IB Math AI SL cheat sheet

Standard form: a × 10^k with 1 ≤ a < 10; multiply/divide by adding/subtracting exponents, then renormalise.

Percentage change = (new − old)/old × 100%; a 20% increase then 20% decrease gives a net 4% decrease, not 0%.

Simple interest: I = Prt. Compound interest/finance: FV = PV(1 + r/n)^(nt), with r as a decimal and n compounds per year.

Depreciation uses the same compound formula with (1 − r) in place of (1 + r).

Arithmetic sequence: u_n = u_1 + (n − 1)d; S_n = n/2 (2u_1 + (n − 1)d) = n/2 (u_1 + u_n).

Geometric sequence: u_n = u_1 r^(n−1); S_n = u_1 (r^n − 1)/(r − 1) for r ≠ 1; S_∞ = u_1/(1 − r) only when |r| < 1.

Logs: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a; solving a^x = b gives x = log b / log a.

Binomial theorem: (a + b)^n = Σ C(n,k) a^(n−k) b^k; the general term is C(n,k) a^(n−k) b^k, and C(n,k) = n!/(k!(n−k)!).

GDC decision rule: use the finance (TVM) solver for loans, annuities and amortisation; use the equation solver or a graph for exponential equations that cannot be done by hand.

Domain restrictions to check: denominator ≠ 0, even-root radicand ≥ 0, log argument > 0.

Quadratic: axis of symmetry x = −b/(2a); vertex y-value found by substitution; discriminant Δ = b² − 4ac gives 2, 1 or 0 real roots as Δ > 0, = 0, < 0.

Completed square form y = a(x − h)² + k gives vertex (h, k); y = a(x − p)(x − q) gives x-intercepts p and q.

Exponential model y = ka^x: a > 1 growth, 0 < a < 1 decay; k is the initial value at x = 0.

Transformations: y = f(x − h) + k shifts right h and up k; y = af(x) stretches vertically by a; y = f(−x) reflects in the y-axis.

Inverse function: swap x and y and solve; graph of f⁻¹ is f reflected in y = x; domain of f⁻¹ = range of f.

Composite f∘g means apply g first; it exists only where range of g lies inside domain of f.

Asymptotes: for rational f(x) = p(x)/q(x), a zero of q not cancelled by p gives a vertical asymptote; compare degrees for horizontal asymptotes.

GDC: use the graph window to find zeros, vertices and intersections rather than solving by hand when the question allows.

Right triangle: SOH-CAH-TOA; Pythagoras a² + b² = c²; exact values sin30° = 1/2, cos30° = √3/2, tan45° = 1.

Sine rule: a/sin A = b/sin B = c/sin C; use for a matching side–angle pair. Ambiguous case (two triangles) arises when given two sides and a non-included angle.

Cosine rule: a² = b² + c² − 2bc cos A; use when given SAS or SSS.

Area of any triangle = ½ab sin C, with C the included angle.

Arc length s = rθ and sector area A = ½r²θ, with θ in radians; 180° = π rad.

3D: angle between a line and a plane is found from a right triangle using the perpendicular projection; the diagonal of a cuboid is √(l² + w² + h²).

Volume and surface area: sphere V = 4/3 πr³, S = 4πr²; cone V = 1/3 πr²h, curved S = πrl.

Trig equations: solve for the principal value, then use periodicity (180° for tan, 360° for sin and cos) to list all solutions in the required interval.

Sine/cosine graphs: y = a sin(bx) + d has amplitude |a|, period 360°/b (or 2π/b), vertical shift d.

Mean = Σx/n; for grouped data use midpoints and frequencies. Standard deviation measures spread; adding a constant shifts the mean but leaves SD unchanged, multiplying by k multiplies SD by |k|.

Median and quartiles: with n values, median position is (n + 1)/2; IQR = Q3 − Q1; an outlier is commonly below Q1 − 1.5 IQR or above Q3 + 1.5 IQR.

Box plots show median, quartiles, range and outliers; histograms have area proportional to frequency, so frequency density = frequency / class width.

Regression line ŷ = ax + b minimises the sum of squared vertical deviations; r is the correlation coefficient, with −1 ≤ r ≤ 1 and r² the coefficient of determination.

Interpolation inside the data range is reliable; extrapolation beyond it is not, because the linear relationship may not continue.

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive means P(A ∩ B) = 0.

Conditional probability: P(A|B) = P(A ∩ B)/P(B); independent events satisfy P(A ∩ B) = P(A)P(B).

Binomial: X ~ B(n, p), P(X = k) = C(n,k) p^k (1 − p)^(n−k); mean = np, variance = np(1 − p).

Normal: X ~ N(μ, σ²); use standardisation z = (x − μ)/σ or the GDC normalcdf/normalpdf/invNorm for probabilities and inverse problems.

Derivative definition: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; f′(a) is the gradient of the tangent at x = a.

Power rule: d/dx (ax^n) = anx^(n−1); derivative of a constant is 0; differentiate term by term.

Increasing where f′(x) > 0, decreasing where f′(x) < 0; stationary points solve f′(x) = 0.

Second derivative test: f″(x) > 0 gives a local minimum, f″(x) < 0 a local maximum, f″(x) = 0 is inconclusive.

Tangent at x = a: y − f(a) = f′(a)(x − a); normal gradient is −1/f′(a).

Integration reverses differentiation: ∫ax^n dx = a x^(n+1)/(n+1) + C for n ≠ −1.

Definite integral ∫ from a to b f(x) dx gives the signed area between curve and x-axis; total area needs separate integrals where the curve crosses the axis.

Area between two curves = ∫ (upper − lower) dx between the intersection x-values.

Kinematics: velocity is the derivative of displacement, acceleration the derivative of velocity; displacement = ∫v dt, distance travelled = ∫|v| dt.