ; as they do, the secants rotate. A bright pink line — the tangent — is the line they settle onto. Labels: "secants (blue)" and "tangent (pink)."

Why the topic needs it: the secant slope is computable (two real points, honest division). The tangent slope is what we want but can't compute directly. The bridge between them is the limit.

6. The difference quotient — the star fraction

Before writing the fraction, be crystal-clear about the two points whose slope we are taking:

Why the topic needs it: this fraction is the object the limit acts on. Master its two pieces (rise on top, run $h$ on bottom) and the derivative formula is no longer scary.

7. The limit symbol $h \to 0 lim$ — "sneaks up on"

Why the topic needs it: the limit is the machine that dodges $\frac{0}{0}$ . It lets us reach the instant (the single point) legally — but only when the two sides agree. Everything rigorous about " $\to 0$ " and one-sided limits lives in Limits — formal definition.

8. The prime symbol $f^{'} (a)$ — the name of the answer

Why the topic needs it: it is the final packaged answer — one number that is both the tangent's slope and the instantaneous rate. Letting $a$ roam over all inputs upgrades $f^{'} (a)$ into a whole new function, the Derivative as a function.

How the foundations feed the topic

Read this map bottom-up as a build order. The coordinate plane is the stage; on it lives a function $f (x)$ with its domain of legal inputs. Choosing a spot $a$ and a step $h$ (with $a + h$ still legal) gives us two points, whose slope — the same rise-over-run that measures steepness — becomes the average rate of change and the secant line. Feeding the difference quotient into a two-sided limit (which needs $f$ continuous at $a$ ) finally produces the derivative $f^{'} (a)$ .

Equipment checklist

Cover the right-hand side and test yourself. If any answer is fuzzy, reread that section.

What does

(x, y)

tell you to do on the plane?

Go right

x

, then up

y

, and mark the dot.

What does

f (3)

mean?

Run the rule

f

on the input

3

; it is NOT

f

times

3

What is the domain of a function?

The set of inputs the machine legally accepts.

Before forming

f (a + h)

, what must you check?

That BOTH

a

and

a + h

lie in

f

's domain (choose

h

small enough that

a + h

stays legal).

What roles do

a

and

h

play?

a

is the fixed spot;

h

is a small nonzero step to a nearby point

a + h

(right if

h > 0

, left if

h < 0

Slope in words and as a fraction?

Steepness

=

rise over run

= \frac{change in y}{change in x}

What does a negative slope look like?

A line going downhill as you move right.

What is the average rate of change geometrically?

The slope of the secant line joining

(a, f (a))

and

(a + h, f (a + h))

Name the two points whose slope the difference quotient takes.

(a, f (a))

and

(a + h, f (a + h))

Secant vs tangent?

Secant cuts the curve at two points; tangent grazes it at one, matching its steepness.

When does a tangent fail to give a finite slope?

At a corner (two-sided limits disagree) or a vertical tangent (slope blows up to infinity).

Write the difference quotient and name its top and bottom.

\frac{f ( a + h ) - f ( a )}{h}

; top is the rise, bottom

h

is the run.

Why can't we just set

h = 0

It gives

\frac{0}{0}

, undefined; we take a limit instead.

What must be true of the two one-sided limits for

f^{'} (a)

to exist?

The

h \to 0^{+}

and

h \to 0^{-}

approaches must sneak up on the same number.

What continuity condition must

f

satisfy at

a

for

f^{'} (a)

to exist?

f

must be continuous at

a

(no jump, gap, or hole) — differentiable implies continuous.

What does

f^{'} (a)

stand for?

The slope of the tangent to

f

x = a

— equivalently the instantaneous rate of change there.

Connections

Average rate of change — Section 4 is its full unpacking.
Limits — formal definition — makes the " $h \to 0$ " and one-sided limits of Section 7 rigorous.
Derivative as a function — what $f^{'} (a)$ becomes when $a$ varies.
Power Rule — the shortcut that replaces these limits later.
Differentiability and continuity — why $f$ must be continuous at $a$ , and why corners/jumps kill the derivative.
Velocity and acceleration — the "rate" costume of Section 4.