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IB Math AI SL formulas not in the formula booklet

The exam gives you a formula booklet, but it leaves these 12 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 and algebra

Depreciation

\(\displaystyle V=V_0\Big(1-\frac r{100}\Big)^n\)

Use it for: Value-loss questions (the booklet's compound interest formula needs the sign changed).

Use the booklet's compound interest formula with a negative rate, or learn this form.

Worked example

A machine bought for £18 000 drops in value by 12% a year. What is it worth 4 years later?

  1. \(V=18000\times0.88^4\)
  2. \(=18000\times0.59969\ldots\)

Answer: £10 794.52

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

Functions and models

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 \(x+3y-5=0\))

Asymptote of an exponential model

\(\displaystyle f(x)=ka^{x}+c\ \text{ or }\ ka^{-x}+c:\) \(\displaystyle \text{horizontal asymptote }y=c\)

Use it for: Long-term behaviour of exponential models.

Worked example

For \(f(x)=5\times2^{-x}+3\), state the horizontal asymptote and the \(y\)-intercept.

  1. As \(x\to\infty\), \(2^{-x}\to0\), so \(f(x)\to3\).
  2. \(f(0)=5+3=8\)

Answer: Asymptote \(y=3\), \(y\)-intercept 8

Sine model in degrees

\(\displaystyle f(x)=a\sin(bx)+d:\) \(\displaystyle \text{amplitude }|a|,\) \(\displaystyle \text{period }\frac{360^\circ}b,\) \(\displaystyle \max=d+|a|\)

Use it for: Modelling tides, temperatures and Ferris wheels.

Worked example

\(T=6\sin(30t)+14\) models the temperature, in °C, \(t\) months into the year. Find the amplitude, the period and the maximum temperature.

  1. Amplitude \(=6\)
  2. Period \(=\frac{360}{30}=12\) months
  3. Maximum \(=14+6=20\)

Answer: Amplitude 6, period 12 months, maximum 20 °C

Geometry

Slant height of a cone

\(\displaystyle l=\sqrt{r^2+h^2}\)

Use it for: Cone surface area when you are given the vertical height.

Worked example

A party hat is a cone 15 cm tall with base radius 8 cm. Work out the area of card it uses (its curved surface).

  1. \(l=\sqrt{64+225}=17\)
  2. Curved surface area \(=\pi rl=136\pi\)

Answer: \(136\pi\approx427\) cm²

Perpendicular bisector

\(\displaystyle m_\perp=-\frac1{m_{AB}},\) \(\displaystyle \text{through the midpoint of }AB\)

Use it for: Voronoi diagrams and "equidistant from two points" questions.

Worked example

Find the perpendicular bisector of \(A(1,2)\) and \(B(5,6)\).

  1. Midpoint \((3,4)\)
  2. \(m_{AB}=\frac{6-2}{5-1}=1\), so the gradient is \(-1\)
  3. \(y-4=-(x-3)\)

Answer: \(y=-x+7\)

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.)

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

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 \((5,-21)\)

Keep going

Questions

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

The ones to learn by heart are: Depreciation; Upper and lower bounds; Perpendicular gradients; Asymptote of an exponential model; Sine model in degrees; Slant height of a cone; Perpendicular bisector; Outliers; Binomial probability; Expected frequency and degrees of freedom; Tangent and normal at a point; Stationary points and their nature. 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.