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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Optimization Explorer</title>
<meta name="description" content="Calculus I: Optimization Explorer.">
<link rel="preconnect" href="https://fonts.googleapis.com">
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Literata:ital,opsz,wght@0,7..72,400;0,7..72,500;0,7..72,600;1,7..72,400;1,7..72,500&family=Public+Sans:wght@400;500;600;700&family=IBM+Plex+Mono:wght@400;500&family=STIX+Two+Text:ital,wght@0,400;0,500;1,400;1,500&family=Kalam:wght@400&display=swap">
<script>try{if(localStorage.getItem('rize-calc1-mode')==='light')document.documentElement.setAttribute('data-theme','light')}catch(e){}</script>
<style>
/* Calc I theme: Slate Board. Reading on a page; worked lines and figures on a dark slate board in chalk colors.
Dark is the default; [data-theme="light"] is the student's toggle. Color roles: --inside = input / inside of a composite,
--outside = output / outside, --rate = rate (speed, slope, derivative), --k-* = a specific value plugged in. */
:root{color-scheme:dark;
--bg:#121513;--surface:#1A1E1C;--ink:#E7EAE5;--muted:#9DA59F;--line:#2D332F;--soft:#202522;--grid:#24302B;
--inside:#93BDF0;--inside-tint:#1A2738;--outside:#F0A877;--outside-tint:#38251A;--rate:#E59BC6;--rate-tint:#34192A;
--k-tint:#5C4E10;--k-line:#D9BC2E;--ok:#8FCB9C;--ok-tint:#18301E;--bad:#F29C93;--bad-tint:#3A1A17;--focus:#93BDF0;
--b-bg:#1F2824;--b-ink:#ECEFE8;--b-muted:#AEB8B1;--b-line:#33403A;--b-in:#8FC3FF;--b-in-t:#22364A;--b-out:#FFB27A;--b-out-t:#4A3222;--b-rate:#F7A3CF;--b-rate-t:#45263A;--b-val:#FFE680;--b-val-t:rgba(255,230,128,.20);
--serif:'Literata','Iowan Old Style',Georgia,serif;--sans:'Public Sans','Segoe UI',system-ui,sans-serif;--mono:'IBM Plex Mono',ui-monospace,Menlo,Consolas,monospace;
--math:'STIX Two Text','Cambria Math',Georgia,serif;--anno:'Kalam','Segoe Print','Comic Neue',cursive;}
:root[data-theme="light"]{color-scheme:light;
--bg:#F3F4F1;--surface:#FFFFFF;--ink:#1A1D1B;--muted:#5B625E;--line:#D6DAD5;--soft:#ECEEEA;--grid:#E3E7E2;
--inside:#23599E;--inside-tint:#E2EAF5;--outside:#A64B12;--outside-tint:#F7E6DA;--rate:#9A2E6E;--rate-tint:#F4E2EC;
--k-tint:#FFE680;--k-line:#C9A800;--ok:#2A6B3B;--ok-tint:#E1F0E4;--bad:#A8261D;--bad-tint:#F8E2DF;--focus:#23599E;
--b-bg:#26302C;--b-ink:#ECEFE8;--b-muted:#B4BEB7;--b-line:#3A4741;--b-in:#8FC3FF;--b-in-t:#2A4056;--b-out:#FFB27A;--b-out-t:#55392A;--b-rate:#F7A3CF;--b-rate-t:#4E2B42;--b-val:#FFE680;--b-val-t:rgba(255,230,128,.20);}
/* the board: worked lines and figures re-scope the tokens, so every component inside turns to chalk without its own rules */
.workbox,.pfig,.overview,.gwrap,.partsfig,.board,.gv{--bg:var(--b-bg);--surface:var(--b-bg);--ink:var(--b-ink);--muted:var(--b-muted);--line:var(--b-line);--soft:var(--b-line);
--inside:var(--b-in);--inside-tint:var(--b-in-t);--outside:var(--b-out);--outside-tint:var(--b-out-t);--rate:var(--b-rate);--rate-tint:var(--b-rate-t);--k-tint:var(--b-val-t);--k-line:var(--b-val);
background:var(--b-bg);color:var(--b-ink);border-color:transparent}
.pfig,.overview,.gwrap,.partsfig,.gv{border-radius:12px;padding:12px}
.pfig svg text,.overview svg text,.gwrap svg text,.partsfig svg text,.gv svg text{font-family:var(--anno)}
/* Colors and fonts come from the course theme (interactive/themes/<name>.css), injected by build.py. */
*{box-sizing:border-box}
body{margin:0;background:var(--bg);color:var(--ink);font-family:var(--serif);font-size:21px;line-height:1.6;-webkit-font-smoothing:antialiased;text-rendering:optimizeLegibility}
a{color:var(--inside);text-underline-offset:3px}a:hover{color:var(--ink)}
:focus-visible{outline:3px solid var(--focus);outline-offset:2px;border-radius:4px}
.page{padding:0 24px 160px}
.col{max-width:760px;margin-left:auto;margin-right:auto}
p{margin:0 0 16px;text-wrap:pretty}
h1,h2,h3{font-family:var(--serif);font-weight:500;letter-spacing:-0.01em;margin:0}
.cap{font-family:var(--sans);font-size:14px;font-weight:700;letter-spacing:.09em;text-transform:uppercase;color:var(--muted);margin:0 0 8px}
.muted{color:var(--muted)}
/* math typesetting: always inline flow, never flex items */
.mn,.mv,.mf,.mo,.ms{font-family:var(--math)}
.mv{font-style:italic;padding-right:.05em}
.mo{white-space:nowrap}
.mo.eq{white-space:normal}
.ms{font-size:.7em;line-height:1;vertical-align:.65em}
.ms.msv{font-style:italic}
.em{font-style:italic}
.in{color:var(--inside)}
.mn.in,.mv.in,.mf.in,.mo.in{text-decoration:underline;text-decoration-thickness:2px;text-underline-offset:5px;text-decoration-skip-ink:none}
.pr.in,.pr.out{font-weight:600}
.out{color:var(--outside)}
.k{color:var(--ink) !important;background:var(--k-tint);box-shadow:inset 0 -1.5px 0 var(--k-line);border-radius:4px;padding:0 .14em;margin:0 .02em;font-weight:500;text-decoration:none !important}
sup.k{padding:0 .1em}
/* bar */
.bar{position:sticky;top:0;z-index:20;background:var(--bg);border-bottom:1px solid var(--line);margin:0 -24px;padding:0 24px}
.bar-in{max-width:1120px;margin:0 auto;display:flex;align-items:center;gap:20px;min-height:60px;position:relative}
.brand{font-family:var(--serif);font-size:20px;font-weight:600;color:var(--ink);text-decoration:none}
.toc{display:flex;gap:20px;flex:1;justify-content:center;font-family:var(--sans);font-size:16px}
.toc a{color:var(--muted);text-decoration:none;padding:6px 0}
.toc a:hover{color:var(--ink)}
.rulesbtn{margin-left:auto;font-family:var(--sans);font-size:16px;font-weight:600;color:var(--ink);background:var(--surface);border:1px solid var(--line);border-radius:999px;min-height:42px;padding:6px 18px;cursor:pointer;display:flex;gap:8px;align-items:center;position:relative;z-index:22}
.rulesbtn .n{color:var(--muted);font-weight:500}
.scrim{position:fixed;inset:0;z-index:21;background:transparent;border:0;padding:0;margin:0;cursor:default}
.drawer{position:absolute;right:0;top:calc(100% + 8px);z-index:23;width:min(460px,calc(100vw - 32px));max-height:72vh;overflow:auto;background:var(--surface);border:1px solid var(--line);border-radius:14px;padding:18px 22px;box-shadow:0 18px 50px rgba(20,18,12,.18)}
.rules{list-style:none;margin:0;padding:0}
.rules li{display:grid;grid-template-columns:minmax(0,9.5em) minmax(0,1fr);gap:12px;padding:12px 0;border-top:1px solid var(--line);align-items:baseline}
.rules li:first-child{border-top:0}
.rname{font-family:var(--sans);font-size:16px;color:var(--muted)}
.rules li.new .rname{color:var(--ink);font-weight:600}
.rf{font-size:22px}
.newtag{grid-column:2;font-family:var(--sans);font-size:14px;color:var(--muted)}
/* hero */
.hero{padding:88px 0 24px}
.kicker{font-family:var(--sans);font-size:15px;font-weight:600;letter-spacing:.1em;text-transform:uppercase;color:var(--muted);margin-bottom:14px}
.hero h1{font-size:clamp(48px,8vw,84px);line-height:1;font-weight:500;margin-bottom:18px}
.dek{font-size:24px;line-height:1.5;color:var(--muted)}
.emblem{margin:52px 0 0;padding:44px 20px 32px;background:var(--surface);border:1px solid var(--line);border-radius:18px;display:flex;flex-direction:column;align-items:center;gap:26px}
.emb-row{font-family:var(--serif);font-size:clamp(56px,11vw,108px);line-height:1.1;display:flex;align-items:flex-start}
.emb-out{color:var(--outside)}
.emb-in{color:var(--inside);background:var(--inside-tint);border-radius:12px;padding:0 .14em;border-bottom:4px solid var(--inside);animation:breathe 4.5s ease-in-out infinite}
.emb-sup{font-size:.55em;color:var(--outside);margin-left:.04em}
.emb-legend{display:flex;flex-wrap:wrap;gap:12px 28px;justify-content:center;font-family:var(--sans);font-size:17px}
.leg{display:flex;align-items:center;gap:10px}
.sw{width:30px;height:18px;border-radius:5px;flex:none}
.sw-in{background:var(--inside-tint);border-bottom:3px solid var(--inside)}
