Skill plan for IB Diploma Programme - Analysis and Approaches HL
IXL provides skill alignments with recommended IXL skills for each topic. Find the IXL skills that are right for you below!
Topic
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1
Number and algebra
SL 1.1
SL 1.1
Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer.
Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer.
SL 1.2
SL 1.2
Arithmetic sequences and series.
Arithmetic sequences and series.
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of sigma notation for sums of arithmetic sequences.
Use of sigma notation for sums of arithmetic sequences.
Applications.
Applications.
Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
SL 1.3
SL 1.3
Geometric sequences and series.
Geometric sequences and series.
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of sigma notation for the sums of geometric sequences.
Use of sigma notation for the sums of geometric sequences.
Applications.
Applications.
SL 1.4
SL 1.4
Financial applications of geometric sequences and series:
Financial applications of geometric sequences and series:
Compound interest
Compound interest
Annual depreciation
Annual depreciation
SL 1.5
SL 1.5
Laws of exponents with integer exponents.
Laws of exponents with integer exponents.
Introduction to logarithms with base 10 and e.
Introduction to logarithms with base 10 and e.
Numerical evaluation of logarithms using technology.
Numerical evaluation of logarithms using technology.
SL 1.6
SL 1.6
Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.
Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.
The symbols and notation for equality and identity.
The symbols and notation for equality and identity.
SL 1.7
SL 1.7
Laws of exponents with rational exponents.
Laws of exponents with rational exponents.
Laws of logarithms.
Laws of logarithms.
logaxy = logax + logay
logaxy = logax + logay
logax/y = logax – logay
logax/y = logax – logay
logaxm = mlogax for a, x, y > 0
logaxm = mlogax for a, x, y > 0
Change of base of a logarithm.
Change of base of a logarithm.
logax = logbx/logba, for a, b, x > 0
logax = logbx/logba, for a, b, x > 0
Solving exponential equations, including using logarithms.
Solving exponential equations, including using logarithms.
SL 1.8
SL 1.8
Sum of infinite convergent geometric sequences.
Sum of infinite convergent geometric sequences.
SL 1.9
SL 1.9
The binomial theorem:
The binomial theorem:
expansion of (a + b)n, n is a natural number.
expansion of (a + b)n, n is a natural number.
Use of Pascal's triangle and nCr.
Use of Pascal's triangle and nCr.
AHL 1.10
AHL 1.10
Counting principles, including permutations and combinations.
Counting principles, including permutations and combinations.
Extension of the binomial theorem to fractional and negative indices, ie (a + b)n, n is a rational number.
Extension of the binomial theorem to fractional and negative indices, ie (a + b)n, n is a rational number.
AHL 1.11
AHL 1.11
Partial fractions.
Partial fractions.
AHL 1.12
AHL 1.12
Complex numbers: the number i such that i 2 = –1.
Complex numbers: the number i such that i 2 = –1.
Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
The complex plane.
The complex plane.
AHL 1.13
AHL 1.13
Modulus–argument (polar) form:
Modulus–argument (polar) form:
z = r(cosθ + isinθ) = rcisθ
z = r(cosθ + isinθ) = rcisθ
Euler form:
Euler form:
z = reiθ
z = reiθ
Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.
Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.
AHL 1.14
AHL 1.14
Complex conjugate roots of quadratic and polynomial equations with real coefficients.
Complex conjugate roots of quadratic and polynomial equations with real coefficients.
De Moivre's theorem and its extension to rational exponents.
De Moivre's theorem and its extension to rational exponents.
Powers and roots of complex numbers.
Powers and roots of complex numbers.
AHL 1.15
AHL 1.15
Proof by mathematical induction.
Proof by mathematical induction.
Proof by contradiction.
Proof by contradiction.
Use of a counterexample to show that a statement is not always true.
Use of a counterexample to show that a statement is not always true.
AHL 1.16
AHL 1.16
Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.
Solve by graphing
Solve by substitution
Solve by elimination
Solve using augmented matrices
Understand solutions
Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.
