Questions from this class

What is calculus, and what will I be able to do?

This came up more than once in our first week, and the syllabus does not answer it.

This notebook began in Week 1. Everything on it is here because someone in this class asked for it, so it starts small. That is the idea.

Begin with the experiment

Put two gates on the track.

Optional practice · no submission required

This activity is not graded, and the site does not record your work. Experiment freely: run the ball, squeeze the gates, and watch the sequence in the log.

Experiment 01 / Average velocity

THE BALL + THE GATES

Drag either gate, or focus it and use arrow keys. Shift + arrow moves one metre.

0 mTrack / 100 m100 m

Try these in order, or choose your own

The gate positions determine Δs before the run. The crossing times — and so the measured value of Δt — are not shown until the ball reaches each gate.

The delay between the two stamps represents the interval whose value is t_B − t_A.

Both positions are measured from the same reference point. Δs is the difference between them.

Because the ball moves only forward here, its average speed and average velocity have the same value.

Active Calculus §1.1 writes this without the Δ, as AV[a,b] = (s(b) − s(a)) / (b − a). Same subtraction over the same subtraction.

Run the ball once, then move Gate B toward Gate A and compare the log.

Run log

newest at bottom

No runs yet. Start wide, then squeeze the gates.

Open the full experiment

What you just discovered

These measurements already contain the derivative.

Call the time $t$ and the position $s (t)$ — the same letter Active Calculus uses for a position function. Gate A is crossed at time $t$ and gate B at $t + h$ , so

Δ s = s (t + h) - s (t), Δ t = h,

and the readout is

\frac{Δ s}{Δ t} = \frac{s (t + h) - s (t)}{h} .

Squeezing the gates is making $h$ small. At the 25-metre mark, the averages approach $10 m/s$ even though the two gates never occupy the same place:

s^{'} (t) = lim_{h \to 0} \frac{s (t + h) - s (t)}{h} .

A position function is one example of a function, so from Section 1.3 the book writes the same limit for any $f$ :

f^{'} (a) = lim_{h \to 0} \frac{f (a + h) - f (a)}{h} .

Nothing changes but the letter. When the moving object is gone, $s$ becomes $f$ and the derivative is whatever that function's rate of change happens to measure.

The notation describes the same measurement process in mathematical terms. Here, the limit is the value approached as the distance between the gates tends to zero.

Active Calculus §1.1 writes this without the Δ, as ${AV}_{[a, b]} = \frac{s (b) - s (a)}{b - a}$ . It is the same subtraction over the same subtraction. Δ is only a name for the difference.

This is the opening idea in

Active Calculus §1.1, “How do we measure velocity?”

The question ledger

Questions from this course

A row appears because a real question was asked. Questions may be paraphrased; they are never invented.

Available

What is calculus, and what will I be able to do?

From our Week 1 discussion · Approximately 6 minutes

In preparation

“How does finding the instantaneous slope of a curve lead down to spacetime being bent like water?”

Asked in MATH A251 · In preparation

There are 2 entries so far. I will add to this list as questions arise.

Request an explanation

Tell me which step is unclear.

This page exists because the question kept coming up. Send the next one and I can answer it once for everyone. Deadlines, grades, and submissions all stay in Blackboard.

— Jeff

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