IB Math AI HL cheat sheet

Scientific notation: a×10^k with 1≤|a|<10; for IB, give 3 significant figures unless the answer is exact or the question specifies otherwise.

Log laws: log(ab)=log a+log b, log(a/b)=log a−log b, log(a^n)=n log a, log_b x=ln x/ln b; change of base is required for equations with mixed bases.

Solve a^x=b by taking logs: x=log b/log a; solve log equations only after stating domain restrictions (argument >0).

Arithmetic sequence: u_n=u_1+(n−1)d, S_n=n/2(2u_1+(n−1)d). Geometric: u_n=u_1 r^(n−1), S_n=u_1(1−r^n)/(1−r), S_∞=u_1/(1−r) for |r|<1.

Compound interest and depreciation use geometric models; nominal annual rate r compounded k times per year gives effective annual rate (1+r/k)^k−1.

Binomial theorem: (a+b)^n=Σ C(n,r) a^(n−r) b^r; the general term is C(n,r) a^(n−r) b^r, useful for finding a specific coefficient.

Complex numbers: i²=−1, modulus |z|=√(a²+b²), argument arg z=arctan(b/a) adjusted by quadrant; z=r cis θ, and de Moivre gives z^n=r^n cis(nθ).

Counting: permutations nPr=n!/(n−r)!, combinations nCr=n!/(r!(n−r)!); use combinations when order is irrelevant, and inclusion–exclusion for 'at least one' events.

A function is a rule assigning exactly one output per input; the vertical line test distinguishes functions from relations, and the horizontal line test decides whether an inverse is a function.

Composite functions: (f∘g)(x)=f(g(x)) requires the range of g to lie inside the domain of f; the domain of f∘g is the set of x in dom g with g(x) in dom f.

Inverse functions: swap x and y and solve; the graph of f⁻¹ is the reflection of y=f(x) in y=x, and dom f⁻¹=range f.

Quadratic: vertex form y=a(x−h)²+k gives vertex (h,k); axis of symmetry x=−b/(2a) and discriminant b²−4ac determine the number of real roots.

Transformations: y=af(b(x−c))+d applies horizontal stretch 1/b, reflection in axes when a or b is negative, then translation by (c,d).

Exponential and logarithmic graphs are inverses; y=e^x has asymptote y=0, while y=ln x has vertical asymptote x=0 and domain x>0.

Rational functions have vertical asymptotes where the denominator is zero (after cancelling) and horizontal asymptotes determined by comparing degrees of numerator and denominator.

Use a GDC to solve equations and to sketch graphs, but state the window and report solutions to the required accuracy; intersections should be verified algebraically where feasible.

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

Sine rule: a/sin A=b/sin B=c/sin C; cosine rule: a²=b²+c²−2bc cos A. Use the cosine rule when given two sides and the included angle or all three sides.

Area of a triangle = ½ab sin C; for a segment, subtract the triangle area from the sector area.

Trig identities: sin²θ+cos²θ=1, tan θ=sin θ/cos θ, and double-angle forms sin 2θ=2 sin θ cos θ, cos 2θ=cos²θ−sin²θ.

General solutions: if sin θ=k then θ=arcsin k+2πn or θ=π−arcsin k+2πn; if cos θ=k then θ=±arccos k+2πn; if tan θ=k then θ=arctan k+πn.

Vectors: magnitude |v|=√(x²+y²+z²), scalar product a·b=|a||b|cos θ; perpendicular vectors have a·b=0.

Lines in 3D: r=a+tb; angle between lines uses the direction vectors, and the shortest distance from a point to a line uses the perpendicular component.

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

Mean of a frequency table: x̄=Σfx/Σf; variance σ²=Σf(x−x̄)²/Σf, and the standard deviation is its square root.

For grouped data, use midpoints; for linear transformations y=ax+b, mean becomes a x̄+b and standard deviation becomes |a|σ.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); independent events satisfy P(A∩B)=P(A)P(B); conditional probability P(A|B)=P(A∩B)/P(B).

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

Normal distribution: standardise with z=(x−μ)/σ; use the GDC for normalcdf and invNorm, and remember the symmetry P(X<μ−a)=P(X>μ+a).

Hypothesis testing: state H₀ and H₁, choose significance level, compute the p-value, and reject H₀ when p<α; distinguish one-tailed from two-tailed tests.

Correlation: Pearson's r measures linear association only; the coefficient of determination r² gives the proportion of variance explained by the regression line.

Chi-squared test for independence uses expected frequencies (row total×column total)/grand total and degrees of freedom (r−1)(c−1).

Derivative from first principles: f′(x)=lim(h→0)[f(x+h)−f(x)]/h; the derivative gives the gradient of the tangent and instantaneous rate of change.

Power rule: d/dx(x^n)=nx^(n−1); d/dx(e^x)=e^x, d/dx(ln x)=1/x, d/dx(sin x)=cos x, d/dx(cos x)=−sin x.

Product rule (uv)′=u′v+uv′; quotient rule (u/v)′=(u′v−uv′)/v²; chain rule dy/dx=dy/du·du/dx.

Stationary points occur where f′(x)=0; use the sign change of f′ or f″ to classify as maximum, minimum, or point of inflection.

Optimisation problems: form an expression in one variable using the constraint, differentiate, set to zero, and verify the nature of the stationary point.

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

Definite integrals give signed area; the area between curves is ∫|f(x)−g(x)| dx, and volumes of revolution about the x-axis use π∫y² dx.

Kinematics: velocity is the derivative of displacement and acceleration is the derivative of velocity; displacement is the integral of velocity, while distance travelled integrates |v|.