SL AI — Formula sheet

Applications & Interpretation SL · use this alongside your official IB booklet.

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Prior learning

Area of a parallelogram

A = b \cdot h

Area of a triangle

A = \frac{1}{2} (b \cdot h)

Area of a trapezium

A = \frac{1}{2} (a + b) h

Area of a circle

A = π r^{2}

Circumference of a circle

C = 2 π r

Volume of a cuboid

V = l \cdot w \cdot h

Volume of a cylinder

V = π r^{2} h

Volume of a prism

V = A \cdot h (A = cross-section area)

Area of curved surface of cylinder

A = 2 π r h

Distance between two points

d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}

Midpoint of two points

(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Topic 1 — Number & Algebra

The n-th term of an arithmetic sequence

u_{n} = u_{1} + (n - 1) d

Sum of n terms of an arithmetic sequence

S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})

The n-th term of a geometric sequence

u_{n} = u_{1} \cdot r^{n - 1}

Sum of n terms of a geometric sequence

S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}, r \neq 1

Compound interest

F V = P V {(1 + \frac{r}{100 k})}^{k n}

k = compounding periods per year, n = years

Exponents

a^{p} \cdot a^{q} = a^{p + q}, \frac{a^{p}}{a^{q}} = a^{p - q}, (a^{p})^{q} = a^{p q}

Logarithms — power law

\log_{a} (x^{n}) = n \log_{a} x

% error

ε = | \frac{v_{A} - v_{E}}{v_{E}} | \times 100 %

Topic 2 — Functions

Equation of a straight line

y = m x + c or a x + b y + d = 0

Gradient of a straight line

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Axis of symmetry of a quadratic

x = - \frac{b}{2 a}

Topic 3 — Geometry & Trigonometry

Length of an arc

ℓ = 2 π r \cdot \frac{θ}{360^{\circ}}

Area of a sector

A = π r^{2} \cdot \frac{θ}{360^{\circ}}

Surface area of a sphere

A = 4 π r^{2}

Volume of a sphere

V = \frac{4}{3} π r^{3}

Volume of a cone

V = \frac{1}{3} π r^{2} h

Curved surface area of a cone

A = π r l

Volume of a pyramid

V = \frac{1}{3} A_{base} \cdot h

Right-angled triangle (SOHCAHTOA)

\sin θ = \frac{O}{H}, \cos θ = \frac{A}{H}, \tan θ = \frac{O}{A}

Sine rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Cosine rule

c^{2} = a^{2} + b^{2} - 2 a b \cos C

Area of a triangle (SAS)

A = \frac{1}{2} a b \sin C

Topic 4 — Statistics & Probability

Interquartile range

IQR = Q_{3} - Q_{1}

Mean of a data set

\bar{x} = \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{n}, n = \sum f_{i}

Probability of an event

P (A) = \frac{n (A)}{n (U)}

Complementary events

P (A^{'}) = 1 - P (A)

Combined events

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Mutually exclusive events

P (A \cap B) = 0

Conditional probability

P (A ∣ B) = \frac{P (A \cap B)}{P (B)}

Independent events

P (A \cap B) = P (A) P (B)

Expected value (discrete)

E (X) = \sum x P (X = x)

Topic 5 — Calculus

Derivative of x^n

f (x) = x^{n} ⟹ f^{'} (x) = n x^{n - 1}

Integral of x^n

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1

Area between curve and x-axis

A = \int_{a}^{b} y d x

Trapezoidal rule (approx. area)

A \approx \frac{h}{2} (y_{0} + y_{n} + 2 (y_{1} + y_{2} + \dots + y_{n - 1})), h = \frac{b - a}{n}