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Math AA HL

5 topics83 outcomesLocal syllabus version: 2021

Number and Algebra

16 outcomes

Open
  1. SL 1.1Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer (scientific notation); calculator notation is not acceptable
  2. SL 1.2Arithmetic sequences and series: formulae for the nth term and the sum of the first n terms; use of sigma notation; applications
  3. SL 1.3Geometric sequences and series: formulae for the nth term and the sum of the first n terms; use of sigma notation; applications
  4. SL 1.4Financial applications of geometric sequences and series: compound interest and annual depreciation; calculating the real value of an investment with interest and inflation
  5. SL 1.5Laws of exponents with integer exponents; introduction to logarithms with base 10 and e; awareness that a^x = b is equivalent to log_a(b) = x; numerical evaluation of logarithms using technology
  6. SL 1.6Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof; symbols and notation for equality and identity
  7. SL 1.7Laws of exponents with rational exponents; laws of logarithms (product, quotient, power rules); change of base of a logarithm; solving exponential equations including using logarithms
  8. SL 1.8Sum of infinite convergent geometric sequences; use of |r| < 1 and modulus notation
  9. SL 1.9The binomial theorem: expansion of (a + b)^n for n ∈ ℕ; use of Pascal's triangle and nCr; finding specific terms
  10. AHL 1.10Counting principles including permutations and combinations; extension of the binomial theorem to fractional and negative indices: (a + b)^n for n ∈ ℚ HL only
  11. AHL 1.11Partial fractions: maximum of two distinct linear terms in the denominator, degree of numerator less than degree of denominator HL only
  12. AHL 1.12Complex numbers: the number i where i² = −1; Cartesian form z = a + bi; real part, imaginary part, conjugate, modulus and argument; the complex plane (Argand diagram) HL only
  13. AHL 1.13Modulus–argument (polar) form z = r(cosθ + i sinθ) = r cisθ; Euler form z = re^{iθ}; sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation HL only
  14. AHL 1.14Complex conjugate roots of quadratic and polynomial equations with real coefficients; De Moivre's theorem and its extension to rational exponents; powers and roots of complex numbers HL only
  15. AHL 1.15Proof by mathematical induction; proof by contradiction; use of a counterexample to show a statement is not always true HL only
  16. AHL 1.16Solutions of systems of linear equations (maximum of three equations in three unknowns) using algebraic and matrix/row-reduction methods; unique solution, infinite solutions, or no solution (inconsistent) HL only

