∞ Grade 12
For curious 17 and 18 year olds in their last school year. Calculus, vectors and probability derived rather than memorised, the ideas waiting at the start of university, and where all of it shows up in the world. Three tracks: Sharpen (the course, with reasons), Around the corner (new mathematics), Connections (where it shows up). One sheet a night, ten to fifteen minutes, and a starred problem if you want more.
Sharpen the final-year course, derived not memorised
Limits, the derivative rules with pictures, optimisation, curve sketching, integration by substitution and parts, volumes, differential equations, Taylor series, lines and planes, complex polynomials, the binomial and normal distributions, p-values, implicit and parametric curves.
What a limit is
sin x / x → 1, by squeezing it between two areas. Limits are about what happens nearby, never at the point, and that is why 0/0 is not an answer.
The product rule is a picture
A rectangle grows by du on one side and dv on the other. The corner piece vanishes, and (uv)′ = u′v + uv′ falls out.
The chain rule: gears
If y turns three times for each turn of u, and u twice for each turn of x, y turns six times for x. That is the chain rule, and why it multiplies.
The derivatives of sin, eˣ and ln x
Why sin′ = cos from the unit circle, why eˣ is its own slope, and why ln x has slope 1/x because it is the inverse.
The cheapest can
Fix the volume, minimise the metal. Set the derivative to zero and find that the best can is as tall as it is wide. Then check it is a minimum.
Reading a curve from its derivatives
Where f′ is zero, where f″ changes sign, and what happens far away. Sketch x³ − 3x and x e⁻ˣ without plotting a single point.
Related rates
A ladder slides, a ripple spreads, a balloon inflates. Differentiate the relationship, not the numbers, and the rates connect.
Substitution: the chain rule backwards
Spot the inner function and its derivative, change variable, and ∫ 2x cos(x²) dx becomes ∫ cos u du.
Integration by parts
The product rule read backwards. ∫ x eˣ, ∫ ln x (the trick is to let dv = dx), and the integral that comes back to itself.
Volumes by slicing
Stack thin discs and integrate. The sphere's 4/3 πr³ from circles, the cone from similar triangles, and a vase of your own design.
The equation that says "rate ∝ amount"
dy/dt = ky solved by separating variables. Radioactive decay, a cooling coffee, and why half-lives never reach zero.
Where the power series come from
Match derivatives at zero, one at a time. eˣ = 1 + x + x²/2 + …, sin x = x − x³/6 + …, and why your calculator can compute them.
Lines and planes in space
A line is a point plus a direction; a plane is a point plus a normal. Distances fall out of the dot product.
Conjugate roots and x⁴ + 4
Why non-real roots of real polynomials come in pairs, how to factor x⁴ + 4 over the reals, and the fundamental theorem of algebra stated.
The binomial distribution
n trials, probability p. Mean np, variance np(1 − p), and why variances add but standard deviations do not.
Why averages are normal
Add enough independent anythings and the sum is bell-shaped. The central limit theorem by picture, and the standard error σ/√n.
What a p-value is, and is not
The chance of data this extreme if nothing is going on. Compute one for a coin, see why 0.05 is a convention, and what a test cannot tell you.
Implicit differentiation
Differentiate both sides of x² + y² = 25 and the slope of a circle appears. Then the derivative of xˣ by taking logs first.
Partial fractions
Split 1/(x² − 1) into two simple pieces, integrate each, and get a logarithm. The same trick sums telescoping series.
Curves by parameter
x and y each as functions of t. The cycloid traced by a wheel, its slope dy/dx = (dy/dt)/(dx/dt), and the area under one arch.
Around the corner new ideas
ε–δ, e is irrational, counting primes, the Chinese remainder theorem, sums of two squares, generating functions, eigenvectors, least squares, gradient descent, a neuron, Laplace's rule, the coupon collector, Poisson, optimal stopping, Kelly, Fourier coefficients, winding numbers, the catenary, oscillators, predator and prey, fixed points, the halting problem, Hall's theorem, derangements, −1/12, inversion, knots, Lagrange multipliers.
