I Used a College Theorem on My Exam. The Grader Drew a Line Through It.

The scout and the bricklayer

The first time I used a theorem I was not supposed to use, I lost four points.

The answer was right. The method was not. I had written the name of a university calculus rule, and the grader drew a line under it and wrote: No basis. Four points gone. Not because I was wrong. Because I had walked into the exam with a foreign passport.

Mr. Okafor told me afterward: a theorem you learn outside the syllabus is a scout. It goes ahead and tells you where the enemy is. But the scout does not build the bridge. You build the bridge with the tools in your own bag.

Three rules

Use anything in multiple choice. Be careful in written answers.

In a multiple-choice question, nobody sees your road. Use a helicopter. Use a theorem from another country. The only thing that matters is the letter you shade.

In a written answer, the grader looks at the road. If the road is not on the syllabus map, the grader has two choices: give you credit anyway, or mark the step as unsupported. Different provinces, different years, different moods. You cannot bet your future on a mood.

Use the outside theorem to see the answer. Walk the syllabus road to write it.

If you can prove it in five minutes, learn it. If you cannot, use it only to check.

A theorem is worth learning if you can rebuild it with the tools you already have. Prove it in five minutes using classroom methods, and you can also write it in an exam without writing its name. Build it on the spot, like an extra line in a geometry diagram.

If you cannot prove it, if its foundation is a concept you have not learned, then it is not a tool. It is a rumor. Use it to check your answer, not to argue your case.

The value is the angle, not the conclusion.

Students who collect off-syllabus theorems can recite the conclusions better than their teachers. Then a new problem appears, and they freeze. The conclusion is a photograph. The angle is a way of seeing. A photograph can be memorized. A way of seeing cannot. The exam changes the photograph every year.

Calculus

Calculus is where the scouts gather. It is the part of high school that stands closest to the university.

Taylor, also known as the tangent line.

In university, Taylor expansion says a smooth curve can be approximated by a polynomial. In high school, the same idea appears in a smaller dress: the exponential curve stays above its tangent line at zero.

The scout tells you: when an exponential function fights a polynomial, think of the tangent line. The exponential is above it.

You do not write "Taylor." You build a helper function. Show that at the starting point the helper is zero. Take its derivative. Show the derivative is positive. The helper grows. The inequality holds.

The scout gave you the direction. The classroom gave you the pen.

Lagrange.

The mean value theorem says that between two points on a smooth curve, there is a place where the slope equals the average slope between the points.

This turns a difference between two values into a derivative at one point. If you want to show that the difference between two sine values is never larger than the difference between their angles, Lagrange whispers the answer in one breath.

Do not write it. It stands on Rolle's theorem, which is not in your syllabus. Use it to see. Build the classroom bridge.

The log-mean inequality.

It appears in problems about shifted extrema. The problem gives two points almost symmetric around a peak and asks about their sum. The log-mean inequality lets you squeeze the expression and see the answer fall out.

Think with it. Prove without it.

L'Hôpital.

Everyone knows this one. It handles limits that look like zero over zero or infinity over infinity. Fast. Elegant. The most dangerous scout in the forest.

It depends on the concept of a limit, and limits are university country. Write "by L'Hôpital's rule" in a written answer, and the grader does not see a clever student. The grader sees a student who did not take the classroom road.

Use it in multiple choice. Use it to check your parameter range. Put it away.

Inequalities

Cauchy.

The two-dimensional version says the product of two sums of squares is at least the square of the sum of products. Ugly sentence. Simple idea: it finds the maximum of a combination of sine and cosine.

The auxiliary angle formula also does it. Cauchy does it faster. In some provinces, Cauchy belongs to an optional topic. In others, it is off-syllabus. Ask your teacher which side of the line you are standing on.

Weighted sum.

Cauchy wearing a different coat. Useful for fractions. If positive numbers add to one and you want the minimum of the sum of their reciprocals, this scout gives the answer in a blink. It is a special case of Cauchy, so prove it on the spot with Cauchy, and it is no longer off-syllabus.

