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In general, Applications and Interpretation covers a broad range of materials, but Analysis and Approaches covers a narrower range in greater depth. Below lists the main topics in each course, by color code from 2014-2020 syllabus Studies, SL and HL (including Further HL). New topics are italicized. HL includes all SL topics of the same course. Common topics in all new Math courses: Algebra: scientific notation, arithmetic/geometric progression, compound interest, annual depreciation, exponents/logarithms Functions: lines, graph of a function, domain/range, definition of inverse, identifying extremas, intercepts, symmetry, zeros, asymptotes Geometry/Trig: distances and midpoints, volume and surface area of solids, angle between 2 lines or line and a plane, solving triangles, angles of elevation/depression, bearings, arcs/sectors Stats: sampling, outliers, data presentation (tables, graphs, central tendency/spread, cumulative frequency, box and whisker, model class), linear correlation of bivariate data, Pearson's correlation coefficient, scatter plot, equation of the regression line of y on x (y given x), Probability: complements, Venn/tree diagrams, combined events, conditional probability, discrete random variables and expected value, binomial/normal distributions Calculus: increasing/decreasing functions, calculus of polynomials, tangents/normals, solving simple ODE with a boundary condition, definite integrals, first derivative test, optimization Topics in Applications and Interpretation SL and HL: Algebra: amortization and annuities Functions: modelling Geometry/Trig: equations of perpendicular bisectors, Voronoi diagrams Probability/Stats: Spearman's rank, null and alternate hypotheses, significant levels, p-values, chi-squared test for independence, for goodness of fit, one-tailed/two-tailed t tests Calculus: Approximations with trapezoidal rule Topics only in Applications and Interpretation HL: Algebra: adding sinusoidal functions of different arguments, matrices, eigenvalues/eigenvectors Functions: scaling with log notations, log-log and semi-log plots Geometry/Trig: matrix transformations, determinants, graph theory/trees, adjacency matrices, walks, Eulerian/Hamiltonian, MST, Kruskal's and Prim's algorithms, Chinese postman problem algorithm, Travelling salesman problem, nearest neighbour and deleted vertex algorithms Probability/Stats: design of data collection, non-linear regression, sum of square residuals, coefficient of determination, unbiased estimates of mean and standard deviation, central limit theorem, confidence intervals of the mean, Poisson distribution, critical values, test for proportion using binomial distribution, test for population mean using Poisson distribution, bivariate normal distribution and p-value, Types I and II errors, transition matrices and to solve system of linear equations, Markhov chains Calculus: setting up rates of change ODEs, slope fields, Euler's method of a system of 2 ODEs or a second-order ODE, phase portrait Topics in Analysis and Approaches SL and HL: (topics also in Applications and Interpretation HL are in an enlarged font) Algebra: binomial theorem/Pascal's triangle Functions: quadratic functions/equations/inequalities, reciprocal, rational functions, graphs of exponential/logarithmic functions, transformations Geometry/Trig: radians, sine rule ambiguous case, sin and cos in relation to unit circle, exact sin and cos values, Pythagorean identity, graphs of trig functions, solving trig equations graphically and analytically, quadratic trig equations Probability/Stats: equation of the regression line of x on y (x given y), standardized normal distribution Calculus: common derivatives/antiderivatives, chain/product/quotient rules, rates of change, second derivative test, points of inflexion, definite integrals, area enclosed by a curve and the axes, area between curves, kinematics Topics only in Analysis and Approaches HL: (topics also in Applications and Interpretation HL are in an enlarged font) Algebra: permutations/combinations, partial fractions, complex numbers, De Moivre's theorem, proofs by induction/contradiction/counterexample, analytical solutions of system of 3 equations Functions: factor/remainder theorems, sum and product of roots of polynomials, odd/even functions, inequalities, absolute value functions, square of functions Geometry/Trig: reciprocal trig functions, compound angle identities, symmetries in trig functions, vectors: unit/base vectors, position vectors, normalizing, vector equation of a line, dot/cross products, components, system of 2 lines, vector equations of a plane, intersections of line/plane, of 2 planes, of 3 planes Probability/Stats: Bayes' theorem, continuous random variables, linear transformations of a random variable Calculus: informal ideas of continuity and differentiability, derive with first principles, higher derivatives, l'Hopital rule, Maclaurin series, related rates, implicit differentiation, trig antideratives, integrate with partial fractions, integration by substitution, repeated use of integration by parts, first order ODE, Euler's method for a single ODE, separation of variables, homogeneous ODE, integrating factor, calculus of series TL;DR ~~~~~~~~~~~~~~~~~~ 1) Vectors are now HL only. 2) Rank of difficulty: current Studies < AI SL < AA SL (easier than current SL if better at stats than vectors) < AA HL < current HL <= AI HL AI SL is mostly Studies. AI HL is studies but with about 30-40% of Further Maths HL. AA SL is pretty much SL. and AA HL is HL with only 20-30% of the Calculus option. 3) AI HL for the fast learner. AA HL for the inquisitive learner. If you are not that strong at math, the recommendation is AA SL. AI SL is still quite a bit easier than AA SL and only choose it if you are fine with dealing with lots of data.