.sw-out{border:2px solid var(--outside)}
.leg b{font-weight:600}
.leg-in b{color:var(--inside)}.leg-out b{color:var(--outside)}
@keyframes breathe{0%,100%{box-shadow:0 0 0 0 rgba(30,88,163,0)}50%{box-shadow:0 0 0 8px rgba(30,88,163,.10)}}
/* sections */
.sec{padding-top:160px;margin-bottom:12px}
.sec-top{display:flex;align-items:baseline;gap:18px;border-top:2px solid var(--ink);padding-top:18px}
.sec-n{font-family:var(--serif);font-size:60px;line-height:.9;color:var(--outside);font-weight:400}
.sec h2{font-size:clamp(34px,5vw,48px);line-height:1.1}
.lede{margin-top:12px;font-size:21px;color:var(--muted)}
.prose{margin-top:68px}
.prose h3{font-size:28px;margin-bottom:12px}
/* buttons */
.btn{font-family:var(--sans);font-size:17px;font-weight:600;min-height:48px;padding:10px 20px;border-radius:10px;border:1.5px solid var(--ink);background:var(--ink);color:var(--bg);cursor:pointer;white-space:nowrap}
.btn:hover{opacity:.9}
.btn.ghost{background:transparent;color:var(--ink)}
.btn.sm{min-height:44px;padding:8px 16px;font-size:16px}
.btn:disabled{opacity:.35;cursor:not-allowed}
.link{font-family:var(--sans);font-size:16px;font-weight:600;background:none;border:0;color:var(--inside);text-decoration:underline;text-underline-offset:3px;min-height:44px;padding:0 4px;cursor:pointer}
/* cards */
.prob,.scard,.exam{background:var(--surface);border:1px solid var(--line);border-radius:18px;padding:36px 36px 30px;margin:44px auto 0;max-width:760px}
.prob-head{display:flex;flex-wrap:wrap;align-items:baseline;gap:6px 12px;margin-bottom:18px}
.plabel{font-family:var(--sans);font-weight:700;font-size:16px;color:var(--muted)}
.ptask{font-family:var(--sans);font-size:17px;color:var(--muted)}
.pmath{font-size:29px;line-height:1.35}
.is-done .plabel::after{content:' \2713';color:var(--ok)}
.locked{font-family:var(--sans);font-size:16px;color:var(--muted);margin:0}
.is-locked{background:transparent;border-style:dashed}
.ansrow{display:flex;flex-wrap:wrap;align-items:center;gap:10px}
.lhs{font-size:24px;white-space:nowrap}
.minput{flex:1 1 200px;min-width:0;font-family:var(--mono);font-size:19px;color:var(--ink);background:var(--bg);border:1.5px solid var(--line);border-radius:10px;min-height:54px;padding:10px 14px;width:100%}
.minput:focus{border-color:var(--inside);outline:none;box-shadow:0 0 0 3px var(--inside-tint)}
.pal{display:grid;grid-template-columns:repeat(auto-fill,minmax(52px,1fr));gap:6px;margin-top:10px}
.key{font-family:var(--serif);font-size:19px;min-height:46px;border-radius:8px;border:1px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer;padding:0 4px}
.key:hover{border-color:var(--ink)}
.fb{font-family:var(--sans);font-size:17px;line-height:1.55;margin-top:10px}
.fb .mn,.fb .mv,.fb .mf,.fb .mo{font-size:20px}
.pv{font-family:var(--sans);font-size:16px;color:var(--muted);margin-top:6px}
.pv .mn,.pv .mv,.pv .mf,.pv .mo{font-size:19px}
.fb-ok{color:var(--ok);font-weight:600}
.fb-no::before,.fb-err::before,.fb-stuck::before{content:'';display:inline-block;width:8px;height:8px;border-radius:2px;background:var(--outside);margin-right:9px;vertical-align:2px}
.fb-err::before{background:var(--muted)}
.fb-stuck::before{background:var(--inside)}
.aft{display:flex;gap:10px;flex-wrap:wrap}
.sol{margin-top:12px;padding:16px 18px;background:var(--soft);border-radius:10px;font-size:20px}
.repair{margin-top:18px;padding:20px;background:var(--soft);border-radius:12px}
.repair .diag{font-size:19px}
.cap2{font-family:var(--sans);font-size:15px;font-weight:600;color:var(--muted);margin:18px 0 6px}
.repair .pmath{font-size:24px}
.hintd{margin:0 0 16px;font-family:var(--sans);font-size:17px}
.hintd summary{font-weight:600;color:var(--inside);cursor:pointer;min-height:40px;display:flex;align-items:center}
.hintd p{margin:6px 0 4px;font-size:17px}
.hintd .mn,.hintd .mv,.hintd .mf,.hintd .mo{font-size:20px}
/* step rows */
.ctitle{font-family:var(--sans);font-size:19px;margin:0 0 18px}
.ctitle .mn,.ctitle .mv,.ctitle .mf,.ctitle .mo{font-size:27px}
.ctitle .ms{font-size:19px}
.srows{display:flex;flex-direction:column}
.srow{display:grid;grid-template-columns:28px minmax(0,10em) minmax(0,1fr);gap:8px 18px;align-items:start;padding:20px 0;border-top:1px solid var(--line)}
.srow.fixrow{grid-template-columns:28px minmax(0,9em) minmax(0,1fr) 120px}
.snum{font-family:var(--sans);font-weight:700;font-size:16px;color:var(--muted);padding-top:15px}
.slabel{font-family:var(--sans);font-size:18px;line-height:1.35;padding-top:13px}
.slabel .mn,.slabel .mv,.slabel .mf,.slabel .mo{font-size:21px}
.sval{min-width:0}
.vrow{display:flex;align-items:flex-start;gap:10px}
.vrow > .sbox,.vrow > .minput{flex:1 1 auto;min-width:0}
.sbox{display:block;font-size:clamp(18px,2.9vw,24px);white-space:nowrap;overflow-x:auto;line-height:1.45;min-height:54px;padding:8px 16px;background:var(--bg);border:1.5px solid var(--line);border-radius:10px;width:100%;text-align:left;color:var(--ink)}
.sbox.given{border-style:dashed}
.sbox.good{border-color:var(--ok);background:var(--ok-tint)}
.sbox.bad{border-color:var(--bad);background:var(--bad-tint)}
.sbox.dim{opacity:.55}
.gtag,.etag{display:inline-block;margin-top:6px;font-family:var(--sans);font-size:13px;font-weight:600;letter-spacing:.04em;text-transform:uppercase;color:var(--muted)}
.bmark{width:22px;flex:none;color:var(--ok);font-weight:700;font-family:var(--sans);padding-top:14px}
.srow.ok .minput{border-color:var(--ok)}
.srow .fb{margin-top:8px}
.rowacts{display:flex;gap:10px;flex-wrap:wrap;margin-top:10px}
.fixbtn{font-family:var(--sans);font-size:15px;font-weight:600;min-height:44px;width:100%;margin-top:5px;padding:6px 10px;border-radius:10px;border:1.5px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer}
.fixbtn:hover{border-color:var(--ink)}
.fixbtn:disabled{opacity:.35;cursor:not-allowed}
.status{display:flex;align-items:flex-start;gap:12px;margin-top:20px;padding:16px 18px;border-radius:12px;font-family:var(--sans);font-size:17px;line-height:1.5}
.status .mn,.status .mv,.status .mf,.status .mo{font-size:20px}
.status .dot{flex:none;width:14px;height:14px;border-radius:50%;margin-top:5px}
.st-bad{background:var(--bad-tint)}.st-bad .dot{background:var(--bad)}
.st-ok{background:var(--ok-tint)}.st-ok .dot{background:var(--ok)}
.note{margin-top:20px;padding-top:16px;border-top:1px solid var(--line);font-size:20px}
/* machine */
.machine{margin:44px 0 0;background:var(--surface);border:1px solid var(--line);border-radius:18px;padding:40px 24px 32px;text-align:center}
.mbig{font-size:clamp(36px,6vw,50px);line-height:1.3;margin:0 0 28px}
.chain{display:flex;flex-direction:column;align-items:center;gap:8px;margin:0 auto 40px}
.cline{white-space:nowrap;font-size:clamp(22px,4.4vw,32px);line-height:1.35;padding:6px 18px;border-radius:10px}
.cline.last{background:var(--soft);font-weight:500}
.carr{height:30px;display:flex;align-items:center;justify-content:center}
.carr svg{display:block}
.arrp{fill:none;stroke:var(--muted);stroke-width:2;stroke-linecap:round;stroke-linejoin:round}
.cnote{font-family:var(--sans);font-size:15px;color:var(--muted);margin-left:10px}
.pipe{display:flex;align-items:center;justify-content:center;gap:10px;flex-wrap:nowrap}
.pnode{display:flex;flex-direction:column;align-items:center;min-width:64px}
.pk{font-family:var(--sans);font-size:13px;font-weight:700;letter-spacing:.06em;text-transform:uppercase;color:var(--muted)}
.pv2{font-family:var(--serif);font-size:30px;font-variant-numeric:tabular-nums}
.pin-n .pv2{color:var(--inside)}
.pku{text-transform:none;font-family:var(--serif);font-style:italic;font-weight:500;font-size:16px;letter-spacing:0}
.pbox{display:flex;flex-direction:column;align-items:center;padding:10px 16px;border-radius:12px;min-width:104px}
.pbox .pf{font-size:26px}