What "approaches" really means
For every ε there is a δ. The definition that made calculus rigorous, used once to prove that 2x → 6, and once to show why 1/x has no limit at 0.
e is irrational
Suppose e = p/q, multiply the series by q!, and an integer lands strictly between 0 and 1. Fourier's one-paragraph proof.
How many primes are there below x?
About x / ln x. Gauss guessed it as a teenager, it took a century to prove, and you can check it against a table tonight.
The Chinese remainder theorem
A number that leaves remainder 2 mod 3, 3 mod 5 and 2 mod 7. Why there is exactly one answer mod 105, and how to find it.
Which primes are sums of two squares
5, 13, 17, 29 are; 3, 7, 11, 19 are not. The rule is the remainder mod 4, and the reason runs through Gaussian integers.
Counting with polynomials
Multiply (1 + x + x²)(1 + x³) and the coefficients count ways to make change. Generating functions turn counting into algebra.
The directions a matrix leaves alone
Most vectors get turned; eigenvectors only get stretched. Find them for a 2×2 matrix, and use them to write Fibonacci in closed form.
Least squares is a projection
The best-fit line is the shadow of the data on a plane. Derive the normal equations from "the error is perpendicular to the fit".
Gradient descent
Follow the slope downhill in small steps. How a machine finds the bottom of a bowl it cannot see, and when it overshoots.
A neuron is a dot product and a threshold
Weights, inputs, a sum and a switch. Train one by hand to recognise a pattern, and see what a single neuron cannot learn.
Will the sun rise tomorrow?
Laplace's rule: after n successes and no failures, bet (n + 1)/(n + 2). Where it comes from, by integrating over every possible probability.
The coupon collector
How many packets to complete a set of 50 stickers? About 50 × (1 + 1/2 + … + 1/50) ≈ 225. The harmonic numbers, with a reason.
The Poisson distribution
Rare events, many chances: the binomial collapses to one formula with one parameter. Horse kicks, typos, and why the mean equals the variance.
When to stop looking
Interview candidates one by one, no going back. Reject the first 37%, then take the next best. Why 1/e, and what it assumes.
How much to bet
A favourable coin, repeated. Bet too much and you go broke; bet the Kelly fraction and your wealth grows fastest. The logarithm decides.
Fourier coefficients by integration
Multiply a square wave by sin kx and integrate: every harmonic except the odd ones vanishes. The formula, the computation, and the overshoot that never dies.
Why every polynomial has a root
Walk round a huge circle and p(z) winds round the origin n times; shrink the circle to a point and it winds zero times. Somewhere in between, it crossed zero.
The hanging chain
A chain hangs in a cosh, not a parabola. Hyperbolic functions, why cosh² − sinh² = 1, and the equation that forces the shape.
y″ = −y
The equation of a spring, a pendulum and a tuning fork. Why sin and cos are its solutions, what the frequency is, and what resonance does.
Foxes and rabbits
Two coupled equations, no closed form, and a loop in the phase plane. Lotka–Volterra, read qualitatively.
Stir your coffee: a point stays still
The intermediate value theorem proves a fixed point in one dimension in two lines. Brouwer's theorem says the same in two, and why that is harder.
The program that cannot exist
Suppose a program could decide whether any program halts. Feed it itself, twisted, and it contradicts itself. Turing's diagonal.
Hall's marriage theorem
When can every job be filled by a qualified person? Exactly when every k jobs have at least k qualified people between them.
Derangements and 1/e
n people, n hats, nobody gets their own. The count is n!/e to the nearest whole number, and the probability settles at 0.3679.
1 + 2 + 3 + … = −1/12?
Not as a sum. What the famous "equation" actually says, what the zeta function is, and why the viral video's trick is not a proof.