Solve by graphing
Solve by substitution
Solve by elimination
Solve using augmented matrices
Understand solutions
2
Functions
SL 2.1
SL 2.1
Different forms of the equation of a straight line.
Different forms of the equation of a straight line.
Gradient; intercepts.
Gradient; intercepts.
Parallel lines m1 = m2.
Parallel lines m1 = m2.
Perpendicular lines m1 × m2 = –1.
Perpendicular lines m1 × m2 = –1.
SL 2.2
SL 2.2
Concept of a function, domain, range and graph.
Concept of a function, domain, range and graph.
Function notation, for example f(x) , v(t) , C(n).
Function notation, for example f(x) , v(t) , C(n).
The concept of a function as a mathematical model.
The concept of a function as a mathematical model.
Informal concept that an inverse function reverses or undoes the effect of a function.
Informal concept that an inverse function reverses or undoes the effect of a function.
Inverse function as a reflection in the line y = x, and the notation f–1(x).
Inverse function as a reflection in the line y = x, and the notation f–1(x).
SL 2.3
SL 2.3
The graph of a function; its equation y = f(x).
Linear functions
Absolute value functions
Piecewise-defined functions
Quadratic functions
Polynomial functions
Radical functions
Exponential and logarithmic functions
The graph of a function; its equation y = f(x).
Linear functions
Absolute value functions
Piecewise-defined functions
Quadratic functions
Polynomial functions
Radical functions
Exponential and logarithmic functions
Creating a sketch from information given or a context, including transferring a graph from screen to paper.
Creating a sketch from information given or a context, including transferring a graph from screen to paper.
Using technology to graph functions including their sums and differences.
Using technology to graph functions including their sums and differences.
SL 2.4
SL 2.4
Determine key features of graphs.
Linear functions
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Rational functions
Exponential and logarithmic functions
Determine key features of graphs.
Linear functions
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Rational functions
Exponential and logarithmic functions
Finding the point of intersection of two curves or lines using technology.
Finding the point of intersection of two curves or lines using technology.
SL 2.5
SL 2.5
Composite functions.
Composite functions.
Identity function. Finding the inverse function f–1(x).
Identity function. Finding the inverse function f–1(x).
SL 2.6
SL 2.6
The quadratic function f(x) = ax2 + bx + c: its graph, y-intercept (0, c). Axis of symmetry.
The quadratic function f(x) = ax2 + bx + c: its graph, y-intercept (0, c). Axis of symmetry.
The form f(x) = a(x – p)(x – q), x-intercepts (p, 0) and (q, 0).
The form f(x) = a(x – p)(x – q), x-intercepts (p, 0) and (q, 0).
The form f(x) = a(x – h)2 + k, vertex (h, k).
The form f(x) = a(x – h)2 + k, vertex (h, k).
SL 2.7
SL 2.7
Solution of quadratic equations and inequalities.
Quadratic equations
Quadratic inequalities
Solution of quadratic equations and inequalities.
Quadratic equations
Quadratic inequalities
The quadratic formula.
The quadratic formula.
The discriminant Δ = b2 – 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
The discriminant Δ = b2 – 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
SL 2.8
SL 2.8
The reciprocal function f(x) = 1/x, x ≠ 0: its graph and self-inverse nature.
The reciprocal function f(x) = 1/x, x ≠ 0: its graph and self-inverse nature.
Rational functions of the form f(x) = ax + b/cx + d and their graphs.
Rational functions of the form f(x) = ax + b/cx + d and their graphs.
Equations of vertical and horizontal asymptotes.
Equations of vertical and horizontal asymptotes.
SL 2.9
SL 2.9
Exponential functions and their graphs:
Exponential functions and their graphs:
f(x) = ax, a > 0, f(x) = ex
f(x) = ax, a > 0, f(x) = ex
Logarithmic functions and their graphs:
Logarithmic functions and their graphs:
f(x) = logax, x > 0, f(x) = lnx, x > 0.
f(x) = logax, x > 0, f(x) = lnx, x > 0.
SL 2.10
SL 2.10
Solving equations, both graphically and analytically.