.pin{background:var(--inside-tint);border-bottom:3px solid var(--inside)}
.pin .pk{color:var(--inside)}
.pout{border:2px solid var(--outside)}
.pout .pk{color:var(--outside)}
.parr{flex:none;width:24px;height:2px;background:var(--muted);position:relative}
.parr::after{content:'';position:absolute;right:-1px;top:-4px;border-left:8px solid var(--muted);border-top:5px solid transparent;border-bottom:5px solid transparent}
.pick{display:flex;align-items:center;justify-content:center;flex-wrap:wrap;gap:8px;margin:0 0 26px;font-family:var(--sans);font-size:16px;color:var(--muted)}
.chip-s{font-family:var(--serif);font-size:19px;min-width:48px;min-height:46px;padding:0 16px;border-radius:999px;border:1px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer}
.chip-s.on{background:var(--k-tint);color:var(--ink);border-color:var(--k-line);box-shadow:inset 0 -2px 0 var(--k-line);font-weight:600}
.mu{margin:28px 0 0;padding-top:20px;border-top:1px solid var(--line);font-size:21px}
/* highlighter */
.hl{margin-top:36px;background:var(--surface);border:1px solid var(--line);border-radius:18px;overflow:hidden}
.hl-tabs{display:grid;grid-template-columns:repeat(auto-fit,minmax(110px,1fr));border-bottom:1px solid var(--line)}
.hl-tab{font-size:19px;min-height:56px;background:transparent;border:0;border-right:1px solid var(--line);color:var(--muted);cursor:pointer;padding:6px 8px;position:relative}
.hl-tab:last-child{border-right:0}
.hl-tab.cur{background:var(--bg);color:var(--ink)}
.hl-tab.cur::after{content:'';position:absolute;left:12px;right:12px;bottom:0;height:3px;background:var(--ink)}
.hl-tab.done{color:var(--ink)}
.hl-tab.done::before{content:'\2713';position:absolute;top:3px;right:7px;font-family:var(--sans);font-size:11px;color:var(--ok)}
.hl-stage{padding:34px 26px 32px;text-align:center}
.hl-count{font-family:var(--sans);font-size:15px;color:var(--muted)}
.hl-formula{font-size:clamp(40px,7vw,62px);line-height:1.3;margin:4px 0 18px}
.hl-formula.solved .in{animation:lift .7s cubic-bezier(.2,.8,.2,1) both;background:var(--inside-tint);border-radius:4px}
.hl-formula.solved .out{animation:fadein .7s .15s both}
.hl-ask{font-size:21px}
.chips{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px;max-width:460px;margin:0 auto}
.chip{font-size:26px;min-height:62px;padding:6px 14px;border-radius:12px;border:1.5px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer;transition:border-color .15s,transform .15s}
.chip:hover{border-color:var(--inside);transform:translateY(-1px)}
.chip.wrong{border-color:var(--outside);border-style:dashed}
.hl .fb{max-width:500px;margin:18px auto 0;text-align:left}
.hl-out{max-width:500px;margin:0 auto;text-align:left}
.hl-line{margin:0 0 8px;font-size:23px}
.hl-note{font-size:19px;color:var(--muted);margin-top:10px}
.hl-out .btn{margin-top:14px}
.hl-done{margin-top:20px;font-size:20px}
/* rule */
.rule{margin:44px 0 0;padding:48px 32px 40px;background:var(--surface);border:2px solid var(--ink);border-radius:18px;text-align:center;position:relative}
.rule-name{font-size:clamp(40px,6.5vw,56px);line-height:1.05;font-weight:500;margin:0 0 6px}
.memo{display:flex;justify-content:center;margin:14px 0 16px}
.memo span{position:relative;display:inline-block;font-family:var(--sans);font-size:14px;font-weight:700;letter-spacing:.08em;text-transform:uppercase;color:var(--outside);background:var(--outside-tint);border:1.5px solid var(--outside);border-radius:999px;padding:5px 16px}
.memo span::after{content:'';position:absolute;left:50%;bottom:-9px;margin-left:-8px;border-left:8px solid transparent;border-right:8px solid transparent;border-top:8px solid var(--outside)}
.rule-eq{white-space:nowrap;overflow-x:auto;font-size:clamp(19px,4.4vw,42px);line-height:1.4;padding:22px 12px;background:var(--bg);border-radius:12px}
.rule-words{margin:26px auto 0;max-width:30em;font-size:22px}
.rule-u{margin:14px auto 0;max-width:32em;font-family:var(--sans);font-size:17px;color:var(--muted)}
.rule-u .mn,.rule-u .mv,.rule-u .mf,.rule-u .mo{font-size:20px}
/* why */
.whyd{margin:44px auto 0;max-width:760px;border:1px solid var(--line);border-radius:14px;background:var(--surface);padding:4px 24px}
.whyd summary,.hintd summary{list-style:none}
.whyd summary::-webkit-details-marker,.hintd summary::-webkit-details-marker{display:none}
.whyd summary{font-family:var(--sans);font-size:18px;font-weight:600;cursor:pointer;min-height:58px;display:flex;align-items:center;gap:10px}
.opt{font-weight:400;color:var(--muted)}
.chev{margin-left:auto;width:12px;height:12px;border-right:2.5px solid currentColor;border-bottom:2.5px solid currentColor;transform:translateY(-3px) rotate(45deg);transition:transform .2s;flex:none}
details[open] > summary .chev{transform:translateY(3px) rotate(225deg)}
.hintd summary{gap:10px}
.hintd summary .chev{margin-left:0;width:9px;height:9px;border-width:2px}
.whyd ul{margin:4px 0 18px;padding-left:1.2em;font-size:19px}
.whyd li{margin-bottom:10px}
/* forms */
.forms{margin-top:28px;background:var(--surface);border:1px solid var(--line);border-radius:18px;padding:8px 26px}
.frow{display:grid;grid-template-columns:minmax(0,1fr) auto minmax(0,1.3fr);gap:14px;align-items:baseline;padding:14px 0;border-top:1px solid var(--line);font-size:26px}
.frow:first-child{border-top:0}
.fl{text-align:right}
.feq{color:var(--muted)}
/* exam */
.exam-q{font-size:23px}
.msol{margin:16px 0 0;padding-left:1.3em;font-size:21px}
.msol li{margin-bottom:10px}
.self{border:1px solid var(--line);border-radius:12px;padding:12px 18px 8px;margin:20px 0 0}
.self legend{font-family:var(--sans);font-size:15px;font-weight:600;color:var(--muted);padding:0 6px}
.ck{display:flex;align-items:center;gap:12px;min-height:46px;font-family:var(--sans);font-size:18px;cursor:pointer}
.ck input{width:22px;height:22px;accent-color:var(--ink);flex:none}
/* close */
.closer h3{font-size:28px;margin:8px 0 14px}
.closer .rules{background:var(--surface);border:1px solid var(--line);border-radius:18px;padding:4px 22px}
.reads{list-style:none;padding:0;margin:0;display:flex;flex-direction:column;gap:10px;font-size:20px}
.reveal{animation:rise .5s cubic-bezier(.2,.8,.2,1) both}
@keyframes rise{from{opacity:0;transform:translateY(8px)}to{opacity:1;transform:none}}
@keyframes lift{0%{background:transparent}40%{background:var(--inside-tint);box-shadow:0 0 0 6px var(--inside-tint)}100%{background:var(--inside-tint)}}
@keyframes fadein{from{opacity:.3}to{opacity:1}}
@media (prefers-reduced-motion: reduce){*{animation:none !important;transition:none !important}}
@media (max-width:860px){.toc{display:none}}
@media (max-width:900px){.srow,.srow.fixrow{grid-template-columns:24px minmax(0,1fr)}.snum{padding-top:2px}.slabel{padding-top:0}.sval,.fixcol{grid-column:2}.fixbtn{width:auto}}
@media (max-width:640px){
body{font-size:19px}
.page{padding:0 16px 110px}
.bar{margin:0 -16px;padding:0 16px}
.hero{padding:52px 0 16px}
.dek{font-size:20px}
.prob,.scard,.exam{padding:22px 16px 18px}
.pmath{font-size:25px}
.pipe{flex-direction:column;gap:6px}
.parr{width:2px;height:18px}
.parr::after{right:-4px;top:auto;bottom:-1px;border-left:5px solid transparent;border-right:5px solid transparent;border-top:8px solid var(--muted);border-bottom:0}
.hl-stage{padding:26px 14px}
.hl-tabs{grid-template-columns:repeat(3,minmax(0,1fr))}
.hl-tab{border-bottom:1px solid var(--line)}
.sbox{font-size:23px}
.frow{font-size:21px;gap:8px}
.rules li{grid-template-columns:minmax(0,1fr)}
.newtag{grid-column:1}
.sec{padding-top:96px}
.sec-n{font-size:46px}
.rule{padding:30px 16px 26px}
}
/* wrong-answer links */
.fixlink{display:inline-block;margin-left:2px;font-weight:600;color:var(--inside);text-underline-offset:3px}
.sec,.scard,[id^="blk-"],[id^="card-"],#ex-avg,#ex-short{scroll-margin-top:84px}
.flash{animation:flash 1.6s ease-out}