Functions

16 outcomes

Open
  1. SL 2.1Different forms of the equation of a straight line: gradient-intercept (y = mx + c), general (ax + by + d = 0), point-gradient form; gradient; intercepts; parallel lines (m1 = m2) and perpendicular lines (m1 × m2 = −1)
  2. SL 2.2Concept of a function, domain, range and graph; function notation f(x), v(t), C(n); the concept of a function as a mathematical model; informal concept of inverse function; inverse notation f⁻¹(x); reflection in y = x
  3. SL 2.3The graph of a function y = f(x); creating a sketch from information given or a context; using technology to graph functions including sums and differences; all axes and key features labelled
  4. SL 2.4Key features of graphs: maximum and minimum values, intercepts, symmetry, vertex, zeros/roots, vertical and horizontal asymptotes using technology; finding the point of intersection of two curves or lines
  5. SL 2.5Composite functions (f ∘ g)(x) = f(g(x)); identity function; finding the inverse function f⁻¹(x); existence of an inverse for one-to-one functions
  6. SL 2.6The quadratic function f(x) = ax² + bx + c: graph, y-intercept, axis of symmetry; factored form f(x) = a(x−p)(x−q) with x-intercepts; vertex form f(x) = a(x−h)² + k with vertex (h, k)
  7. SL 2.7Solution of quadratic equations and inequalities using factorization, completing the square, and the quadratic formula; the discriminant Δ = b² − 4ac and the nature of the roots (two distinct real, two equal real, no real roots)
  8. SL 2.8The reciprocal function f(x) = 1/x and its self-inverse nature; rational functions f(x) = (ax + b)/(cx + d) and their graphs; equations of vertical and horizontal asymptotes
  9. SL 2.9Exponential functions f(x) = a^x, f(x) = e^x and their graphs; logarithmic functions f(x) = log_a x, f(x) = ln x and their graphs; exponential and logarithmic functions as inverses of each other
  10. SL 2.10Solving equations both graphically and analytically; use of technology to solve equations where there is no appropriate analytic approach; applications to real-life situations
  11. SL 2.11Transformations of graphs: translations y = f(x) + b and y = f(x − a); reflections in both axes y = −f(x) and y = f(−x); vertical stretch y = p f(x); horizontal stretch y = f(qx); composite transformations
  12. AHL 2.12Polynomial functions: graphs, equations, zeros, roots and factors; the factor and remainder theorems; sum and product of the roots of polynomial equations HL only
  13. AHL 2.13Rational functions of the form f(x) = (ax + b)/(cx² + dx + e) and f(x) = (ax² + bx + c)/(dx + e); graphs including all asymptotes (horizontal, vertical and oblique) and intercepts HL only
  14. AHL 2.14Odd and even functions; finding the inverse function f⁻¹(x) including domain restriction; self-inverse functions; periodic functions HL only
  15. AHL 2.15Solutions of g(x) ≥ f(x) both graphically and analytically; graphical or algebraic methods for simple polynomials up to degree 3, technology for others HL only
  16. AHL 2.16The graphs of y = |f(x)|, y = f(|x|), y = 1/f(x), y = f(ax + b), y = [f(x)]²; solution of modulus equations and inequalities HL only

Geometry and Trigonometry

18 outcomes

Open
  1. SL 3.1Distance between two points in three-dimensional space and their midpoint; volume and surface area of 3D solids including right pyramid, right cone, sphere, hemisphere and combinations; angle between two intersecting lines or between a line and a plane
  2. SL 3.2Use of sine, cosine and tangent ratios in right-angled triangles; the sine rule a/sinA = b/sinB = c/sinC; the cosine rule c² = a² + b² − 2ab cosC; area of a triangle as (1/2)ab sinC
  3. SL 3.3Applications of right and non-right-angled trigonometry including Pythagoras's theorem; angles of elevation and depression; construction of labelled diagrams from written statements; contexts may include bearings
  4. SL 3.4The circle: radian measure of angles; length of an arc; area of a sector
  5. SL 3.5Definition of cosθ and sinθ in terms of the unit circle; definition of tanθ as sinθ/cosθ; exact values of trig ratios of 0, π/6, π/4, π/3, π/2 and their multiples; extension of the sine rule to the ambiguous case
  6. SL 3.6The Pythagorean identity cos²θ + sin²θ = 1; double angle identities for sine and cosine; relationship between trigonometric ratios
  7. SL 3.7The circular functions sinx, cosx and tanx: amplitude, periodic nature and graphs; composite functions of the form f(x) = a sin(b(x + c)) + d; transformations; real-life contexts such as tide height or Ferris wheel motion
  8. SL 3.8Solving trigonometric equations in a finite interval both graphically and analytically; equations leading to quadratic equations in sinx, cosx or tanx
  9. AHL 3.9Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ; Pythagorean identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ; inverse trig functions arcsin, arccos, arctan: domains, ranges and graphs HL only
  10. AHL 3.10Compound angle identities for sin(A ± B), cos(A ± B), tan(A ± B); derivation of double angle identities from compound angle identities; double angle identity for tan HL only
  11. AHL 3.11Relationships between trigonometric functions and the symmetry properties of their graphs, e.g. sin(π − θ) = sinθ, cos(π − θ) = −cosθ, tan(π − θ) = −tanθ HL only
  12. AHL 3.12Concept of a vector; position vectors; displacement vectors; representation using directed line segments; base vectors i, j, k; components; algebraic and geometric approaches to sum/difference, zero vector, scalar multiplication, magnitude and unit vectors HL only
  13. AHL 3.13The scalar (dot) product of two vectors: definition, properties, angle between two vectors v · w = |v||w|cosθ; perpendicular vectors (v · w = 0); parallel vectors (|v · w| = |v||w|) HL only
  14. AHL 3.14Vector equation of a line in two and three dimensions r = a + λb; parametric form and Cartesian form; the angle between two lines; simple applications to kinematics HL only
  15. AHL 3.15Coincident, parallel, intersecting and skew lines in 3D; distinguishing between these cases; points of intersection HL only
  16. AHL 3.16The vector product v × w of two vectors: definition, properties; geometric interpretation |v × w| as area of parallelogram (and hence triangle) HL only
  17. AHL 3.17Vector equations of a plane: r = a + λb + μc; r · n = a · n; Cartesian equation ax + by + cz = d HL only
  18. AHL 3.18Intersections of: a line with a plane; two planes; three planes; finding intersections by solving equations; geometrical interpretation of solutions; angle between a line and a plane and between two planes HL only

