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IB Math AI HL formulas not in the formula booklet

The exam gives you a formula booklet, but it leaves these 13 out. Learn them by heart. Each has a short worked example: tap “Worked example” to open it.

The IB gives this booklet to schools rather than posting it, so we checked the list against our own copy, not the IB's file. Your teacher has the official booklet. Official information: IB DP Mathematics page.

Test yourself with flashcards One-page formula sheet

Number, algebra and functions

Upper and lower bounds

\(\displaystyle x\text{ to the nearest }u:\) \(\displaystyle x-\tfrac u2\le\text{value}

Use it for: Rounding and accuracy questions.

Worked example

A square has side 7.4 cm to 1 decimal place. Find the bounds of its perimeter.

  1. Side: \(7.35\le s<7.45\)
  2. Perimeter \(=4s\): \(4\times7.35=29.4\), \(4\times7.45=29.8\)

Answer: \(29.4\le P<29.8\) cm

Perpendicular gradients

\(\displaystyle m_1m_2=-1\) \(\displaystyle \Big(m_2\)​\(\displaystyle {}=-\frac1{m_1}\Big)\)

Use it for: Finding a perpendicular line, a normal, or a perpendicular bisector.

Worked example

Line \(L\) passes through \((2,1)\) and meets \(y=3x-4\) at right angles. Give the equation of \(L\).

  1. The given gradient is \(3\), so the perpendicular gradient is \(-\frac13\).
  2. \(y-1=-\frac13(x-2)\)
  3. \(y=-\frac13x+\frac53\)

Answer: \(y=-\frac13x+\frac53\) (or \(x+3y-5=0\))

Linearising a power model

\(\displaystyle y=ax^b\ \)​\(\displaystyle {}\Rightarrow\ \ln y\)​\(\displaystyle {}=\ln a+b\ln x\)

Use it for: Turning a curve into a straight line to find \(a\) and \(b\).

Worked example

\(y=ax^b\) passes through \((1,3)\) and \((4,24)\). Find \(a\) and \(b\).

  1. At \(x=1\): \(y=a=3\)
  2. \(24\)​\({}=3\times4^b\)​\({}\Rightarrow4^b\)​\({}=8\)​\({}\Rightarrow2^{2b}\)​\({}=2^3\)
  3. \(b=\frac32\)

Answer: \(a=3\), \(b=\frac32\)

Graph theory

Handshaking lemma and trees

\(\displaystyle \sum\deg(v)=2e;\) \(\displaystyle \text{a tree with }n\text{ vertices has }n-1\text{ edges}\)

Use it for: Graph theory: counting edges, and checking whether a graph can exist.

Worked example

A graph has vertex degrees 3, 3, 2, 2, 4. How many edges does it have? How many edges does a tree on 9 vertices have?

  1. Sum of degrees \(=14\), so \(e=7\).
  2. Tree: \(9-1=8\) edges.

Answer: 7 edges; 8 edges

Counting walks with a matrix

\(\displaystyle \text{walks of length }k\text{ from }i\text{ to }j:\ \ (A^k)_{ij}\)

Use it for: Adjacency matrix questions: raise the adjacency matrix \(A\) to the power \(k\) and read off entry \((i,j)\).

Worked example

A graph on vertices \(P,Q,R,S\) has edges \(PQ, PR, QR, QS, RS\). How many walks of length 3 go from \(P\) to \(Q\)?

  1. Adjacency matrix (order \(P,Q,R,S\)): \(A=\begin{pmatrix}0&1&1&0\\1&0&1&1\\1&1&0&1\\0&1&1&0\end{pmatrix}\)
  2. Row \(P\) of \(A^2\) is \((2,1,1,2)\)
  3. \((A^3)_{PQ}\)​\({}=2\cdot1+1\cdot0+1\cdot1+2\cdot1\)​\({}=5\)

Answer: 5

Statistics and probability

Outliers

\(\displaystyle x \(\displaystyle \text{or}\) \(\displaystyle x>Q_3+1.5\times IQR\)

Use it for: Box plots and "is this value an outlier?" questions.

Worked example

A data set has \(Q_1=14\) and \(Q_3=22\). Is 36 an outlier?

  1. \(IQR=22-14=8\)
  2. Fences: \(14-12=2\) and \(22+12=34\)
  3. \(36>34\), so 36 is an outlier.

Answer: Yes: the fences are 2 and 34

Binomial probability

\(\displaystyle X\sim B(n,p):\) \(\displaystyle P(X=r)\)​\(\displaystyle {}=\binom nr p^r(1-p)^{n-r}\)

Use it for: Probability questions with a fixed number of independent trials (often with a GDC as well).

Worked example

A spinner lands on red with probability 0.25. It is spun 12 times. Work out the probability of exactly 3 reds.

  1. \(X\sim B(12,\,0.25)\); \(\binom{12}3=220\)
  2. \(220\times0.25^3\times0.75^9\)

Answer: \(0.258\) (3 s.f.)

