Sanity checks

How to tell whether an answer is wrong without doing the problem again. This is triage: it is not meant to prove you are right, only to catch the ways you are most likely to be wrong.

nine moves, drawn from the checks scattered through the guide. Every one takes seconds rather than minutes — that is the whole point of them. If a check takes as long as the original problem, it is not a check.

If you only use three

Undo it · Special case · Sign and size. Between them they catch most of what actually goes wrong: an antiderivative that does not differentiate back, a formula misremembered under pressure, and a negative area. The other six are worth knowing, but these three are worth having by reflex.

None of this is invented here. Sanjoy Mahajan's Street-Fighting Mathematics (MIT Press, and free to read) teaches the same skill as six tools — dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Three of them are three of these under different names, and lumping is taken from him directly.

His argument is worth having in mind: conventional teaching is about solving exactly stated problems exactly, and leaves out the separate skill of finding out whether an answer is roughly right. That skill is the one that rescues you in an exam.

The three

Undo it one of the three

Differentiate your antiderivative. Substitute your solution back into the differential equation. Integration and differential equations are the only places in the course where the answer checks itself, and it takes one line.

Used in the guide:

Unit 6 — d/dx[x sin x + cos x] = x cos x.
Unit 7 — y = Ce^(x²) satisfies dy/dx = 2xy.

Special case one of the three

Set a parameter to 0, to 1, or to something symmetric. A correct general formula has to collapse to the specific thing you already know. A misremembered one usually does not.

Used in the guide:

Unit 2 — Power rule at n = 1 gives 1; at n = 0 gives 0.
Unit 3 — sin(A + 0) = sin A.
Unit 5 — On x² − 4x over [0,4], equal endpoints force f′(c) = 0 at c = 2.
Unit 10 — Σ_{n=0..15} 1/n! equals e to within 1e-9.

Sign and size one of the three

Should this be positive? Is it inside a bound you can see? Is it roughly the size you expected? Most lost points are not subtle — they are a negative area or an answer ten times too big.

Used in the guide:

Unit 4 — Tangent-line estimate 2.025 exceeds √4.1 = 2.02485.
Unit 5 — A = x(10 − x) peaks at 25, above both endpoint values of 0.
Unit 6 — Average of x² on [0,3] is 3, inside the range [0,9].
Unit 8 — ∫₀¹(x − x²)dx = 1/6 > 0.
Unit 10 — Σ1/n² to 10,000 terms is ≈1.64483 and stops climbing.

The other six

Lumping

Replace the messy thing with a simple thing of about the same size — a curve with a rectangle, a region with a triangle. You are not trying to get the answer, only to find out whether the answer you have is the right size.

Used in the guide:

Unit 6 — ∫₀²x² dx = 8/3 ≈ 2.67, between the bounding rectangles 0 and 8.
Unit 8 — Widest gap between x and x² is 0.25 at x = 0.5; the area 1/6 is the right size.

Known case

Feed your general method something whose answer you know independently — a circle, a cone, a triangle. A method that cannot reproduce a circle is being applied wrongly.

Used in the guide:

Unit 6 — ½[arcsin x + x√(1−x²)] from −1 to 1 is π/2, the unit semicircle.
Unit 8 — ∫₀ʰπx² dx = πh³/3 equals the cone formula ⅓πr²h with r = h.
Unit 9 — Polar arc length of r = a is 2πa.

Limiting behaviour

Push the variable to its extreme. As x or n goes to infinity, where must this end up? Answers that run past a bound, or settle in the wrong place, are wrong before the arithmetic is checked.

Used in the guide:

Unit 1 — An answer implying a polynomial outran an exponential is wrong.
Unit 7 — P = 500/(1+4e^(−0.3t)) starts at 100 and approaches 500.
Unit 10 — n/(n+1) → 1 ≠ 0, so Σ n/(n+1) diverges.

Rate vs amount

Integrate a rate and you must get an amount; differentiate an amount and you must get a rate. Checking the units takes two seconds and catches differentiating the wrong formula entirely.

Used in the guide:

Unit 4 — dV/dt = 4πr²·dr/dt matches the numerical derivative of (4/3)πr³.

Second route

Get the same answer a different way — simplify first, use a different rule, or put a nearby number into the original expression. Two methods agreeing is much stronger evidence than one method feeling right.

Used in the guide:

Unit 1 — x^(1/x) → 1, reachable from the growth hierarchy without L’Hôpital.
Unit 1 — (3.001² − 9)/0.001 = 6.001, confirming lim = 6.
Unit 2 — Quotient rule on x²/x returns 1, matching d/dx[x].
Unit 9 — For x = t, y = t², (dy/dt)/(dx/dt) = 2t agrees with d/dx[x²] = 2x.

Match the picture

Does the number agree with the graph? A slope you computed should look like the slope you can see. This is the check that connects the algebra back to what it means.

Used in the guide:

Unit 3 — Slope of 2^x at 0 is ln 2 ≈ 0.693; of 3^x, ln 3 ≈ 1.099.