How every line is checked

Most answer keys ask you to trust them. This one shows its work about its work.

Each line carries a label

✓ ProvedThe computer algebra system SymPy showed this line equals the one before it: it reduced the difference to zero.

✓ Checked independentlyA separate program proved it another way, usually one written for that line and saved so anyone can rerun it.

≈ Checked numericallyThe two sides agree at every point sampled. That is strong evidence, but it is not a proof, and it is never called one.

ReviewedA sentence (setting up a model, choosing a method) that no computer can check. A reviewer read it and found it sound.

Not checkedNothing has looked at it yet.

The answer is checked a second way

Separately from the solution, every answer is confirmed by a route the solution doesn't take: a derivative is recomputed directly, an antiderivative is differentiated back, an area is found by numerical integration, a differential equation's solution is substituted back into the equation, a maximum is found by sampling thousands of points.

The explanation is judged separately

A solution can be correct line by line and still explain a step badly. A panel of AI models reads every solution and objects to anything unclear or misleading. When they disagree, a stronger model reads it, rules on each objection, and writes a small program to settle any line the algebra system couldn't. Every verdict stays on the problem's receipt, dated, including the ones that were overruled.

A second solution

Many derivative, limit and integral problems also have a second solution, written by a different model that never sees the first. It is checked line by line the same way. When two separate solutions reach the same answer and every line of both is proved, the answer is about as settled as an answer key gets. Where a topic page shows one, it sits under “A second solution, written separately.”

Who does the checking

SymPy, an open-source computer algebra system, checks every equation on every line. Its labels are the ones marked Proved or Checked numerically.
A panel of AI models running on my own computers reads every explanation and objects to anything unclear, misleading or wrong. They are free to run and they disagree often, which is the point.
Claude, Anthropic’s model, is the senior reviewer. It reads what the panel could not settle, rules on each objection, and writes a small SymPy program for any line the main check could not prove. Those programs are kept, and any of them can be rerun.
Me. When I check a problem myself, its receipt says so, by name.

Every receipt names the checker behind each label, with the date.

Badges

Nihil obstat (“nothing stands in the way”): every line is machine-checked (or, for a sentence, read and found sound), the answer is confirmed a second way, and no objection to the explanation stands.

Beautiful: a messy function whose derivative collapses to almost nothing. A program awards it by measuring the collapse, and every Beautiful problem goes to the senior reviewer to read.

Mental math: it can be done in your head.

Interactives and their assumptions

An interactive that treats a river as a vector field, or a hillside as a smooth surface, takes a step no computer can prove. So it does not claim the river or the hill behaves that way. It lists what it assumes, numbered, at the top of the page, and each step that leans on an assumption is tagged with its number. The math that follows is checked the way a problem is, and the receipt at the end shows which assumptions each result rests on. The assumptions themselves are granted, not proved: you decide whether you accept them.

Assumptions stated: every statement the interactive makes about the physical world rests on a numbered assumption or definition listed on the page.

What this doesn’t promise

A proof by SymPy shows that each line follows from the one before. It doesn’t show that a sentence explains the step well, and a numerical check is evidence, not proof. Each label says which of these a line got. If you find a mistake, I want to know; my contact details are on my site.

Where the problems come from

The problems are original. A program builds each one so that its answer is known before any solution is written, and they are organized the way a three-semester calculus sequence is taught. Each topic page works its strongest problems in full. More about the site and who runs it.