Notes from the team building Locus: how the grader works, what we are shipping, and what we are learning about teaching math.
Bernoulli's equation is the work-energy theorem per unit volume, and it holds under four hypotheses that get quietly dropped. Torricelli derived, a Venturi worked, and an honest look at airplane lift.
Computer algebra system grading asks whether a typed answer is equivalent to the key, not whether it matches. Here is what the Locus grader accepts, what it refuses, and the cost of one guard.
An object circling at constant speed still accelerates. The centripetal acceleration is v squared over r, aimed at the centre: the derivation, a banked curve worked, and the force that supplies it.
The chain rule says the derivative of a composition is the outside derivative read at the inside, times the inside derivative. Here is the proof, a worked product, and the factor people drop.
The Doppler effect for sound gives a different frequency when the source moves than when the listener moves at the same speed. Here is the sign rule, both derivations, and the numbers.
Eigenvalues and eigenvectors are the scalars and nonzero directions with Ax = lambda x. A worked two by two, the trace and determinant check, repeated roots, and rotations.
The equivalent resistance of resistors in series and parallel follows from charge and energy conservation. Here are both derivations, a ladder worked from the inside out, and two checks.
The half-life formula comes in two shapes, N = N0 exp(-lambda t) and N = N0 times one half to the power t/T, and they are one statement. Here is the link between them, activity in becquerels, and carbon-14 dating.
The ideal gas law is pV = nRT with R = 8.314 J per mole per kelvin, and it fails loudly if you feed it Celsius. Worked examples, the combined form, molar volume derived rather than quoted.
Implicit differentiation finds a slope on a curve you never solved for. Differentiate both sides, attach dy/dx to every y-term, then solve. Two worked curves and the term people drop.
Integration by parts trades one integral for another, uv minus the integral of v du. Where the formula comes from, how LIATE guesses u, and the case where the integral comes back.
The four kinematic equations each leave out one variable, and that is the selection rule. Derivations from constant acceleration, a two-stage worked example, and the sign trap on a reversing object.
L'Hopital's rule applies to 0/0 and infinity over infinity, and to nothing else. Here are the hypotheses, the other indeterminate forms rewritten as quotients, and two ways the rule goes wrong.
The logarithm rules for products, quotients and powers are the exponent laws written backwards. Here each one is derived, plus change of base, and the two mistakes that cost solutions.
The photoelectric effect broke wave theory on four counts at once: instant emission, a threshold frequency, intensity that changes current but not energy, and a stopping potential linear in frequency.
The quadratic formula gives x = (-b +/- the square root of b squared - 4ac) over 2a. Here is the full derivation by completing the square, the discriminant, a worked example, and the three sign traps.
Separable differential equations are the ones whose right side factors as f(x)g(y). The method, why multiplying by dx is legal, and the constant solutions it quietly destroys.
Snell's law says n1 sin(theta1) = n2 sin(theta2), with both angles measured from the normal. Here is where it comes from, a worked refraction, and the critical angle it predicts.
Standard deviation is the square root of the mean squared deviation from the mean. Here is the calculation done both ways on one data set, and what Bessel's n-1 correction is really fixing.
A Taylor series is the one power series whose every derivative matches the function at the centre. The coefficient formula, the standard series, and a worked error bound.
Time dilation says a moving clock runs slow by the factor gamma, one over the square root of 1 - v squared over c squared. The light-clock derivation takes six lines, and it tells you which clock reads the proper time.
U substitution is the chain rule run backwards: name the inside u, replace dx with du, integrate. Here is how to choose u, both ways to finish a definite integral, and the factor-of-twelve error.
On the unit circle the point at angle t has coordinates (cos t, sin t). That single fact generates the exact values, the quadrant signs and reference angles, with no table to memorise.
The acceleration of a block on an incline is a = g(sin theta - mu cos theta). Here is the free-body diagram in words, a frictionless example, a rough example, and the sign mistake that eats most of the marks.
A charged particle crossing a magnetic field moves in a circle of radius r = mv/(qB). Here is the formula, a calculator, and worked numbers for an electron and a proton.
The Rydberg formula puts Lyman alpha at 121.5 nm and Balmer alpha at 656.1 nm. The same energy levels, scaled by Z squared, put the He+ ground state at -54.4 eV.
On level ground the range of a projectile is R = v0 squared times sin(2 theta) over g, which peaks at 45 degrees. Here is the derivation, a worked example, and what changes when the launch point is above the landing point.
Locus has no multiple-choice math. Every numeric or symbolic answer is typed, and a computer algebra system decides whether it is right.
A missed problem should come back later, with the same skill and different numbers. What to bring back, when to bring it back, and why the correction is where the learning happens.
Locus gives every student a rating and every problem a rating, on the same scale. It is the Elo system from chess, pointed at a different kind of match.