Linear equations
Absolute value equations
Polynomial equations
Radical equations
Variation equations
Exponential and logarithmic equations
Trigonometric equations
Solving equations, both graphically and analytically.
Linear equations
Absolute value equations
Polynomial equations
Radical equations
Variation equations
Exponential and logarithmic equations
Trigonometric equations
Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
Applications of graphing skills and solving equations that relate to real-life situations.
Applications of graphing skills and solving equations that relate to real-life situations.
SL 2.11
SL 2.11
Transformations of graphs.
Transformations of graphs.
Translations:
Translations:
y = f(x) + b; y = f(x – a).
y = f(x) + b; y = f(x – a).
Reflections (in both axes):
Reflections (in both axes):
y = – f(x); y = f( – x).
y = – f(x); y = f( – x).
Vertical stretch with scale factor p:
Vertical stretch with scale factor p:
y = pf(x).
y = pf(x).
Horizontal stretch with scale factor 1/q :
Horizontal stretch with scale factor 1/q :
y = f(qx).
y = f(qx).
Composite transformations.
Composite transformations.
AHL 2.12
AHL 2.12
Polynomial functions, their graphs and equations; zeros, roots and factors.
Equations
Roots and factors
Graphs
Domain and range
Polynomial functions, their graphs and equations; zeros, roots and factors.
Equations
Roots and factors
Graphs
Domain and range
The factor and remainder theorems.
The factor and remainder theorems.
Sum and product of the roots of polynomial equations.
Sum and product of the roots of polynomial equations.
AHL 2.13
AHL 2.13
Rational functions of the form f(x) = ax + b/cx² + dx + e, and f(x) = ax² + bx + c/dx + e.
Rational functions of the form f(x) = ax + b/cx² + dx + e, and f(x) = ax² + bx + c/dx + e.
AHL 2.14
AHL 2.14
Odd and even functions.
Odd and even functions.
Finding the inverse function, f–1(x), including domain restriction.
Finding the inverse function, f–1(x), including domain restriction.
Self-inverse functions.
Self-inverse functions.
AHL 2.15
AHL 2.15
Solutions of g(x) ≥ f(x), both graphically and analytically.
Linear inequalities
Quadratic inequalities
Higher-degree inequalities
Solutions of g(x) ≥ f(x), both graphically and analytically.
Linear inequalities
Quadratic inequalities
Higher-degree inequalities
AHL 2.16
AHL 2.16
The graphs of the functions, y = |f(x)| and y = f(|x|), y = 1/f(x), y = f(ax + b), y = [f(x)]2.
The graphs of the functions, y = |f(x)| and y = f(|x|), y = 1/f(x), y = f(ax + b), y = [f(x)]2.
Solution of modulus equations and inequalities.
Absolute value equations
Absolute value inequalities
Solution of modulus equations and inequalities.
Absolute value equations
Absolute value inequalities
3
Geometry and trigonometry
SL 3.1
SL 3.1
The distance between two points in three-dimensional space, and their midpoint.
The distance between two points in three-dimensional space, and their midpoint.
Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.
Surface area
Volume
Surface area and volume
Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.
Surface area
Volume
Surface area and volume
The size of an angle between two intersecting lines or between a line and a plane.
The size of an angle between two intersecting lines or between a line and a plane.
SL 3.2
SL 3.2
Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
The sine rule:
The sine rule:
a/sinA = b/sinB = c/sinC
a/sinA = b/sinB = c/sinC
The cosine rule:
The cosine rule:
c2 = a2 + b2 – 2abcosC; cosC = a² + b² – c²/2ab.
c2 = a2 + b2 – 2abcosC; cosC = a² + b² – c²/2ab.
Area of a triangle as 1/2absinC.
Area of a triangle as 1/2absinC.
SL 3.3
SL 3.3
Applications of right and non-right angled trigonometry, including Pythagoras's theorem.
Applications of right and non-right angled trigonometry, including Pythagoras's theorem.
Angles of elevation and depression.
Angles of elevation and depression.
Construction of labelled diagrams from written statements.
Construction of labelled diagrams from written statements.