Sum of independent Poisson variables

\(\displaystyle X\)​\(\displaystyle {}\sim\mathrm{Po}(m_1),\ Y\)​\(\displaystyle {}\sim\mathrm{Po}(m_2)\ \text{independent}\)​\(\displaystyle {}\Rightarrow X+Y\)​\(\displaystyle {}\sim\mathrm{Po}(m_1+m_2)\)

Use it for: Combining two Poisson counts (two shops, two time periods).

Worked example

Calls arrive at two desks independently: \(X\sim\mathrm{Po}(2.1)\) and \(Y\sim\mathrm{Po}(1.4)\) per hour. Find \(P(X+Y\le2)\).

  1. \(X+Y\sim\mathrm{Po}(3.5)\)
  2. GDC (Poisson cdf, mean 3.5, upper bound 2)

Answer: \(0.321\) (3 s.f.)

Expected frequency and degrees of freedom

\(\displaystyle f_e\)​\(\displaystyle {}=\frac{\text{row total}\times\text{column total}}{\text{grand total}},\) \(\displaystyle \nu=(r-1)(c-1)\)

Use it for: Every chi-squared test for independence.

Worked example

In a 3 × 4 contingency table, a cell's row total is 40, its column total is 30 and the grand total is 120. Find the expected frequency and the degrees of freedom.

  1. \(f_e\)​\({}=\frac{40\times30}{120}\)​\({}=10\)
  2. \(\nu=(3-1)(4-1)=6\)

Answer: \(f_e=10\), 6 degrees of freedom

Degrees of freedom: goodness of fit

\(\displaystyle \nu=c-1-p\)

Use it for: Chi-squared goodness-of-fit tests, with \(c\) classes (after combining) and \(p\) parameters estimated from the data.

Worked example

A goodness-of-fit test for a Poisson model uses 6 classes, and the mean is estimated from the data. Find the degrees of freedom.

  1. One parameter (the mean) is estimated.
  2. \(\nu=6-1-1\)

Answer: 4

Calculus

Tangent and normal at a point

\(\displaystyle \text{tangent: }y-f(a)=f'(a)(x-a);\) \(\displaystyle m_{\text{normal}}\)​\(\displaystyle {}=-\frac1{f'(a)}\)

Use it for: Any "find the equation of the tangent / normal" question.

Worked example

Find the tangent and the normal to \(y=x^3-2x\) at \(x=1\).

  1. \(y(1)=1-2=-1\)
  2. \(\frac{dy}{dx}=3x^2-2=1\) at \(x=1\)
  3. Tangent: \(y+1\)​\({}=1(x-1)\)​\({}\Rightarrow y=x-2\)
  4. Normal gradient \(-1\): \(y+1\)​\({}=-(x-1)\)​\({}\Rightarrow y\)​\({}=-x\)

Answer: Tangent \(y=x-2\), normal \(y=-x\)

Stationary points and their nature

\(\displaystyle f'(x)=0;\) \(\displaystyle f''(x)<0\)​\(\displaystyle {}\Rightarrow\text{max},\) \(\displaystyle f''(x)>0\)​\(\displaystyle {}\Rightarrow\text{min}\)

Use it for: Optimisation and curve sketching.

Worked example

Locate the stationary points of \(f(x)=x^3-9x^2+15x+4\) and say which is a maximum.

  1. \(f'(x)\)​\({}=3x^2-18x+15\)​\({}=3(x-1)(x-5)\)​\({}=0\)​\({}\Rightarrow x\)​\({}=1,\ 5\)
  2. \(f''(x)=6x-18\): \(f''(1)=-12<0\), \(f''(5)=12>0\)
  3. \(f(1)=11\), \(f(5)=-21\)

Answer: Maximum \((1,11)\), minimum \((5,-21)\)

Integrating a function of ax + b

\(\displaystyle \int f(ax+b)\,dx\)​\(\displaystyle {}=\frac1aF(ax+b)+C\)

Use it for: Integrals like \((3x-1)^4\), \(e^{2x+1}\) or \(\cos(5x)\).

Worked example

Find \(\int(3x-1)^4\,dx\).

  1. Integrate the outside: \(\frac{(3x-1)^5}5\)
  2. Divide by the inside's \(x\)-coefficient, 3.

Answer: \(\frac{(3x-1)^5}{15}+C\)

Keep going

Questions

Which IB Math AI HL formulas are not in the formula booklet?

The ones to learn by heart are: Upper and lower bounds; Perpendicular gradients; Linearising a power model; Handshaking lemma and trees; Counting walks with a matrix; Outliers; Binomial probability; Sum of independent Poisson variables; Expected frequency and degrees of freedom; Degrees of freedom: goodness of fit; Tangent and normal at a point; Stationary points and their nature; Integrating a function of ax + b. Each one is on this page with a worked example.

Do I get a formula booklet in the exam?

Yes. You get a clean copy of the IB formula booklet in every IB Mathematics exam, so you only need to memorise what it leaves out.

What is the best way to memorise them?

Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.

Is this page a copy of the official booklet?

No. We link to the official IB Mathematics: applications and interpretation formula booklet; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.

Our own list, wording and examples, written and checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.