@keyframes flash{0%{box-shadow:0 0 0 4px var(--k-line)}100%{box-shadow:0 0 0 4px transparent}}
.backpill{position:fixed;left:50%;transform:translateX(-50%);bottom:calc(18px + env(safe-area-inset-bottom,0px));z-index:30;font-family:var(--sans);font-weight:700;font-size:16px;padding:12px 20px;border-radius:999px;border:0;background:var(--ink);color:var(--bg);box-shadow:0 6px 20px rgba(0,0,0,.25);cursor:pointer}
/* the button must vanish when hidden, whatever display a theme gives buttons */
.backpill[hidden]{display:none}
@media (prefers-reduced-motion: reduce){.flash{animation:none;outline:3px solid var(--k-line)}}
/* light and dark toggle */
.modebtn{margin-left:auto;font-family:var(--sans);font-size:15px;font-weight:600;color:var(--ink);background:transparent;border:1.5px solid var(--line);border-radius:999px;padding:6px 14px;cursor:pointer;white-space:nowrap}
.modebtn:hover{border-color:var(--muted)}
.toc a{white-space:nowrap}
@media (max-width:1080px){.toc{display:none}}
/* figure labels: words take the theme's label face, numbers the math face so a 1 never reads as a slash */
.pfig svg text.n,.overview svg text.n,.gwrap svg text.n,.partsfig svg text.n{font-family:var(--math)}
/* ================= v2 (F01, 2026-10-10) =================
Engine-generic rules that used to live in Unit 2's page.css, copied verbatim and in their original order,
so every page gets fractions, choices, cards, tables and the Memorize box without a page.css of its own. */
/* stacked fractions: inline-block in ordinary flow, never flex or grid */
.fr{display:inline-block;vertical-align:middle;text-align:center;line-height:1.08;margin:0 .1em;font-size:.86em}
.fr > .fn,.fr > .fd{display:block;padding:0 .14em;white-space:nowrap}
.fr > .fn{border-bottom:1.5px solid currentColor;padding-bottom:.06em}
.fr > .fd{padding-top:.06em}
.fr.rate{color:var(--rate);background:var(--rate-tint);border-radius:5px;padding:.08em .06em;box-shadow:inset 0 -2px 0 var(--rate)}
.fr.op{font-size:.8em}
.fr .mv{padding-right:0}
.dd{font-style:normal}
.rt{color:var(--rate);font-weight:600}
.k .fr{background:transparent}
.ml{white-space:nowrap}
.mline{display:block;white-space:nowrap;padding:3px 0}
.mline + .mline{border-top:1px dashed var(--line)}
.marr{display:block;color:var(--muted);font-family:var(--sans);font-size:15px;line-height:1;padding:2px 0 2px 6px}
/* emblem */
.emb-math{font-size:clamp(30px,6.4vw,64px);line-height:1.5;white-space:nowrap;text-align:center}
.emb-sub{font-size:clamp(19px,3.4vw,30px);color:var(--muted);text-align:center;padding:0 8px}
.sw-rate{background:var(--rate-tint);border-bottom:3px solid var(--rate)}
.sw-k{background:var(--k-tint);border-bottom:3px solid var(--k-line)}
.leg-rate b{color:var(--rate)}
/* choice lists */
.choices{display:flex;flex-direction:column;gap:10px;margin-top:6px}
.choice{display:block;width:100%;text-align:left;font-family:var(--serif);font-size:20px;line-height:1.45;min-height:56px;padding:12px 18px;border-radius:12px;border:1.5px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer}
.choice:hover{border-color:var(--ink)}
.choice.picked-ok{border-color:var(--ok);background:var(--ok-tint)}
.choice.picked-no{border-color:var(--outside);border-style:dashed}
.choice:disabled{cursor:default}
.choice .mn,.choice .mv,.choice .mf,.choice .mo{font-size:22px}
.sbox.multi{white-space:normal;overflow-x:auto}
.why{font-family:var(--sans);font-size:15px;color:var(--muted);margin-top:6px}
.why .mn,.why .mv,.why .mf,.why .mo{font-size:17px}
.mcrow{display:flex;flex-direction:column;gap:8px}
.mcrow .choice{min-height:52px;font-size:20px;padding:8px 14px}
.ptext{font-size:22px;line-height:1.5;margin:0 0 18px}
.pfig{margin:0 0 18px}
.pfig svg{display:block;width:100%;max-width:320px;height:auto;margin:0 auto}
.pfig svg text{fill:var(--ink)}
@media (max-width:640px){.ptext{font-size:20px}.choice{font-size:19px}}
.k + sup.ms{margin-left:-.06em}
svg text{font-family:var(--serif);fill:var(--ink)}
svg .lab{font-style:normal;font-size:19px}
svg .num{font-size:16px;fill:var(--muted)}
.marr{font-size:19px;padding:2px 0 2px 10px}
.ml.nb .eq{white-space:nowrap}
.marr{font-size:22px;padding:2px 0 2px 18px}
.ml.nb{display:inline-block}
.in{color:var(--inside)}
span.in{text-decoration:underline;text-decoration-thickness:2px;text-underline-offset:5px;text-decoration-skip-ink:none}
.fr span.in{text-underline-offset:3px}
.out{color:var(--outside)}
.sw-out2{background:var(--outside-tint);border-bottom:3px solid var(--outside)}
.emb-math{font-size:clamp(26px,5.6vw,52px)}
.emb-sub{max-width:28em;margin:0 auto}
.disp{text-align:center;font-size:clamp(28px,5vw,40px);margin:18px 0 8px}
.ctitle{font-family:var(--sans);font-size:17px;font-weight:700;color:var(--muted);margin:0 0 14px}
.ctitle .mn,.ctitle .mv,.ctitle .mo,.ctitle .pr{font-size:19px;font-weight:400}
.workbox{font-size:clamp(18px,2.9vw,23px);padding:12px 18px}
.workbox .mline{padding:6px 0}
.note.small,.small-sans{font-family:var(--sans);font-size:16px;color:var(--muted)}
.srow.tr{grid-template-columns:minmax(0,1fr);padding:14px 0}
.srow.tr .lhs{min-width:9.5em}
.tables{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:18px;margin:28px 0 0}
.avt{width:100%;border-collapse:collapse;background:var(--surface);border:1px solid var(--line);border-radius:14px;font-family:var(--sans);font-size:17px}
.avt caption{caption-side:top;text-align:left;font-weight:700;color:var(--muted);padding:0 0 8px;font-size:15px}
.avt th,.avt td{padding:10px 14px;border-top:1px solid var(--line);text-align:left}
.avt thead th{border-top:0;font-size:14px;color:var(--muted);font-weight:600}
.avt td.nv{font-family:var(--serif);font-size:21px;font-variant-numeric:tabular-nums;white-space:nowrap}
.avt .ip{display:inline-block;min-width:2ch;text-align:right}
.avt tr.near td{background:var(--soft);font-weight:600}
.avt .tbd{font-family:var(--sans);font-size:14px;color:var(--muted)}
.avt.small{margin:6px 0 4px;width:auto}
.avt.small td,.avt.small th{text-align:center;font-family:var(--serif);font-size:19px}
.rule-sub{font-size:20px;margin:18px 0 10px}
.rule-words{font-size:20px}
.zoombtns{display:flex;flex-wrap:wrap;gap:8px;margin-bottom:12px}
.zb{font-family:var(--sans);font-size:16px;font-weight:600;min-height:44px;padding:6px 16px;border-radius:999px;border:1.5px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer}
.zb.on{background:var(--ink);color:var(--bg);border-color:var(--ink)}
.gwrap svg{display:block;width:100%;height:auto}
.gread{font-size:21px;margin:12px 0 4px}
@media (max-width:640px){.tables{grid-template-columns:minmax(0,1fr)}.graphcard{padding:16px 12px 12px}.cardlist{padding:18px 16px 8px}}
.mline.cont{border-top:0 !important;padding-top:0;padding-left:1.2em}
code{font-family:var(--mono);font-size:.92em;background:var(--soft);border-radius:5px;padding:1px 6px;color:var(--ink);font-style:normal}
.rule .rule-eq{font-size:clamp(28px,6.4vw,44px)}
@media (max-width:640px){.avt.small td,.avt.small th{padding:8px 6px;font-size:17px}}
@media (max-width:640px){.srow.tr .minput{flex:1 1 140px}.srow.tr .ansrow{flex-wrap:wrap}.srow.tr .lhs{width:100%}}
.ml.nb{padding:0 .08em}
/* punctuation right after inline math rides inside the math's box, so it never starts a line (F03); same spacing as before */
.ml.nb.mlp{padding-right:0}
.mpu{padding-left:.08em}
.mine{font-family:var(--mono);font-size:18px}
.moreq + .moreq{border-top:1px solid var(--line);margin-top:18px;padding-top:18px}
/* ---------- v2 notation (lesson-core.js mth) ---------- */
.mn.me{font-style:normal}
.msb{font-size:.7em;line-height:1;vertical-align:-.3em}
.lim{display:inline-flex;flex-direction:column;align-items:center;vertical-align:baseline;line-height:1.15;margin-right:.2em}
.lim-s{font-size:.6em;line-height:1;white-space:nowrap;margin-top:.02em}