SL 3.4
SL 3.4
The circle: radian measure of angles; length of an arc; area of a sector.
The circle: radian measure of angles; length of an arc; area of a sector.
SL 3.5
SL 3.5
Definition of cosθ, sinθ in terms of the unit circle.
Definition of cosθ, sinθ in terms of the unit circle.
Definition of tanθ as sinθ/cosθ.
Definition of tanθ as sinθ/cosθ.
Exact values of trigonometric ratios of 0, π/6, π/4, π/3, π/2 and their multiples.
Exact values of trigonometric ratios of 0, π/6, π/4, π/3, π/2 and their multiples.
Extension of the sine rule to the ambiguous case.
Extension of the sine rule to the ambiguous case.
SL 3.6
SL 3.6
The Pythagorean identity cos2θ + sin2θ = 1.
The Pythagorean identity cos2θ + sin2θ = 1.
Double angle identities for sine and cosine.
Double angle identities for sine and cosine.
The relationship between trigonometric ratios.
The relationship between trigonometric ratios.
SL 3.7
SL 3.7
The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs.
The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs.
Composite functions of the form f(x) = asin(b(x + c)) + d.
Sine functions
Cosine functions
Sine and cosine functions
Composite functions of the form f(x) = asin(b(x + c)) + d.
Sine functions
Cosine functions
Sine and cosine functions
Transformations.
Transformations.
Real-life contexts.
Real-life contexts.
SL 3.8
SL 3.8
Solving trigonometric equations in a finite interval, both graphically and analytically.
Solving trigonometric equations in a finite interval, both graphically and analytically.
Equations leading to quadratic equations in sinx, cosx or tanx.
Equations leading to quadratic equations in sinx, cosx or tanx.
AHL 3.9
AHL 3.9
Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.
Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.
Pythagorean identities:
Pythagorean identities:
1 + tan2θ = sec2θ
1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ
1 + cot2θ = cosec2θ
The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs.
The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs.
AHL 3.10
AHL 3.10
Compound angle identities.
Compound angle identities.
Double angle identity for tan.
Double angle identity for tan.
AHL 3.11
AHL 3.11
Relationships between trigonometric functions and the symmetry properties of their graphs.
Relationships between trigonometric functions and the symmetry properties of their graphs.
AHL 3.12
AHL 3.12
Concept of a vector; position vectors; displacement vectors.
Concept of a vector; position vectors; displacement vectors.
Representation of vectors using directed line segments.
Representation of vectors using directed line segments.
Base vectors i, j, k.
Base vectors i, j, k.
Components of a vector:
Components of a vector:
v = v1i + v2j + v3k.
v = v1i + v2j + v3k.
Algebraic and geometric approaches to the following:
Algebraic and geometric approaches to the following:
the sum and difference of two vectors
the sum and difference of two vectors
the zero vector 0, the vector –v
the zero vector 0, the vector –v
multiplication by a scalar, kv, parallel vectors
multiplication by a scalar, kv, parallel vectors
magnitude of a vector, |v|; unit vectors, v/|v|
Magnitude of a vector
Unit vectors
magnitude of a vector, |v|; unit vectors, v/|v|
Magnitude of a vector
Unit vectors
position vectors OA = a, OB = b
position vectors OA = a, OB = b
displacement vector AB = b – a
displacement vector AB = b – a
Proofs of geometrical properties using vectors.
Proofs of geometrical properties using vectors.
AHL 3.13
AHL 3.13
The definition of the scalar product of two vectors.
The definition of the scalar product of two vectors.
The angle between two vectors.
The angle between two vectors.
Perpendicular vectors; parallel vectors.
Perpendicular vectors; parallel vectors.
AHL 3.14
AHL 3.14
Vector equation of a line in two and three dimensions:
Vector equation of a line in two and three dimensions:
r = a + λb.
r = a + λb.
The angle between two lines.
The angle between two lines.
Simple applications to kinematics.
Simple applications to kinematics.
AHL 3.15
AHL 3.15
Coincident, parallel, intersecting and skew lines, distinguishing between these cases.