Jensen.

For a convex curve, the value at the average is not greater than the average of the values. For a concave curve, the opposite.

Take the logarithm. Jensen gives you the mean inequality in one line. The mean inequality is not a separate fact. It is Jensen in disguise.

The value of Jensen is not the formula. The value is the word "convexity." Once that word lives in your head, many multi-variable maximum and minimum problems show their shape. You ask: is this thing convex? The answer tells you where to look.

Think with Jensen. Do not write it.

Bernoulli.

The discrete cousin of the tangent line. For a number close to one, raising it to a power gives at least a certain lower bound. Useful for estimation. Easy to prove by induction. Low risk.

Geometry

Heron.

Heron's formula gives the area of a triangle from its three sides. The classroom road is longer: cosine rule to find an angle, then the area formula with sine. Heron skips the angle.

Low risk. A pure algebraic formula. Three sides and you need the area? Use it.

Thirteen, fourteen, fifteen. Half the perimeter is twenty-one. The area is eighty-four. Done.

Ptolemy.

For cyclic quadrilaterals: the product of the diagonals equals the sum of the products of the opposite sides. It does not appear often. When it does, it is smooth.

In a written geometry answer, fall back on similar triangles and congruent triangles. The scout can point. The classroom must build.

Analytic geometry

Determinant area.

A way to compute the area of a triangle from the coordinates of its three vertices. Feed in the coordinates. Get the area.

In conic problems, this is gold. It can find a chord, an area, or a collinearity in one line. Write it as coordinate arithmetic. Low risk.

The "hard-solve theorem."

A famous set of formulas circulating online. The name sounds impressive. It is not a theorem. It is a lookup table. It takes the mechanical process of combining a line and an ellipse and packages the result into a fixed form.

Every step inside that package is classroom arithmetic and Vieta's theorem. So it is not off-syllabus. The danger is different: if you only write the conclusion and skip the combining, the grader sees a jump.

Keep the table in your head. Use it to check your answer. On paper, write "combining the equations…" and show the work.

Symmetric Vieta versus asymmetric Vieta.

This is the real divide in analytic geometry. Not an off-syllabus theorem. A high-level use of a classroom theorem.

When a line meets a curve at two points, Vieta gives two numbers: the sum of the roots and the product of the roots.

If the expression you want is symmetric, if swapping the two points does not change it, substitute the sum and the product and walk to the answer. Regular road.

If the expression is asymmetric, Vieta is not enough. The sum and the product do not tell you which root is larger. They leave a sign undecided. The problem is missing half a piece of information: the order of the two roots.

The exam gives you that half in two ways.

First, the problem itself gives it. A condition like "the two lines are perpendicular" or "one root is twice the other" or "the line passes through a fixed point" can be written as an equation. Combine it with Vieta. The sign is fixed.

Second, the pairing method. Take the asymmetric expression and its mirror image, where the two roots swap. Add them and subtract them. The sum is symmetric, so Vieta handles it. The difference contains the missing sign, and you still have to choose.

That little choice is the price of asymmetry.

Symmetric Vieta is substitution. Asymmetric Vieta is asking for one more equation. When you are stuck, ask: is my expression symmetric? If not, where is the condition I forgot?

Pole, polar, Monge, Apollonius.

The "secret manual" names of analytic geometry. They reveal fixed points and fixed values in one glance.

Do not memorize them as formulas. Treat them as search terms. If you know a problem is about pole and polar, you can predict the shape of the answer, maybe the fixed point lies on the horizontal axis, and then use the regular method to compute it.

Prediction is the safest use of an off-syllabus conclusion.

Sequences

Characteristic roots.

A recurrence like "next term equals two times the current term plus three" can be turned into a geometric sequence. Find a fixed point. Subtract it from both sides. The recurrence becomes simple.

The method is useful. The name is not. Do not write "characteristic root method." Write the construction: subtract, factor, show it is geometric.

Fixed points.

