IB Maths SL Formula Sheet: Algebra, Functions, Trig, Vectors, Stats, Calculus, Study Guides, Projects, Research of Art

Download IB Maths SL Formula Sheet: Algebra, Functions, Trig, Vectors, Stats, Calculus and more Study Guides, Projects, Research Art in PDF only on Docsity! IB Maths SL – Formula Sheet ~ Revision Village ~ Topic 1: Algebra The nth term of an arithmetic sequence 𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑 Sum of an arithmetic sequence 𝑠𝑛 = 𝑛2 (2𝑢1 + (𝑛 − 1)𝑑) = 𝑛2 (𝑢1 + 𝑢𝑛) The nth term of a geometric sequence 𝑢𝑛 = 𝑢1𝑟𝑛−1 Sum of a geometric sequence 𝑠𝑛 = 𝑢1(𝑟𝑛 − 1)𝑟 − 1 = 𝑠𝑛 = 𝑢1(1 − 𝑟𝑛)1 − 𝑟 , 𝑟 ≠ 1 Sum of an infinity geometric sequence 𝑠∞ = 𝑢11 − 𝑟 , |𝑟| < 1 Exponents & logs 𝑎𝑥 = 𝑏 ↔ 𝑥 = log𝑎 𝑏 Log laws log𝑎 𝑏 + log𝑎 𝑐 = log𝑎 𝑏𝑐 log𝑎 𝑏 − log𝑎 𝑐 = log𝑎 𝑏𝑐 log𝑎 𝑏𝑟 = 𝑟 log𝑎 𝑏 log𝑐 𝑏 = log𝑎 𝑏log𝑎 𝑐 Binomial coefficient (𝑛𝑟) = 𝑛!𝑟! (𝑛 − 𝑟)! Binomial expansion (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + (𝑛1) 𝑎𝑛−1𝑏+. . . + (𝑛𝑟) 𝑎𝑛−𝑟𝑏𝑟+. . . + 𝑏𝑛 Topic 2: Functions & Equations Axis of symmetry 𝑥 = −𝑏2𝑎 Log & exponential functions 𝑎𝑥 = 𝑒𝑥𝑙𝑛𝑎 log𝑎 𝑎𝑥 = 𝑥 = 𝑎log𝑎 𝑥 Quadratic formula 𝑥 = −𝑏 ± √𝑏2 − 4𝑎𝑐2𝑎 Discriminant ∆ = 𝑏2 − 4𝑎𝑐 Topic 3: Circular Functions & Trigonometry Length of arc 𝑙 = 𝜃𝑟 Area of sector 𝐴 = 12 𝜃𝑟2 Trig identities tan 𝜃 = sin 𝜃cos 𝜃 cos2 𝜃 + sin2 𝜃 = 1 sin 2𝜃 = 2 sin 𝜃 cos 𝜃 cos 2𝜃 = cos2 𝜃 − sin2 𝜃 = 2 cos2 𝜃 − 1 = 1 − 2 sin2 𝜃 Cosine rule 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶 cos 𝐶 = 𝑎2 + 𝑏2 − 𝑐22𝑎𝑏 Sine rule 𝑎sin 𝐴 = 𝑏sin 𝐵 = 𝑐sin 𝐶 Area of triangle 𝐴 = 12 𝑎𝑏 sin 𝐶 Topic 4: Vectors Magnitude of a vector |𝒗| = √𝑣12 + 𝑣22 + 𝑣32 Dot / Scalar product 𝒗 ∙ 𝒘 = |𝒗||𝒘| cos 𝜃 𝒗 ∙ 𝒘 = 𝑣1𝑤1 + 𝑣2𝑤2 + 𝑣3𝑤3 Angle between 2 vectors cos 𝜃 = 𝒗 ∙ 𝒘|𝒗||𝒘| Vector equation of a line 𝒓 = 𝒂 + 𝑡𝒃 Topic 5: Statistics & Probability Mean of a set of data ?̅? = ∑ 𝑓𝑖𝑥𝑖𝑛𝑖=1∑ 𝑓𝑖𝑛𝑖=1 Probability of an event A 𝑃(𝐴) = 𝑛(𝐴)𝑛(𝑢) Complementary events 𝑃(𝐴) + 𝑃(𝐴′) = 1 Combined events 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) Mutually exclusive events 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) Conditional probability 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) Independent events 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵) Expected value of a discrete random variable X 𝐸(𝑋) = 𝜇 = ∑ 𝑥𝑥 𝑃(𝑋 = 𝑥) Binomial distribution 𝑋~𝐵(𝑛, 𝑝) 𝑃(𝑋 = 𝑟) = (𝑛𝑟) 𝑝𝑟(1 − 𝑝)𝑛−𝑟, 𝑟 = 0,1,2. . 𝑛 Mean 𝐸(𝑋) = 𝑛𝑝 Variance 𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 − 𝑝) Standardized normal variable 𝑧 = 𝑥 − 𝜇𝜎 Topic 6: Calculus Derivative of a function 𝑑𝑦𝑑𝑥 = 𝑓′(𝑥) = limℎ→0 (𝑓(𝑥 + ℎ) − 𝑓(𝑥)ℎ ) Standard derivatives 𝑓(𝑥) = 𝑥𝑛 → 𝑓′(𝑥) = 𝑛𝑥𝑛−1 𝑓(𝑥) = sin 𝑥 → 𝑓′(𝑥) = cos 𝑥 𝑓(𝑥) = cos 𝑥 → 𝑓′(𝑥) = − sin 𝑥 𝑓(𝑥) = tan 𝑥 → 𝑓′(𝑥) = 1cos2 𝑥 𝑓(𝑥) = 𝑒𝑥 → 𝑓′(𝑥) = 𝑒𝑥 𝑓(𝑥) = ln 𝑥 → 𝑓′(𝑥) = 1𝑥 Chain rule 𝑦 = 𝑔(𝑢), 𝑢 = 𝑓(𝑥) → 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑢 × 𝑑𝑢𝑑𝑥 Product rule 𝑦 = 𝑢𝑣 → 𝑑𝑦𝑑𝑥 = 𝑢 𝑑𝑣𝑑𝑥 + 𝑣 𝑑𝑢𝑑𝑥 Quotient rule 𝑦 = 𝑢𝑣 → 𝑑𝑦𝑑𝑥 = 𝑣 𝑑𝑢𝑑𝑥 − 𝑢 𝑑𝑣𝑑𝑥𝑣2 Standard integrals ∫ 𝑥𝑛𝑑𝑥 = 𝑥𝑛+1𝑛 + 1 + 𝐶 , 𝑛 ≠ −1 ∫ 1𝑥 𝑑𝑥 = ln 𝑥 + 𝐶 , 𝑥 > 0 ∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶 ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶 ∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥 + 𝐶 Area under a curve between x = a and x = b 𝐴 = ∫ 𝑦𝑏 𝑎 𝑑𝑥 Volume of revolution about the x-axis from x = a to x = b 𝑉 = ∫ 𝜋𝑦2𝑏 𝑎 𝑑𝑥 Total distance travelled from t1 to t2 distance = ∫ |𝑣(𝑡)|𝑡2𝑡1 𝑑𝑡 www.revisionvillage.com © Revision Village 2019 Voted #1 IB Maths Resource