Coincident, parallel, intersecting and skew lines, distinguishing between these cases.
Points of intersection.
Points of intersection.
AHL 3.16
AHL 3.16
The definition of the vector product of two vectors.
The definition of the vector product of two vectors.
Properties of the vector product.
Properties of the vector product.
Geometric interpretation of | v × w |.
Geometric interpretation of | v × w |.
AHL 3.17
AHL 3.17
Vector equations of a plane:
Vector equations of a plane:
r = a + λb + μc, where b and c are non-parallel vectors within the plane.
r = a + λb + μc, where b and c are non-parallel vectors within the plane.
r ⋅ n = a ⋅ n, where n is a normal to the plane and a is the position vector of a point on the plane.
r ⋅ n = a ⋅ n, where n is a normal to the plane and a is the position vector of a point on the plane.
Cartesian equation of a plane ax + by + cz = d.
Cartesian equation of a plane ax + by + cz = d.
AHL 3.18
AHL 3.18
Intersections of: a line with a plane; two planes; three planes.
Intersections of: a line with a plane; two planes; three planes.
Angle between: a line and a plane; two planes.
Angle between: a line and a plane; two planes.
4
Statistics and probability
SL 4.1
SL 4.1
Concepts of population, sample, random sample, discrete and continuous data.
Concepts of population, sample, random sample, discrete and continuous data.
Reliability of data sources and bias in sampling.
Reliability of data sources and bias in sampling.
Interpretation of outliers.
Interpretation of outliers.
Sampling techniques and their effectiveness.
Sampling techniques and their effectiveness.
SL 4.2
SL 4.2
Presentation of data (discrete and continuous): frequency distributions (tables).
Presentation of data (discrete and continuous): frequency distributions (tables).
Histograms.
Histograms.
Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).
Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).
Production and understanding of box and whisker diagrams.
Production and understanding of box and whisker diagrams.
SL 4.3
SL 4.3
Measures of central tendency (mean, median and mode).
Measures of central tendency (mean, median and mode).
Estimation of mean from grouped data.
Estimation of mean from grouped data.
Modal class.
Modal class.
Measures of dispersion (interquartile range, standard deviation and variance).
Measures of dispersion (interquartile range, standard deviation and variance).
Effect of constant changes on the original data.
Effect of constant changes on the original data.
Quartiles of discrete data.
Quartiles of discrete data.
SL 4.4
SL 4.4
Linear correlation of bivariate data.
Linear correlation of bivariate data.
Pearson's product-moment correlation coefficient, r.
Pearson's product-moment correlation coefficient, r.
Scatter diagrams; lines of best fit, by eye, passing through the mean point.
Scatter diagrams; lines of best fit, by eye, passing through the mean point.
Equation of the regression line of y on x.
Equation of the regression line of y on x.
Use of the equation of the regression line for prediction purposes.
Use of the equation of the regression line for prediction purposes.
Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.
Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.
SL 4.5
SL 4.5
Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.
Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.
The probability of an event A is P(A) = n(A)/n(U).
The probability of an event A is P(A) = n(A)/n(U).
The complementary events A and A' (not A).
The complementary events A and A' (not A).
Expected number of occurrences.
Expected number of occurrences.
SL 4.6
SL 4.6
Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
Combined events:
Combined events:
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = P(A) + P(B) – P(A and B)
Mutually exclusive events:
Mutually exclusive events:
P(A and B) = 0
P(A and B) = 0
Conditional probability:
Conditional probability:
P(A | B) = P(A and B)/P(B)
P(A | B) = P(A and B)/P(B)
Independent events:
Independent events:
P(A and B) = P(A)P(B)
P(A and B) = P(A)P(B)
SL 4.7
SL 4.7
Concept of discrete random variables and their probability distributions.
Concept of discrete random variables and their probability distributions.
Expected value (mean), for discrete data.
Expected value (mean), for discrete data.
Applications.
Applications.
SL 4.8
SL 4.8
Binomial distribution.
Binomial distribution.
Mean and variance of the binomial distribution.
Mean and variance of the binomial distribution.