.lim-s .ms{font-size:.85em;vertical-align:.5em}
.int{white-space:nowrap;margin-right:.1em}
.int-s{font-family:var(--math);font-size:1.7em;line-height:1;vertical-align:-.28em;font-style:normal}
.int-l{display:inline-flex;flex-direction:column;justify-content:space-between;align-items:flex-start;vertical-align:middle;height:2.9em;font-size:.6em;line-height:1;margin-left:-.05em}
.int-b{margin-top:-.35em}.int-a{margin-bottom:-.5em;margin-left:-.35em}
.sum{display:inline-flex;flex-direction:column;align-items:center;vertical-align:middle;line-height:1;margin:0 .14em}
.sum-s{font-family:var(--math);font-size:1.5em;line-height:1}
.sum-a,.sum-b{font-size:.6em;white-space:nowrap;line-height:1.1}
.evb{display:inline-flex;flex-direction:column;justify-content:space-between;align-items:flex-start;vertical-align:middle;height:3.1em;font-size:.6em;line-height:1;border-left:.13em solid currentColor;padding-left:.22em;margin-left:.12em}
.abs > .mn:first-child{padding-right:.05em}.abs > .mn:last-child{padding-left:.05em}
.dif .mv{padding-right:0}
.fr .lim,.fr .int{font-size:1em}
.mf.fs{padding-right:.2em}
/* scripts under lim, beside ∫, around Σ and on the bar keep their small size even where a container fixes math spans in px */
.lim-s *,.int-l *,.sum-a *,.sum-b *,.evb *,sub.msb *{font-size:inherit !important}
/* v2 pages (mount without legacy): superscripts stay small inside choices, feedback, hints and card titles */
.v2 sup.ms *{font-size:inherit !important}
.v2 .ctitle .ms{font-size:.7em}
.rule-line{display:block}
/* ---------- v2 components (components.js) ---------- */
/* SharedBadge, card header */
.shared-badge{display:inline-flex;align-items:center;font-family:var(--sans);font-size:14px;font-weight:700;line-height:1.3;color:var(--ink);background:var(--k-tint);border:1.5px solid var(--k-line);border-radius:999px;padding:4px 12px}
.ctop{display:flex;flex-wrap:wrap;align-items:center;justify-content:space-between;gap:8px 12px;margin:0 0 14px}
.ctop .ctitle{margin:0}
/* VariationCard */
.vc-idea{font-family:var(--sans);font-size:17px;color:var(--muted);margin:-6px 0 8px}
.vc-count{font-family:var(--sans);font-size:15px;font-weight:600;color:var(--muted);margin:0 0 6px}
.vc-list{list-style:none;counter-reset:vc;margin:0;padding:0}
.vc-item{counter-increment:vc;position:relative;padding:22px 0 8px 42px;border-top:1px solid var(--line)}
.vc-item::before{content:counter(vc);position:absolute;left:0;top:22px;width:28px;height:28px;border-radius:50%;border:1.5px solid var(--line);font-family:var(--sans);font-size:15px;font-weight:700;color:var(--muted);display:flex;align-items:center;justify-content:center;box-sizing:border-box}
.vc-item.done::before{content:'\2713';color:var(--ok);border-color:var(--ok);background:var(--ok-tint)}
.vc-item .ptext{font-size:20px}
/* Matcher */
.mt-how{font-family:var(--sans);font-size:16px;color:var(--muted);margin:0 0 12px}
.mt-cols{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px}
.mt-col{display:flex;flex-direction:column;gap:10px;min-width:0}
.mt-i{display:flex;align-items:center;gap:10px;width:100%;text-align:left;font-family:var(--serif);font-size:19px;line-height:1.4;min-height:56px;padding:10px 14px;border-radius:12px;border:1.5px solid var(--line);background:var(--bg);color:var(--ink);cursor:pointer}
.mt-i:hover{border-color:var(--ink)}
.mt-i.sel{border-color:var(--inside);box-shadow:0 0 0 3px var(--inside-tint)}
.mt-i.done{border-color:var(--ok);background:var(--ok-tint);cursor:default}
.mt-n{flex:none;width:26px;height:26px;border-radius:50%;background:var(--ok);color:var(--bg);font-family:var(--sans);font-size:14px;font-weight:700;display:flex;align-items:center;justify-content:center}
.mt-body{flex:1 1 auto;min-width:0}
.mt-body svg{display:block;width:100%;max-width:220px;height:auto;margin:0 auto}
.mt-body svg text{fill:var(--ink);font-family:var(--math);font-size:15px}
.mt-count{font-family:var(--sans);font-size:15px;color:var(--muted);margin:12px 0 0}
/* Slider and Toggle */
.sl{margin:14px 0 8px}
.sl-lab{display:block;font-family:var(--sans);font-size:16px;font-weight:600;color:var(--ink);margin-bottom:2px}
.sl-lab .mn,.sl-lab .mv,.sl-lab .mo{font-size:19px}
.sl-row{display:flex;align-items:center;gap:14px}
.sl-in{flex:1 1 auto;min-width:0;height:44px;accent-color:var(--inside);margin:0}
.sl-out{flex:none;min-width:4.2em;text-align:right;font-family:var(--math);font-size:21px;font-variant-numeric:tabular-nums;color:var(--ink)}
.tg .zb .mn,.tg .zb .mv,.tg .zb .mo{font-size:17px}
/* GraphView (the theme puts .gv on the board) */
.gv{margin:0 0 18px;border-radius:12px;padding:12px}
.gv svg{display:block;width:100%;height:auto;max-width:560px;margin:0 auto;overflow:visible}
.gv svg text{fill:var(--ink);font-size:16px}
.gv svg text.num{fill:var(--muted)}
.gv svg text.n{font-family:var(--math)}
.gv svg .gv-lab{font-size:16px}
.gv svg .gv-sign{font-size:22px;font-weight:700;fill:var(--ink)}
.gv-drag{cursor:grab;outline:none;touch-action:none}
.gv-drag:focus-visible .gv-dot,.gv-drag:focus .gv-dot{stroke:var(--focus);stroke-width:4}
.gv-read{font-family:var(--sans);font-size:17px;text-align:center;margin:10px 0 0;min-height:1.6em}
.gv-read .mn,.gv-read .mv,.gv-read .mo,.gv-read .mf,.gv-read .pr{font-size:21px}
.gv-cap{font-family:var(--sans);font-size:16px;color:var(--muted);margin:8px 0 0}
.gv .zoombtns{justify-content:center}
/* TableBuilder */
.tb-wrap{overflow-x:auto;margin:18px 0 0;max-width:100%}
.avt.tb{width:auto;min-width:min(100%,300px)}
.tb td,.tb th{white-space:nowrap}
.tb th[scope="row"]{font-family:var(--sans);font-size:16px;font-weight:600;color:var(--muted);border-top:1px solid var(--line)}
.tb tbody tr:first-child th[scope="row"]{border-top:0}
.tb td.nv{font-family:var(--serif);font-size:20px;font-variant-numeric:tabular-nums}
.tb th .mn,.tb th .mv,.tb th .mf,.tb th .mo{font-size:19px}
.dn{display:inline-flex;white-space:nowrap}
.dn-i{display:inline-block;text-align:right}
.dn-f{display:inline-block;text-align:left}
.tb-across td,.tb-across th{text-align:center;padding:10px 10px}
.tb-cell{display:inline-flex;align-items:center;gap:6px}
.minput.tb-in{flex:none;width:6.2em;min-height:44px;font-size:17px;padding:6px 8px}
.tb-chk.btn.sm{min-height:44px;padding:6px 10px;font-size:15px}
.tb-ok{color:var(--ok);font-weight:700;font-family:var(--sans)}
.tb-fb .fb{margin-top:10px}
/* MemorizeBox: an entry may carry a caption after its equation */
.rule-cap{margin:12px auto 0;max-width:30em;font-size:19px;color:var(--muted)}
.rule-eq + .rule-sub,.rule-cap + .rule-sub,.rule-cap + .rule-eq,.rule-eq + .rule-eq{margin-top:26px}
/* FadedRows */
.srows.faded .workbox{margin:14px 0 4px}
.srows.faded .srow.tr:first-child{border-top:0}
.locked{margin-top:14px}
/* Optional */
.optd-body{padding:0 0 14px}
.optd-body > .scard:first-child{margin-top:8px}
@media (max-width:640px){
.mt-cols{gap:8px}
.mt-i{font-size:17px;padding:8px 10px;gap:8px}
.gv{padding:8px}
.vc-item{padding-left:36px}
.shared-badge{font-size:13px}
.sl-out{font-size:19px;min-width:3.6em}
}
/* teaching frame (F02, 2026-10-11): Where We Are, What You Will Know, Definition, New Notation. Theme tokens only. */
.vh{position:absolute;width:1px;height:1px;padding:0;margin:-1px;overflow:hidden;clip:rect(0 0 0 0);white-space:nowrap;border:0}
.know,.defn,.notn{scroll-margin-top:84px}
.where p{font-size:22px;line-height:1.55;margin:0 0 18px}
.prose.where{margin-top:28px}
.tb-lab{font-family:var(--sans);font-size:13px;font-weight:700;letter-spacing:.09em;text-transform:uppercase;color:var(--muted);margin:0 8px 6px 0}
p.tb-lab{display:block}
.tb-read{font-size:20px;line-height:1.5;margin:18px 0 0}
.tb-read .tb-lab{display:inline}
/* What You Will Know: the Memorize box's shape and display, dashed and in the preview color */