Statistics and Probability

14 outcomes

Open
  1. SL 4.1Concepts of population, sample, random sample, discrete and continuous data; reliability of data sources and bias in sampling; interpretation of outliers (data more than 1.5 × IQR from nearest quartile); sampling techniques: simple random, convenience, systematic, quota, stratified
  2. SL 4.2Presentation of data: frequency distributions and tables; histograms; cumulative frequency graphs; use of cumulative frequency to find median, quartiles, percentiles, range and IQR; box-and-whisker diagrams; comparing two distributions
  3. SL 4.3Measures of central tendency: mean, median and mode; estimation of mean from grouped data; modal class; measures of dispersion: IQR, standard deviation and variance; effect of constant changes on original data
  4. SL 4.4Pearson's product-moment correlation coefficient r; linear correlation of bivariate data; scatter diagrams; positive, zero, negative; strong, weak, no correlation; equation of the regression line of y on x; use for prediction; distinguish correlation from causation
  5. SL 4.5Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space and event; probability of an event P(A) = n(A)/n(U); complementary events A and A′; expected number of occurrences
  6. SL 4.6Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes; combined events P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive events; conditional probability P(A|B) = P(A ∩ B)/P(B); independent events P(A ∩ B) = P(A)P(B)
  7. SL 4.7Concept of discrete random variables and their probability distributions; expected value (mean) E(X) for discrete data; applications including fair games
  8. SL 4.8Binomial distribution B(n, p): situations where it is an appropriate model; mean and variance of the binomial distribution; probabilities found using technology
  9. SL 4.9The normal distribution and curve; properties of the normal distribution; diagrammatic representation; normal probability calculations using technology; inverse normal calculations
  10. SL 4.10Equation of the regression line of x on y; use for prediction; awareness that a y-on-x line cannot always be used to predict x from a given y value
  11. SL 4.11Formal definition and use of the conditional probability formula P(A|B) = P(A ∩ B)/P(B); testing for independence: P(A|B) = P(A) = P(A|B′) for independent events
  12. SL 4.12Standardization of normal variables (z-values): z = (x − μ)/σ; the standardized value gives the number of standard deviations from the mean; inverse normal calculations where mean and standard deviation are unknown using z-values
  13. AHL 4.13Use of Bayes' theorem for a maximum of three events HL only
  14. AHL 4.14Variance of a discrete random variable; continuous random variables and their probability density functions; mode and median of continuous random variables; mean, variance and standard deviation of discrete and continuous random variables; effect of linear transformations E(aX + b) = aE(X) + b, Var(aX + b) = a²Var(X) HL only