SL 4.9
SL 4.9
The normal distribution and curve.
The normal distribution and curve.
Properties of the normal distribution.
Properties of the normal distribution.
Diagrammatic representation.
Diagrammatic representation.
Normal probability calculations.
Normal probability calculations.
Inverse normal calculations.
Inverse normal calculations.
SL 4.10
SL 4.10
Equation of the regression line of x on y.
Equation of the regression line of x on y.
Use of the equation for prediction purposes.
Use of the equation for prediction purposes.
SL 4.11
SL 4.11
Formal definition and use of the formulae:
Formal definition and use of the formulae:
P(A | B) = P(A and B)/P(B) for conditional probabilities, and P(A | B) = P(A) = P(A | B ') for independent events
P(A | B) = P(A and B)/P(B) for conditional probabilities, and P(A | B) = P(A) = P(A | B ') for independent events
SL 4.12
SL 4.12
Standardization of normal variables (z-values).
Standardization of normal variables (z-values).
Inverse normal calculations where mean and standard deviation are unknown.
Inverse normal calculations where mean and standard deviation are unknown.
AHL 4.13
AHL 4.13
Use of Bayes' theorem for a maximum of three events.
Use of Bayes' theorem for a maximum of three events.
AHL 4.14
AHL 4.14
Variance of a discrete random variable.
Variance of a discrete random variable.
Continuous random variables and their probability density functions.
Continuous random variables and their probability density functions.
Mode and median of continuous random variables.
Mode and median of continuous random variables.
Mean, variance and standard deviation of both discrete and continuous random variables.
Mean, variance and standard deviation of both discrete and continuous random variables.
The effect of linear transformations of X.
The effect of linear transformations of X.
5
Calculus
SL 5.1
SL 5.1
Introduction to the concept of a limit.
Introduction to the concept of a limit.
Derivative interpreted as gradient function and as rate of change.
Derivative interpreted as gradient function and as rate of change.
SL 5.2
SL 5.2
Increasing and decreasing functions.
Increasing and decreasing functions.
Graphical interpretation of f '(x) > 0, f '(x) = 0, f '(x) < 0.
Graphical interpretation of f '(x) > 0, f '(x) = 0, f '(x) < 0.
SL 5.3
SL 5.3
Derivative of f(x) = axn is f '(x) = anxn –1, n is an integer.
Derivative of f(x) = axn is f '(x) = anxn –1, n is an integer.
The derivative of functions of the form f(x) = axn + bxn – 1 … where all exponents are integers.
The derivative of functions of the form f(x) = axn + bxn – 1 … where all exponents are integers.
SL 5.4
SL 5.4
Tangents and normals at a given point, and their equations.
Tangents and normals at a given point, and their equations.
SL 5.5
SL 5.5
Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn – 1 + …, where n is an integer, n ≠ – 1.
Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn – 1 + …, where n is an integer, n ≠ – 1.
Anti-differentiation with a boundary condition to determine the constant term.
Anti-differentiation with a boundary condition to determine the constant term.
Definite integrals using technology.
Definite integrals using technology.
Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0.
Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0.
SL 5.6
SL 5.6
Derivative of xn (n is a rational number), sinx, cosx, ex and lnx.
Derivative of xn (n is a rational number), sinx, cosx, ex and lnx.
Differentiation of a sum and a multiple of these functions.
Differentiation of a sum and a multiple of these functions.
The chain rule for composite functions.
The chain rule for composite functions.
The product and quotient rules.
The product and quotient rules.
SL 5.7
SL 5.7
The second derivative.
The second derivative.
Graphical behaviour of functions, including the relationship between the graphs of f , f ' and f ''.
Graphical behaviour of functions, including the relationship between the graphs of f , f ' and f ''.
SL 5.8
SL 5.8
Local maximum and minimum points.
Local maximum and minimum points.
Testing for maximum and minimum.
Testing for maximum and minimum.
Optimization.
Optimization.
Points of inflexion with zero and non-zero gradients.
Points of inflexion with zero and non-zero gradients.
SL 5.9
SL 5.9
Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.
Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.