.know{margin:36px 0 0;padding:34px 32px 26px;background:var(--surface);border:2px dashed var(--inside);border-radius:18px}
.know-name{font-size:clamp(32px,5vw,42px);line-height:1.1;text-align:center}
.know-tag{display:flex;justify-content:center;margin:12px 0 8px}
.know-tag span{display:inline-block;font-family:var(--sans);font-size:14px;font-weight:700;letter-spacing:.08em;text-transform:uppercase;color:var(--inside);background:var(--inside-tint);border:1.5px dashed var(--inside);border-radius:999px;padding:5px 16px}
.know-list{list-style:none;margin:14px 0 0;padding:0;counter-reset:know}
.know-list li{position:relative;counter-increment:know;padding:16px 0 16px 48px;border-top:1px solid var(--line)}
.know-list li:first-child{border-top:0}
.know-list li::before{content:counter(know);position:absolute;left:0;top:16px;width:32px;height:32px;border-radius:50%;background:var(--inside-tint);color:var(--inside);font-family:var(--sans);font-size:16px;font-weight:700;line-height:32px;text-align:center}
.know-text{font-size:21px;line-height:1.5;margin:0}
.know-eq{margin:10px 0 0;padding:14px 12px;background:var(--bg);border-radius:12px;text-align:center;font-size:clamp(22px,4.6vw,32px);line-height:1.4;overflow-x:auto}
.know-eq:first-child{margin-top:0}
.know-cap{margin:10px 0 0;font-size:19px;color:var(--muted)}
/* Definition */
.defn{margin:40px 0;background:var(--surface);border:2px solid var(--inside);border-radius:16px;overflow:hidden}
.defn-head{display:block;padding:16px 26px 14px;background:var(--inside-tint);border-bottom:2px solid var(--inside)}
.defn-kind{display:block;font-family:var(--sans);font-size:13px;font-weight:700;letter-spacing:.1em;text-transform:uppercase;color:var(--inside);margin-bottom:2px}
.defn-term{display:block;font-weight:700;font-size:clamp(26px,4.4vw,34px);line-height:1.15;color:var(--ink)}
.defn-body{padding:22px 26px 24px}
.defn-words{font-size:22px;line-height:1.55;margin:0 0 16px}
.defn-eq{padding:18px 12px;background:var(--bg);border-radius:12px;text-align:center;font-size:clamp(24px,5vw,38px);line-height:1.4;overflow-x:auto}
.defn-not{margin:18px 0 0;padding:14px 18px;background:var(--outside-tint);border-left:4px solid var(--outside);border-radius:0 10px 10px 0}
.defn-not .tb-lab{color:var(--outside)}
.defn-not p:not(.tb-lab){font-size:20px;line-height:1.5;margin:0}
.defn-not p + p:not(.tb-lab){margin-top:8px}
/* New Notation: the symbol large, then its parts; side by side when there is room, stacked at phone width */
.notn{margin:40px 0;padding:22px 26px 24px;background:var(--surface);border:2px solid var(--ink);border-radius:16px}
.notn-head{font-family:var(--sans);font-size:13px;font-weight:700;letter-spacing:.1em;text-transform:uppercase;color:var(--muted);margin:0 0 12px}
.notn-sym{padding:22px 12px;background:var(--bg);border-radius:12px;text-align:center;font-size:clamp(34px,7.5vw,54px);line-height:1.3;overflow-x:auto}
.notn-parts{display:grid;grid-auto-flow:column;grid-auto-columns:minmax(0,1fr);gap:16px;margin:18px 0 0}
.notn-part{display:flex;flex-direction:column;align-items:center;text-align:center}
.notn-piece{display:inline-block;min-width:2.4em;padding:6px 14px;border:2px solid var(--inside);background:var(--inside-tint);border-radius:10px;font-size:26px;line-height:1.3}
.notn-part:nth-child(3n+2) .notn-piece{border-color:var(--outside);background:var(--outside-tint)}
.notn-part:nth-child(3n) .notn-piece{border-color:var(--rate);background:var(--rate-tint)}
.notn-mean{margin:0;padding-top:16px;position:relative;font-size:19px;line-height:1.45}
.notn-mean::before{content:'';position:absolute;top:2px;left:50%;height:10px;border-left:2px solid var(--line)}
.notn-ex{margin:20px 0 0;padding:14px 18px;background:var(--soft);border-radius:12px}
.notn-ex p:not(.tb-lab){font-size:20px;line-height:1.5;margin:0}
.notn-ex-row{display:grid;grid-template-columns:7.5em minmax(0,1fr);gap:10px;align-items:baseline}
.notn-ex-row + .notn-ex-row{margin-top:8px !important}
.notn-ex-k{font-family:var(--sans);font-size:15px;font-weight:600;color:var(--muted)}
@media (max-width:640px){
.where p{font-size:20px}
.know{padding:26px 16px 18px}
.know-list li{padding-left:42px}
.know-text{font-size:19px}
.defn-head{padding:14px 16px 12px}
.defn-body{padding:18px 16px 20px}
.defn-words{font-size:20px}
.notn{padding:18px 16px 20px}
.notn-parts{grid-auto-flow:row;grid-auto-columns:auto;grid-template-columns:auto minmax(0,1fr);gap:12px 14px;align-items:center}
.notn-part{display:contents}
.notn-piece{font-size:22px;text-align:center;justify-self:stretch}
.notn-mean{padding-top:0;font-size:18px;text-align:left}
.notn-mean::before{display:none}
.notn-ex-row{grid-template-columns:minmax(0,1fr);gap:2px}
}
/* Optimization Explorer: page-only styles. Theme tokens only; tests/lint.js and the build refuse literal colors. */
/* page-local figures (the fence, the sheet and the box): wider than the default .pfig, labels in one size */
.pfig.ofig svg { max-width: 420px; }
.ofig svg text { font-size: 16px; }
.ofig-row { display: flex; flex-wrap: wrap; gap: 10px 14px; justify-content: center; align-items: flex-end; }
.pfig.ofig .ofig-row svg { flex: 1 1 260px; max-width: 360px; margin: 0; }
/* the draggable handle inside a figure */
.ofig-h { cursor: grab; outline: none; touch-action: none; }
.ofig-h:focus-visible .ofig-dot, .ofig-h:focus .ofig-dot { stroke: var(--focus); stroke-width: 4; }
/* live readouts under a figure */
.oread { display: flex; flex-wrap: wrap; justify-content: center; gap: 4px 20px; margin: -6px 0 16px; font-family: var(--sans); font-size: 17px; }
.oread span { white-space: nowrap; }
/* steps that open one at a time */
.ostep { border-top: 1px solid var(--line); padding: 14px 0 6px; }
.ostep-k { font-family: var(--sans); font-size: 14px; letter-spacing: .02em; color: var(--muted); margin: 0 0 4px; }
.ostep.ok .ostep-k { color: var(--ok); }
.ostep .choice { text-align: left; }
/* nudge buttons under a figure's readout */
.onudge { display: flex; justify-content: center; gap: 12px; margin: -8px 0 16px; }
.onudge .btn { min-width: 96px; min-height: 44px; }
</style>
</head>
<body>
<div class="page" id="app"></div>
<script>
/*
* mathkit.js: the math engine from the chain rule page, usable on its own.
*
* Works in the browser (window.MathKit) and in Node (require('./mathkit.js')).
*
* const mk = new MathKit();
* mk.same(mk.cf('15(3x+1)^4'), mk.cf('5(3x+1)^4*3')) // true: equivalence by sampling
* mk.parse('sin x^2') // tree, or {err: 'friendly message'}
* mk.pr(mk.simp(mk.diff(mk.parse('ln(x^2+4)').ast)), 'x').s // '1/(x^2 + 4) * 2x'
* mk.rich('Find `f(x) = «(»{3x + 1}«)^5»` at `x = ‹2›`') // typeset segments (see math-typesetting.md)
* mk.judge({ ans: '2x', wrongs: [['x^2', 'forgot to differentiate']] }, studentText)
*
* CLI (validate answer keys while authoring):
* node mathkit.js same "15(3x+1)^4" "5(3x+1)^4*3"
* node mathkit.js diff "sqrt(ln(x^2+5))"
* node mathkit.js parse "e^2x" # shows how an ambiguous input is read
*
* Scope: one variable, + - * / ^, implicit multiplication, sin cos tan sec csc cot
* arcsin arccos arctan ln log sqrt exp abs, |..| absolute value bars, e, pi.
* Equivalence is numeric: two expressions agree at 10 sample points (relative tolerance
* 1e-6, at least 3 points where both are defined). Good for answer checking; not a proof.
*
* The variable (v2, 2026-10-10). The second argument to tok/parse/compile/cf is either
* - nothing or false: the variable is x;
* - true: the variable is u (kept for older pages);
* - a letter, such as 't': the variable is that letter;
* - an object { v: 't', params: ['c', 'y'] }: the variable is v, and each letter in params
* is read as a named constant whose value is supplied at evaluation time:
* compile(src, o).f(x, { c: 1, y: 2 }). Missing params evaluate to 0.
