The Surgical Calculus Catch-Up Plan

Polar Coordinates, from zero

The searchlight-beam story: naming points, the two quirks, converting, graphing, pie-slice area — with check-yourself boxes.

Polar Coordinates

The exam-day map: worked examples, classic traps, the curve-tracer animation, prerequisite kit, exercise table.

Polar Problem Clinic

For when the pieces work but full problems don't: the arc-length recipe, the limits routine, and six worked twins of the standard §11.5 tutorial shapes.

Limits & Continuity, from zero

The block with no lecture, taught properly: the 0/0 toolbox, one-sided limits, continuity, the ε–δ game, asymptotes.

Vector Functions & Motion, from zero

Curves as moving points, velocity and speed, arc length, curvature and the normal vector — with a live motion widget. §13.1–13.4.

Limits & Vector Functions

The full module map — including Part B (vector functions, curvature), which builds on the limits block.

Functions of Several Variables, from zero

Domains as regions, level curves as contour maps, open/closed vocabulary, and why "the limit exists" now means every path agrees — with the two-path test.

Several Variables: Limits & Continuity

The compressed map: domain/range/level-curve routine, two-path proofs, polar and squeeze arguments, traps, exercise table.

Partial Derivatives, from zero

Slope depends on direction: freeze one letter, differentiate the other — with a surface-slice widget, Clairaut, differentiability, and the chain rule as a dependency diagram.

Partial Derivatives & Chain Rule

The compressed map: second partials, chain-rule routes, implicit −Fₓ/Fₔ — three audited worked examples, traps, exercise table.

Directional Derivatives, from zero

The slope in any direction: the limit definition first, why u must be a unit vector, the gradient shortcut, steepest and zero-change directions, the gradient ⟂ level curves, tangent planes and normal lines. A widget lets you rotate u at a point and watch |∇f| cos φ. It also covers the classic "every directional derivative exists yet f is discontinuous" example.

Directional Derivatives & Tangent Planes

Rule cards, a steepest/flattest/given-rate table, and three past-paper questions worked (2023 Q2a with unit-vector answers, 2025 Q1 in full, 2023 Q2b), plus the 2024 Q3 parts. Traps, prerequisite kit, practice. In the older papers this block was worth 15–30 of 60 marks.

Maxima, Minima & Saddles, from zero

Critical points, bowls vs saddles, the D test (and what D = 0 means). Then the four-step routine for the absolute max/min on a region, with a boundary-walk widget over the past-paper regions. Then Lagrange multipliers, including the divide-by-zero trap and two constraints.

Extrema & Lagrange Multipliers

The routine card, 2024 Q2(a) and 2023 Q3 quoted and solved, each with a figure and a candidate table, a Lagrange example cross-checked by parametrising, seven classic traps and practice. A whole past-paper question every year before 2025.

Same skeleton, different skin

For "the quiz will be twisted": the 40-minute plan for Tuesday, the nine Quiz 2 routines (Lagrange added for the mid-sem), each shown in three twisted versions, and a name-the-routine drill of questions not on the sheets.

Hottest and coldest points on an ellipse

The slow version of W3's Q7: copying the given partials, where 4 cos 2t comes from, why there are four values of t, cap vs cup, and turning each t into a point.

Five directions, one gradient

The slow version of W4's Q10: the quotient rule colour-coded, the perpendicular trick, and part (d)'s quadratic written out line by line.

Recover the gradient, and when ∇f · u lies

Two measured rates give the gradient; then a piecewise function where the formula fails, with the walk into the origin and the blow-up graph animated.

Tutorial 6 Q11–Q12, from zero

The two hardest directional-derivative questions, assuming nothing: recovering the gradient from two measured rates, and why ∇f · u gives the wrong answer at a piecewise point (with a turn-the-direction widget).

Finding multivariable limits

Approaching a point from every direction, then the toolkit: substitute and use a known limit, polar coordinates ("r times something bounded"), squeeze, and ε–δ as a challenge and response.

When limits fail, and continuity

Continuity as "no jump", then the two-path test. The twist in Q4: every straight line gives 0, yet the parabola y = x² gives 1, so the limit doesn't exist.

