Rework the derivatives chapter: intuition-first prose + four figures - #7
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Work in progress on "The Slope of a Function" — the intuitive build-up before any calculation. - Slope stated as a *ratio* rather than a distance. It was "how far do we move up ... that distance is called the slope", which names a quantity that changes with step size and so cannot describe the line. Now "how far up for each single step forward", with a sentence on why the comparison, not either measurement alone, is what matters. - Graph and function reframed as siblings — two representations of the same motion — rather than the function representing the graph. Credits Descartes and, independently, Fermat for the curves/equations correspondence, noting the modern function concept came later with Leibniz and Euler. - Rate-of-change examples corrected: they were all unintentionally second-order (speed and growth are already rates). Now position → speed → acceleration gives first and second derivative in one breath, plus the "prices still rising, but more slowly" example that needs both at once. - First figure in these docs: an @setup/@example block drawing one line with three differently-sized steps and identical ratios, carrying the ratio-not-a-distance point visually. Plotting code hidden; Documenter embeds it as SVG. - x and y as $x$ and $y$ throughout; typos fixed; broken analytic-geometry link repaired (it used parentheses instead of square brackets, so it rendered as literal text). Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Introduced in this branch's own Slope section commit, written before the domain migration landed on main (PR #4). URL only, no prose change.
…n difference quotient
Continues "The Slope of a Function" and starts "Computing Derivatives".
- Second figure in the chapter: a saturating curve $5(1 - e^{-x/3})$ with three
tangent segments (slopes ≈ 1.01, 0.52, 0.19) getting visibly shallower. Makes the
first-vs-second-derivative point concrete — the rate of change stays positive (still
rising) while shrinking (rising more slowly), i.e. the "prices still rising, but
easing off" sentence drawn. Tangents distinguished by position and numeric slope
labels, never by colour. New @setup/@example `slope2` namespace so it can't clobber
figure 1's plot.
- Lead-in prose keeps the description in graph language ("positive", "declining")
rather than the technical positive/negative of a derivative, which isn't defined
until later.
- Constant function: its slope is now shown as the difference quotient
$\frac{f(x+h)-f(x)}{(x+h)-x} = \frac{c-c}{h} = \frac{0}{h} = 0$, replacing an
ambiguous $y/x$ (which reads as a ratio of coordinates, not of changes). This mirrors
the Linear and Quadratic subsections immediately below — same machine, simplest input
— and the numerator $c - c = 0$ is literally the "rise is always 0" stated in words.
- Minor: mismatched quote mark and a couple of typos fixed; "rate of the rate of
change" for terminology consistency.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
…first, two figures, explicit limit
Rebuilds "The Quadratic Power Function" around building intuition before formalism,
and adds the two figures that carry the approximation→limit idea.
- Drops "secant" jargon from this first pass: the slope of x² is built as the line
through two nearby points, x and x+h. (The term returns, properly named, in the later
"The Derivative as a Limit" section — intuition first, terminology when precise.)
- Makes the move from approximation to limit explicit: worked instances (x = 1 → 2 + h,
x = 2 → 4 + h) show the slope genuinely varies with x, then a numeric table
(h = 1/10, 1/100, 1/1000 → 2.1, 2.01, 2.001) makes h → 0 something you can watch.
- Two figures:
- The parabola with tangents at x = 0.5, 1.5, 2.5 (slopes 1, 3, 5) — the slope plainly
grows with x, cashing "the slope is not constant".
- The line through (x, x²) and (x+h, (x+h)²) drawn for h = 1, 0.5, 0.25, its slope
(3, 2.5, 2.25) closing in on the tangent's slope 2 as h → 0. Labels distinguished by
position and value, never colour.
- The general "…turn into the general definition of the derivative" sentence moved to
"The General Power Function"; a concrete lead-in now introduces the specific f'(x) = 2x.
- LaTeX / mechanical corrections along the way: the "x = h" → "x + h" point error;
U+2215 division slash → "/"; a sentence fragment and a garbled limit sentence rewritten;
"tangents"-as-a-verb → the tangent line; the dangling "Fig. 6.4" (a leftover Walz
reference) removed; assorted typos and $x$/$0$ notation consistency.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
…nce-quotient terminology Fills in "The Derivative as a Limit" and cleans up its vocabulary. - Adds the derivation of the tangent line from the point-slope form, and a worked example that separates the two things Walz runs together: the *general* tangent formula t(x̄) = 2x·x̄ − x² (valid at any point of tangency x), and the *specific* tangent at a fixed point — x = 1 gives t(x̄) = 2x̄ − 1, the very line the secants close in on in the qlimit figure. Makes explicit that x is the fixed point of tangency while x̄ is the tangent's own running input (the bar keeps the two roles of "x" apart) — the exact distinction Walz's own derivation obscures. - Terminology: keep "difference quotient" (the standard English term for the finite ratio) and drop "differential quotient" (the dated Germanic name for the limit — in English it's just the derivative); the definition heading is now "First derivative". 202 / 204 / 206 now speak consistently. - Cross-references the point-slope form to the [slope-intercept form] in the Math Foundations notes (absolute link, matching the existing analytic-geometry reference). - Fixes $y-axis$ (the hyphen had landed inside the math, rendering as "y minus axis") → $y$-axis. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
- 01 Derivatives and Rules.md: mathemeticians → mathematicians and Noneteenth → nineteenth, in the Newton–Leibniz aside. - notebooks/Project.toml: drop LAlatex + BlockArrays (Linear-Algebra / vector-calculus machinery, carried over when the file was copied from that repo) and add Latexify. The forthcoming derivatives notebook needs symbolic-expression → LaTeX rendering, not block-matrix display; Latexify is already in the resolved graph via Calculus → CalculusWithJuliaSquared → Symbolics, so this just promotes it to a direct dependency so the notebook can `using Latexify`. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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Reworks the first differential-calculus chapter (
Differential_Calculus/01 Derivatives and Rules.md) from a terse Walz summary into an intuition-first walk-through, adds four executable figures, and cleans up notation/terminology.Prose
x(worked instancesx=1→2+h,x=2→4+h), then makes the approximation→limit move explicit with a numeric table and the general limitlim(2x+h)=2x.x=1 → t(x̄)=2x̄−1).Figures (executed at build via
@example, embedded as SVG)Also
notebooks/Project.toml: droppedLAlatex+BlockArrays(Linear-Algebra carryover), addedLatexify— prep for the forthcoming derivatives notebook.Builds clean (only the expected SVG-fallback info for the four figures).
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