* Input is lowercased first, so params and variables are lowercase letters.
*/
class MathKit {
constructor() { this.fcache = {}; this.rcache = {}; this.D = { VOC: {} }; }
/* normalize the variable option (see the header) */
vo(uv) {
if (uv && typeof uv === 'object') return { v: String(uv.v || 'x').toLowerCase(), params: (uv.params || []).map((p) => String(p).toLowerCase()) };
if (typeof uv === 'string' && uv) return { v: uv.toLowerCase(), params: [] };
return { v: uv ? 'u' : 'x', params: [] };
}
/* ---------- math reading: text to tree ---------- */
tok(src, uv) {
let bar = 0;
const s = String(src).replace(/[−–—]/g, '-').replace(/[·×∙⋅]/g, '*').replace(/÷/g, '/').replace(/π/g, 'pi').replace(/√/g, 'sqrt').replace(/²/g, '^2').replace(/³/g, '^3').replace(/⁴/g, '^4').replace(/\*\*/g, '^').replace(/[\[{]/g, '(').replace(/[\]}]/g, ')')
.replace(/\|/g, () => (bar++ % 2 === 0 ? ' (abs(' : '))')).toLowerCase();
if (bar % 2) return { err: 'An absolute value bar “|” is missing its partner.' };
const W = ['sqrt', 'arcsin', 'arccos', 'arctan', 'sin', 'cos', 'tan', 'sec', 'csc', 'cot', 'exp', 'abs', 'ln', 'log', 'pi'];
const o = this.vo(uv), vname = o.v;
const out = [];
let i = 0;
while (i < s.length) {
const c = s[i];
if (c === ' ' || c === '\t') { i++; continue; }
if (c === vname) { out.push({ t: 'x' }); i++; continue; }
if (/[0-9.]/.test(c)) {
let j = i;
while (j < s.length && /[0-9.]/.test(s[j])) j++;
const t = s.slice(i, j);
if (t === '.' || (t.match(/\./g) || []).length > 1) return { err: 'The number “' + t + '” has a stray decimal point.' };
out.push({ t: 'num', v: parseFloat(t) });
i = j;
continue;
}
if (/[a-z]/.test(c)) {
const w = W.find((k) => s.startsWith(k, i));
if (w) { i += w.length; if (w === 'pi') out.push({ t: 'pi' }); else out.push({ t: 'fn', v: w === 'log' ? 'ln' : w }); continue; }
if (c === 'e') { out.push({ t: 'e' }); i++; continue; }
if (o.params.indexOf(c) >= 0) { out.push({ t: 'p', v: c }); i++; continue; }
if (vname === 'u' && c === 'x') return { err: 'Write this row using u, not x. Here u stands for the whole inside.' };
return { err: 'This box uses ' + vname + ' as its only variable, so “' + c + '” is not something it can read.' };
}
if ('+-*/^()'.indexOf(c) >= 0) { out.push({ t: c }); i++; continue; }
return { err: 'The symbol “' + c + '” is not something this box can read.' };
}
return { toks: out };
}
parse(src, uv) {
if (!src || !String(src).trim()) return { empty: true };
const r = this.tok(src, uv);
if (r.err) return { err: r.err };
const T = r.toks;
let i = 0;
const peek = () => T[i];
const vn = this.vo(uv).v;
const isF = (t) => !!t && (t.t === 'num' || t.t === 'x' || t.t === 'p' || t.t === 'e' || t.t === 'pi' || t.t === 'fn' || t.t === '(');
const show = (t) => (t.t === 'fn' || t.t === 'p' ? t.v : t.t === 'num' ? String(t.v) : t.t === 'x' ? vn : t.t);
const fail = (m) => { throw new Error(m); };
const B = (k, a, b) => ({ k, a, b });
let expr, term, unary, power, primary;
expr = () => {
let a = term();
while (peek() && (peek().t === '+' || peek().t === '-')) { const op = T[i++].t; a = B(op, a, term()); }
return a;
};
term = () => {
let a = unary();
for (;;) {
const t = peek();
if (t && (t.t === '*' || t.t === '/')) { i++; a = B(t.t, a, unary()); }
else if (isF(t)) a = B('*', a, power());
else break;
}
return a;
};
unary = () => {
const t = peek();
if (t && t.t === '-') { i++; return { k: 'neg', a: unary() }; }
if (t && t.t === '+') { i++; return unary(); }
return power();
};
power = () => {
const b = primary();
if (peek() && peek().t === '^') {
i++;
let e;
if (peek() && peek().t === '-') { i++; e = { k: 'neg', a: power() }; } else e = power();
return B('^', b, e);
}
return b;
};
primary = () => {
const t = T[i];
if (!t) fail('The expression ends too soon. Something is missing at the end.');
if (t.t === 'num') { i++; return { k: 'n', v: t.v }; }
if (t.t === 'x') { i++; return { k: 'v' }; }
if (t.t === 'p') { i++; return { k: 'p', name: t.v }; }
if (t.t === 'e') { i++; return { k: 'c', name: 'e', v: Math.E }; }
if (t.t === 'pi') { i++; return { k: 'c', name: 'pi', v: Math.PI }; }
if (t.t === '(') {
i++;
if (peek() && peek().t === ')') fail('There is an empty pair of parentheses.');
const a = expr();
if (!peek()) fail('A parenthesis is still open. Close it with “)”.');
if (peek().t !== ')') fail('Something is out of place near “' + show(peek()) + '”.');
i++;
return a;
}
if (t.t === 'fn') {
i++;
let fe = null;
if (peek() && peek().t === '^') { i++; fe = primary(); }
let arg;
if (peek() && peek().t === '(') arg = primary();
else {
if (!peek() || !isF(peek()) || peek().t === 'fn') fail(t.v + ' needs something to act on, such as ' + t.v + '(' + vn + ').');
arg = power();
while (peek() && isF(peek()) && peek().t !== 'fn' && peek().t !== '(') arg = B('*', arg, power());
}
const g = { k: 'f', name: t.v, a: arg };
return fe ? B('^', g, fe) : g;
}
if (t.t === ')') fail('There is a “)” without a matching “(”.');
fail('Something is out of place near “' + show(t) + '”.');
};
try {
const ast = expr();
if (i < T.length) {
const t = T[i];
if (t.t === ')') fail('There is a “)” without a matching “(”.');
fail('Something is out of place near “' + show(t) + '”.');
}
return { ast };
} catch (e) {
return { err: e.message };
}
}
ev(n, x, env) {
switch (n.k) {
case 'n': case 'c': return n.v;
case 'v': return x;
case 'p': return env && env[n.name] !== undefined ? env[n.name] : 0;
case '+': return this.ev(n.a, x, env) + this.ev(n.b, x, env);
case '-': return this.ev(n.a, x, env) - this.ev(n.b, x, env);
case '*': return this.ev(n.a, x, env) * this.ev(n.b, x, env);
case '/': return this.ev(n.a, x, env) / this.ev(n.b, x, env);
case '^': {
const b = this.ev(n.a, x, env), e = this.ev(n.b, x, env);
/* a negative base to a fractional power with an odd denominator (x^(1/3), x^(2/3)) is real */
if (b < 0 && !Number.isInteger(e)) { for (let q = 3; q <= 9; q += 2) { const p = e * q; if (Math.abs(p - Math.round(p)) < 1e-9) { const r = Math.pow(-b, e); return Math.round(p) % 2 ? -r : r; } } }
return Math.pow(b, e);
}
case 'neg': return -this.ev(n.a, x, env);
case 'f': {
const v = this.ev(n.a, x, env);
const F = { sin: Math.sin, cos: Math.cos, tan: Math.tan, sec: (q) => 1 / Math.cos(q), csc: (q) => 1 / Math.sin(q), cot: (q) => 1 / Math.tan(q), arcsin: Math.asin, arccos: Math.acos, arctan: Math.atan, exp: Math.exp, abs: Math.abs, ln: Math.log, sqrt: Math.sqrt };
return F[n.name](v);
}
}
return NaN;
}
compile(src, uv) {
const p = this.parse(src, uv);
if (p.empty) return { empty: true };
if (p.err) return { err: p.err };
const ast = p.ast;
return { f: (x, env) => this.ev(ast, x, env), ast };
}
/* does a tree use a named constant (a param)? */
hasParam(n, name) { if (!n) return false; if (n.k === 'p') return name === undefined || n.name === name; return this.hasParam(n.a, name) || this.hasParam(n.b, name); }
cf(s, uv) {
const o = this.vo(uv);
const k = o.v + '|' + o.params.join(',') + ':' + s;
if (!(k in this.fcache)) { const c = this.compile(s, uv); this.fcache[k] = c.f || null; }
return this.fcache[k];
}
same(f, g) {
if (!f || !g) return false;