Partial derivatives & implicit differentiation

From a hill and a slice to dy/dx = −F_x/F_y, then both implicit questions using the opening routine.

The chain rule as a tree

"Rates multiply", then the dependency tree with each path lighting up in turn. Covers ∂w/∂r = 12, the box rates (the diagonals are shrinking), and the polar chain rule. The walkthrough labels the bottom edges without deriving them (x_r = 1, y_r = −sin(r+s), z_r = cos(r+s)); that working is in the tutorial companion, T6 Q3 step 2.

Rates along a curve

Walk a path through a temperature map: df/dt = ∇f · r′(t). The helix candidates, then the ellipse max/min by dT/dt and d²T/dt².

Gradient & directional derivatives

Uphill arrow, D_uf = |∇f| cos φ with a rotating u, and "u must be a unit vector". Then the steepest direction, the zero directions and a given rate on the unit circle, and why the formula fails at a bad point.

Tangent planes, normal lines & intersections

∇F as the surface normal, then the plane and the line. Two surfaces meeting: the tangent direction is ∇F × ∇G, with the determinant laid out.

Critical points & the second-derivative test

Bowl, cap and saddle as 3D surfaces and contour maps, then the D test. Q5's four critical points classified live; Q6 has a saddle and no extremum.

Absolute max & min on a triangle

The candidate routine again, on a triangle. Here the critical points sit on an edge. Max 18, min −32.

Mid-sem past papers, decoded

Three papers quoted as set, 2025 (open book, the closest match) first. Every question has its opening moves, a full worked solution, a figure and traps. A marks table by topic. The two off-syllabus parts (torsion, series) are marked skip, and the handwritten key's unit-vector slip is flagged.

How to start any mid-sem question

A five-move opening routine and eleven question families (polar area, polar tangents, curvature, osculating circle, limits at a special point, gradient, tangent planes, extrema, Lagrange multipliers…). Each has trigger words, first moves, the wrong first move, and the real past-paper openings. Plus which pages to tab for an open-book exam.

Area between polar curves, from the first move

A short narrated lesson on 2023 Q4. The curves draw themselves, a ray sweeps out the region, the integral splits where the inner curve changes. It pauses three times for you to commit to the next move.

Absolute max & min, the candidate routine

2024 Q2(a) on the diamond |x| + |y| ≤ 1: interior point, four edge walks with f plotted along each, corners, then the comparison table. It pauses three times for you to commit to the next move.

Start drill

Fresh question stems in twelve families, including "off syllabus → skip". Name the family, write the first moves, then compare with the model opening. No full solving. 1,200 generated items verified.

Quiz 1, question by question, from zero

The real paper (four variants, key filed): the line-cuts-a-limaçon limits with the tangency root explained and drawn, T and N of ⟨t, e^t⟩, domain and partials of a log of a ratio. Every variant tabulated, a twin per question, and the 40-minute plan for an answer-only paper. Read before the 5 Oct mid-sem.

Doubts, worked from zero

One card per problem you send in, filed on a topic page that matches the notes, with a filterable index. Each card: the question as set, where it lives in the notes, the explanation from first principles with the figure drawn, the trap, and a twin to try. Live: Tutorial 4 Q3 (osculating circle of y = x²) and Q4 (where an ellipse bends most).

Tutorial Companion

Every question from Tutorial Sheets 2–7, each with a hint that includes the opening move and a collapsed, verified full solution. Sheets 6–7 add figures and links to the walkthroughs. Attempt first, then open.

Tutorial Practice

Fresh problems in the same pattern families as the sheets, with answers and "see the method" solutions.

Unit-Circle Flashcards

sin & cos at every standard angle, in radians — tap to reveal, missed cards recycle. The Week 1 trig warm-up.

Daily Drill Sheet

Ten randomized differentiation & integration problems with a timer and answers — the 20-min/day fluency engine.

Formula Flashcards

Every derivative, integral, identity and polar formula this course leans on, as recall cards — filter by category before a study session.

Polar Pipeline Drill

Eight randomized §11.5-style reps per sheet — collapsing arc lengths, standard areas, and the loop-limit / intersection / csc-line set-up moves.