SL 5.10
SL 5.10
Indefinite integral of xn (n is a rational number), sinx, cosx, 1/x and ex.
Indefinite integral of xn (n is a rational number), sinx, cosx, 1/x and ex.
The composites of any of these with the linear function ax + b.
The composites of any of these with the linear function ax + b.
Integration by inspection (reverse chain rule) or by substitution for expressions of the form:
Integration by inspection (reverse chain rule) or by substitution for expressions of the form:
∫ kg '(x)f(g(x))dx.
∫ kg '(x)f(g(x))dx.
SL 5.11
SL 5.11
Definite integrals, including analytical approach.
Definite integrals, including analytical approach.
Areas of a region enclosed by a curve y = f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology.
Areas of a region enclosed by a curve y = f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology.
Areas between curves.
Areas between curves.
AHL 5.12
AHL 5.12
Informal understanding of continuity and differentiability of a function at a point.
Informal understanding of continuity and differentiability of a function at a point.
Understanding of limits (convergence and divergence).
Limits of functions
Limits at vertical asymptotes
Understanding of limits (convergence and divergence).
Limits of functions
Limits at vertical asymptotes
Definition of derivative from first principles f′(x) = limit as h approaches 0 of f(x + h) – f(x)/h.
Definition of derivative from first principles f′(x) = limit as h approaches 0 of f(x + h) – f(x)/h.
Higher derivatives.
Higher derivatives.
AHL 5.13
AHL 5.13
The evaluation of limits of the form limit as x approaches a of f(x)/g(x) and limit as x approaches infinity of f(x)/g(x) using l'Hôpital's rule or the Maclaurin series.
The evaluation of limits of the form limit as x approaches a of f(x)/g(x) and limit as x approaches infinity of f(x)/g(x) using l'Hôpital's rule or the Maclaurin series.
Repeated use of l'Hôpital's rule.
Repeated use of l'Hôpital's rule.
AHL 5.14
AHL 5.14
Implicit differentiation.
Implicit differentiation.
Related rates of change.
Related rates of change.
Optimisation problems.
Optimisation problems.
AHL 5.15
AHL 5.15
Derivatives of tanx, secx, cosecx, cotx, ax, logax, arcsinx, arccosx, arctanx.
Derivatives of tanx, secx, cosecx, cotx, ax, logax, arcsinx, arccosx, arctanx.
Indefinite integrals of the derivatives of any of the above functions.
Indefinite integrals of the derivatives of any of the above functions.
The composites of any of these with a linear function.
The composites of any of these with a linear function.
Use of partial fractions to rearrange the integrand.
Use of partial fractions to rearrange the integrand.
AHL 5.16
AHL 5.16
Integration by substitution.
Integration by substitution.
Integration by parts.
Integration by parts.
Repeated integration by parts.
Repeated integration by parts.
AHL 5.17
AHL 5.17
Area of the region enclosed by a curve and the y-axis in a given interval.
Area 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.
Volumes of revolution about the x-axis or y-axis.
AHL 5.18
AHL 5.18
First order differential equations.
First order differential equations.
Numerical solution of dy/dx= f(x, y) using Euler's method.
Numerical solution of dy/dx= f(x, y) using Euler's method.
Variables separable.
Variables separable.
Homogeneous differential equation dy/dx= f(y/x) using the substitution y = vx.
Homogeneous differential equation dy/dx= f(y/x) using the substitution y = vx.
Solution of y′ + P(x)y = Q(x), using the integrating factor.
Solution of y′ + P(x)y = Q(x), using the integrating factor.
AHL 5.19
AHL 5.19
Maclaurin series to obtain expansions for ex, sinx, cosx, ln(1 + x), (1 + x)p, p is a rational number.
Maclaurin series to obtain expansions for ex, sinx, cosx, ln(1 + x), (1 + x)p, p is a rational number.
Use of simple substitution, products, integration and differentiation to obtain other series.
Use of simple substitution, products, integration and differentiation to obtain other series.
Maclaurin series developed from differential equations.
Maclaurin series developed from differential equations.
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