const P = [-2.3, -1.45, -0.75, -0.2, 0.35, 0.8, 1.3, 1.9, 2.6, 3.4];
let n = 0;
for (let k = 0; k < P.length; k++) {
const a = g(P[k]);
if (!isFinite(a)) continue;
const b = f(P[k]);
if (!isFinite(b)) return false;
if (Math.abs(a - b) > 1e-6 * Math.max(1, Math.abs(a), Math.abs(b))) return false;
n++;
}
return n >= 3;
}
/* ---------- a small symbolic engine for the find-the-mistake card ---------- */
N(v) { return { k: 'n', v }; }
isN(n, v) { return n.k === 'n' && (v === undefined || Math.abs(n.v - v) < 1e-12); }
hasVar(n) { if (n.k === 'v') return true; return !!((n.a && this.hasVar(n.a)) || (n.b && this.hasVar(n.b))); }
diff(n) {
const d = (m) => this.diff(m);
const B = (k, a, b) => ({ k, a, b });
const F = (name, a) => ({ k: 'f', name, a });
switch (n.k) {
case 'n': case 'c': case 'p': return this.N(0);
case 'v': return this.N(1);
case '+': case '-': return B(n.k, d(n.a), d(n.b));
case 'neg': return { k: 'neg', a: d(n.a) };
case '*': return B('+', B('*', d(n.a), n.b), B('*', n.a, d(n.b)));
case '/': return B('/', B('-', B('*', d(n.a), n.b), B('*', n.a, d(n.b))), B('^', n.b, this.N(2)));
case '^':
if (!this.hasVar(n.b)) return B('*', B('*', n.b, B('^', n.a, B('-', n.b, this.N(1)))), d(n.a));
if (!this.hasVar(n.a)) return B('*', n.a.k === 'c' && n.a.name === 'e' ? n : B('*', n, F('ln', n.a)), d(n.b));
return B('*', n, B('+', B('*', d(n.b), F('ln', n.a)), B('/', B('*', n.b, d(n.a)), n.a)));
case 'f': {
const a = n.a, da = d(a);
const G = {
sin: () => F('cos', a), cos: () => ({ k: 'neg', a: F('sin', a) }), tan: () => B('^', F('sec', a), this.N(2)),
sec: () => B('*', F('sec', a), F('tan', a)), exp: () => F('exp', a),
ln: () => B('/', this.N(1), a), sqrt: () => B('/', this.N(1), B('*', this.N(2), F('sqrt', a)))
};
if (!G[n.name]) return null;
return B('*', G[n.name](), da);
}
}
return null;
}
hasNull(n) { if (!n) return true; return !!((n.a !== undefined && n.a !== null && this.hasNull(n.a)) || (n.a === null) || (n.b === null) || (n.b && this.hasNull(n.b))); }
simp(n) {
if (!n) return n;
const S = (m) => this.simp(m);
const B = (k, a, b) => ({ k, a, b });
if (n.k === 'f') return { k: 'f', name: n.name, a: S(n.a) };
if (n.k === 'neg') { const a = S(n.a); if (a.k === 'n') return this.N(-a.v); if (a.k === 'neg') return a.a; return { k: 'neg', a }; }
if (!n.b) return n;
const a = S(n.a), b = S(n.b);
switch (n.k) {
case '+':
if (this.isN(a, 0)) return b; if (this.isN(b, 0)) return a;
if (a.k === 'n' && b.k === 'n') return this.N(a.v + b.v);
if (b.k === 'neg') return B('-', a, b.a);
if (b.k === 'n' && b.v < 0) return B('-', a, this.N(-b.v));
return B('+', a, b);
case '-':
if (this.isN(b, 0)) return a; if (this.isN(a, 0)) return S({ k: 'neg', a: b });
if (a.k === 'n' && b.k === 'n') return this.N(a.v - b.v);
return B('-', a, b);
case '*':
if (this.isN(a, 0) || this.isN(b, 0)) return this.N(0);
if (this.isN(a, 1)) return b; if (this.isN(b, 1)) return a;
if (a.k === 'n' && b.k === 'n') return this.N(a.v * b.v);
if (a.k === 'neg') return S({ k: 'neg', a: B('*', a.a, b) });
if (b.k === 'neg') return S({ k: 'neg', a: B('*', a, b.a) });
if (b.k === 'n' && a.k !== 'n') return S(B('*', b, a));
if (a.k === 'n' && b.k === '*' && b.a.k === 'n') return S(B('*', this.N(a.v * b.a.v), b.b));
if (a.k === 'n' && a.v < 0) return { k: 'neg', a: S(B('*', this.N(-a.v), b)) };
return B('*', a, b);
case '/':
if (this.isN(b, 1)) return a; if (this.isN(a, 0)) return this.N(0);
if (a.k === 'n' && b.k === 'n') return this.N(a.v / b.v);
return B('/', a, b);
case '^':
if (this.isN(b, 1)) return a; if (this.isN(b, 0)) return this.N(1);
if (a.k === 'n' && b.k === 'n') return this.N(Math.pow(a.v, b.v));
return B('^', a, b);
}
return B(n.k, a, b);
}
subst(n, r) {
if (n.k === 'v') return r;
const o = Object.assign({}, n);
if (n.a) o.a = this.subst(n.a, r);
if (n.b) o.b = this.subst(n.b, r);
return o;
}
fnum(v) {
if (Number.isInteger(v)) return String(v);
for (let q = 2; q <= 12; q++) { const p = v * q; if (Math.abs(p - Math.round(p)) < 1e-9) return Math.round(p) + '/' + q; }
return String(Math.round(v * 1000) / 1000);
}
pr(n, vn) {
const P = (m) => this.pr(m, vn);
const wrap = (r, need) => (r.p < need ? '(' + r.s + ')' : r.s);
switch (n.k) {
case 'n': return n.v < 0 ? { s: '-' + this.fnum(-n.v), p: 3 } : { s: this.fnum(n.v), p: (Number.isInteger(n.v) ? 5 : 2) };
case 'c': case 'p': return { s: n.name, p: 5 };
case 'v': return { s: vn, p: 5 };
case 'f': return { s: n.name + '(' + P(n.a).s + ')', p: 5 };
case 'neg': return { s: '-' + wrap(P(n.a), 2), p: 3 };
case '+': return { s: P(n.a).s + ' + ' + wrap(P(n.b), 2), p: 1 };
case '-': return { s: P(n.a).s + ' - ' + wrap(P(n.b), 2), p: 1 };
case '*': {
const l = P(n.a), r = P(n.b);
const implicit = n.a.k === 'n' && n.b.k !== 'n' && r.p >= 2 && !/^[0-9]/.test(r.s);
return { s: wrap(l, 2) + (implicit ? '' : ' * ') + wrap(r, 2), p: 2 };
}
case '/': return { s: wrap(P(n.a), 2) + '/' + wrap(P(n.b), 3), p: 2 };
case '^': {
if (n.a.k === 'f' && n.b.k === 'n' && Number.isInteger(n.b.v) && n.b.v > 0) return { s: n.a.name + '^' + n.b.v + '(' + P(n.a.a).s + ')', p: 5 };
const base = P(n.a), ex = P(n.b);
const simpleEx = (n.b.k === 'n' && Number.isInteger(n.b.v) && n.b.v >= 0) || n.b.k === 'v' || n.b.k === 'c';
return { s: wrap(base, 5) + '^' + (simpleEx ? ex.s : '(' + ex.s + ')'), p: 4 };
}
}
return { s: '?', p: 5 };
}
judge(it, raw) {
let val = String(raw || '');
if (val.indexOf('=') >= 0) val = val.slice(val.lastIndexOf('=') + 1);
if (!val.trim() || /^\s*(idk|i ?d[o']?n'?t ?know|i dont know|dunno|no idea|not sure|stuck|help|\?+)\s*$/i.test(val)) return { st: 'stuck' };
const uv = it.v ? it.v : !!it.u;
const c = this.compile(val, uv);
if (c.err) return { st: 'err', msg: c.err };
const g = this.cf(it.ans, uv);
if (this.same(c.f, g)) return { st: 'ok' };
const W = it.wrongs || [];
for (let k = 0; k < W.length; k++) {
const w = this.cf(W[k][0], uv);
if (w && this.same(c.f, w)) {
const t = W[k][1];
return { st: 'slip', tag: this.D.VOC[t] ? t : null, key: t, msg: this.D.VOC[t] || t };
}
}
if (it.f) { const o = this.cf(it.f, uv); if (o && this.same(c.f, o)) return { st: 'no', msg: 'That is the original function, unchanged. Differentiate it.' }; }
return { st: 'no', msg: '' };
}
/* ---------- typesetting ---------- */
supFlat(g) {
const SUP = { '0': '⁰', '1': '¹', '2': '²', '3': '³', '4': '⁴', '5': '⁵', '6': '⁶', '7': '⁷', '8': '⁸', '9': '⁹', '-': '⁻' };
let s = String(g).replace(/sqrt/g, '√').replace(/pi/g, 'π').replace(/\*/g, '·');
s = s.replace(/\^\(?(-?[0-9]+)\)?/g, (m, d) => d.split('').map((c) => SUP[c] || c).join(''));
s = s.replace(/\s*-\s*/g, (m, off) => (off === 0 ? '−' : ' − ')).replace(/\s*\+\s*/g, ' + ');
return s;
}
supSegs(g, R, out) {
const s = this.supFlat(g);
const parts = s.match(/[a-zA-Z]+|[^a-zA-Z]+/g) || [];
parts.forEach((p) => out.push({ s: p, cls: 'ms' + (/^[a-zA-Z]$/.test(p) ? ' msv' : '') + R, sup: true, nsup: false }));
}
chunks(s) {
const out = [];
let cur = '', role = '', sup = false;
const flush = () => { if (cur) out.push([cur, role, sup]); cur = ''; };
let keep = null;
for (const c of s) {