A fixed point is a number that does not move when you apply the function. For a recurrence, the fixed point is the value that stays the same forever.

For a fractional recurrence, the fixed point method is powerful. Solve for the fixed points. If there are two, a certain ratio becomes geometric. If there is one repeated fixed point, a reciprocal becomes arithmetic.

A student sees a recurrence with a fraction. He tries to list terms. Ugly fractions. Then he remembers the fixed-point scout. He solves for the fixed points: zero and one. He subtracts one, takes the reciprocal, and suddenly the sequence is geometric. The answer appears.

On the exam, he does not write "fixed-point method." He writes: "Subtract one from both sides, then take the reciprocal. This gives a geometric sequence." That is the language the grader accepts.

The fixed-point method and the characteristic-root method are the same idea: find the point that does not move, and lift the whole sequence from there.

Complex numbers, probability, counting

De Moivre.

Connects complex numbers and trigonometry. Makes high powers of complex numbers almost instant. Off-syllabus, but complex numbers are growing in importance. Keep it for multiple choice.

Linearity of expectation.

The most underrated scout in probability. The expected value of a sum is the sum of the expected values, even if the variables are not independent.

That last phrase is the whole treasure. The textbook often requires independence. This does not.

Classic problem: n people sit randomly in n chairs. What is the expected number of people who sit in their correct chair? Computing the distribution is a mess. Use indicator variables. Each person has probability one over n of sitting correctly. Add them up. The expected number is one. It does not depend on n.

Beautiful. Use it in multiple choice, or prove it yourself in a written answer.

Counting knives.

Three quick ones.

Stars and bars counts the number of ways to write a number as a sum of positive integers.

Catalan numbers count valid pairings of parentheses, or stack orders. Three pairs of parentheses have five valid arrangements.

Derangements count the number of ways to put everything in the wrong place. Four letters into four envelopes, all wrong, has nine ways.

Perfect for multiple choice. Know them, and you answer in seconds. Do not know them, and you count by hand.

The ledger

The tangent-line scout: low risk, as long as you do not write the name. Build the helper function.

The Lagrange scout: high risk. Use only to think.

The log-mean scout: high risk. Use only to think, and prove it if needed.

The L'Hôpital scout: highest risk. Only for multiple choice and self-check.

Cauchy: depends on your province. Write the two-dimensional expansion.

Weighted sum: medium risk. Prove it with Cauchy on the spot.

Jensen: high risk. Think, do not write.

Bernoulli: low risk. Prove by induction.

Heron: low risk. Write it directly.

Ptolemy: medium-high risk. Think, do not write.

Determinant area: low risk. Write it as coordinate arithmetic.

The hard-solve table: low risk, but you must write "combining the equations."

Symmetric and asymmetric Vieta: no risk, because it is classroom material. The key is to find the missing equation.

Pole and polar: highest risk. Use as a search term for prediction.

Characteristic roots: medium-high risk. Write the construction, not the name.

Fixed points: high risk. Think, then downgrade to "subtract and take reciprocal."

De Moivre: medium-high risk. Use in multiple choice.

Linearity of expectation: medium risk. Use in multiple choice, or prove it.

Counting knives: low risk. Use in multiple choice.

Last page

The night before the exam, I closed the book. I did not review the names of the scouts. I reviewed the roads.

The next morning, I saw a calculus problem. The exponential curve was fighting a polynomial. The Taylor scout whispered, "Tangent line." I did not write Taylor. I built the helper function. I took the derivative. I showed it was positive. The answer came.

I saw a conic problem. The expression was asymmetric. The Vieta scout whispered, "You are missing a condition." I looked back. There it was: a perpendicular line I had forgotten. I wrote it as an equation. The sign was fixed. The answer came.

I saw a probability problem about people and chairs. The linearity scout whispered, "One." I checked it with a small case. It was one. I wrote the proof in classroom language. The answer came.

The scouts stayed in my head. The bridge stayed on the page.

Are there useful theorems that high school does not teach?

Yes. Many.

Put them in your head. Do not put them on your exam paper.