Calculus

19 outcomes

Open
  1. SL 5.1Introduction to the concept of a limit; derivative interpreted as gradient function and as rate of change; informal understanding of the gradient of a curve as a limit; derivative notation dy/dx, f′(x)
  2. SL 5.2Increasing and decreasing functions; graphical interpretation of f′(x) > 0 (increasing), f′(x) = 0 (stationary), f′(x) < 0 (decreasing)
  3. SL 5.3Derivative of f(x) = ax^n is f′(x) = anx^{n−1} for n ∈ ℤ; derivative of functions of the form f(x) = ax^n + bx^{n−1} + … with integer exponents
  4. SL 5.4Tangents and normals at a given point and their equations; use of both analytic approaches and technology
  5. SL 5.5Introduction to integration as anti-differentiation of f(x) = ax^n + bx^{n−1} + … for n ∈ ℤ, n ≠ −1; anti-differentiation with boundary condition to find the constant term; definite integrals using technology; area under a curve y = f(x) where f(x) > 0
  6. SL 5.6Derivative of x^n (n ∈ ℚ), sinx, cosx, e^x and lnx; differentiation of a sum and a multiple of these functions; the chain rule for composite functions; the product and quotient rules
  7. SL 5.7The second derivative f″(x) / d²y/dx²; graphical behaviour of functions including the relationship between the graphs of f, f′ and f″
  8. SL 5.8Local maximum and minimum points; testing using change of sign of f′ or sign of f″; optimization; points of inflexion with zero and non-zero gradients; concave-up (f″ > 0) and concave-down (f″ < 0)
  9. SL 5.9Kinematic problems involving displacement s, velocity v = ds/dt, acceleration a = dv/dt = d²s/dt²; total distance travelled; displacement and distance as definite integrals of velocity
  10. SL 5.10Indefinite integrals of x^n (n ∈ ℚ), sinx, cosx, 1/x and e^x; composites of any of these with linear function ax + b; integration by inspection (reverse chain rule) or by substitution for ∫k g′(x) f(g(x)) dx
  11. SL 5.11Definite integrals using analytical approach ∫_a^b g′(x) dx = g(b) − g(a); areas of a region enclosed by y = f(x) and the x-axis where f(x) can be positive or negative; areas between curves
  12. AHL 5.12Informal understanding of continuity and differentiability at a point; understanding of limits (convergence and divergence); definition of derivative from first principles f′(x) = lim_{h→0} [f(x+h) − f(x)]/h; higher derivatives d^n y/dx^n and f^{(n)}(x) HL only
  13. AHL 5.13Evaluation of limits of indeterminate forms 0/0 and ∞/∞ using l'Hôpital's rule or Maclaurin series; repeated use of l'Hôpital's rule HL only
  14. AHL 5.14Implicit differentiation; related rates of change; optimisation problems including cases where the optimum is at an endpoint HL only
  15. AHL 5.15Derivatives of tanx, secx, cosecx, cotx, a^x, log_a x, arcsinx, arccosx, arctanx; corresponding indefinite integrals; composites with a linear function; use of partial fractions to rearrange the integrand HL only
  16. AHL 5.16Integration by substitution (substitution provided if not of form ∫k g′(x) f(g(x)) dx); integration by parts ∫u dv = uv − ∫v du; repeated integration by parts HL only
  17. AHL 5.17Area of the region enclosed by a curve and the y-axis in a given interval; volumes of revolution about the x-axis or y-axis HL only
  18. AHL 5.18First order differential equations: numerical solution using Euler's method; variables separable; homogeneous differential equations using substitution y = vx; solution of y′ + P(x)y = Q(x) using the integrating factor HL only
  19. AHL 5.19Maclaurin series to obtain expansions for e^x, sinx, cosx, ln(1+x), (1+x)^p for p ∈ ℚ; use of substitution, products, integration and differentiation to obtain other series HL only