Inversion in a circle
Turn the plane inside out: lines become circles through the centre, and tangent circles stay tangent, and a hard problem becomes easy.
Proving a knot is knotted
The trefoil cannot be undone. Colour its arcs with three colours and watch the colouring survive every move. An invariant in one evening.
The best point on a curve
Maximise f along a constraint: at the optimum the gradients are parallel. Lagrange multipliers with a picture, then the largest box in a sphere.
A derivative for whole numbers
Make up a rule: D(prime) = 1 and D(ab) = a D(b) + b D(a), the product rule. It works on whole numbers, it has a power rule, and a few numbers turn out to be their own derivative.
Spot the error, then cancel it
Estimate an area with 4 rectangles, then 8. The error roughly halves. If you know how the error behaves, you can cancel most of it with one line of arithmetic.
How far apart are two functions?
How far is y = x from y = x squared? You can measure the biggest gap, or the area between them. Different measures can disagree about which functions are close.
What is 1 − 1 + 1 − 1 + …?
The partial sums of 1 − 1 + 1 − 1 + … go 1, 0, 1, 0 for ever, so the series has no sum. Yet two famous methods, averaging and Abel's trick, both assign it the value 1/2.
Why the simulated planet drifts away
Simulate a spring or an orbit with the simplest step-by-step method and it slowly spirals outwards, gaining energy from nowhere. Update the velocity first, then move using the new velocity, and the drift disappears.
Does the order of heads matter?
Two friends flip the same coin 10 times. One writes down every flip; the other only counts the heads. For learning how biased the coin is, the count is all you need, as long as the flips are independent.
Connections space · finance · medicine · control · genetics · images · archaeology · queues · climate
The same mathematics, found in the wild.
Escape velocity
Integrate the inverse-square law from the surface to infinity and the energy needed is finite. 11.2 km/s for Earth, and why a black hole follows.
The rocket equation
Burn fuel, lose mass, gain speed. A differential equation whose answer is a logarithm, and why getting to orbit is so hard.
What an option is worth
One step up or down, no free lunch. Price a stock option by building a portfolio that copies it, and see why the real probability does not appear.
How many patients does a trial need?
Randomisation, the size of the effect, the noise, and the power. The sample-size formula from the standard error, with real numbers.
Cruise control
Measure the error, push back in proportion, and the car settles. Feedback as a differential equation, and why too much gain makes it wobble.
Why recessive genes do not die out
Allele frequencies p and q give genotypes p², 2pq, q². One generation of random mating and the mix is stable. Then a dose of selection.
How a photo gets ten times smaller
Blocks of pixels, the cosine transform, and throwing away the high frequencies nobody sees. Fourier in your phone.
Radiocarbon dating
Carbon-14 halves every 5,730 years. From the ratio left in a sample to its age, with the decay equation, and what makes the answer uncertain.
Why the queue is longer than you think
Arrivals at random, one server. Utilisation ρ, average queue ρ/(1 − ρ), and why a server at 90% has a queue nine times longer than at 50%.
Why the Earth is not frozen
Sunlight in, heat radiated out as T⁴. Balance them and the planet should be −18 °C. The greenhouse effect is the 33 degrees of difference.
Where should the lifeguard enter the water?
A lifeguard runs fast on sand and swims slowly. The quickest route bends where sand meets water, by exactly the rule light follows when it enters water.
Guessing a number from noisy readings
A sensor gives 52, 47, 50, 55, 46 for the same hidden number. Keep a running best guess, correcting it a little with each reading. That loop is the seed of the filters that help navigate spacecraft.
How long do light bulbs last when some never fail?
You test 8 bulbs for 6 months. Three are still glowing at the end, and one was removed early. How do you estimate lifetimes without throwing that information away?
Demos play with it
Twelve new interactive demos go with this grade, from Taylor polynomials to a binomial option tree, alongside the other grades' demos. All of them live in the demo gallery.