Syllabus map · 83 topics · 32 HL only

IB Maths AA (Analysis and Approaches) syllabus map

IB Mathematics: Analysis and Approaches is the more algebraic of the two Diploma maths courses, with an emphasis on exact working, proof and calculus. The map below follows Heptavo's topic list through the five topics: each sub-topic shows what questions on it ask, what you need to be able to do, the terms to use and the classic trap, and HL-only sub-topics are marked HL.

Try 4 questions How it's assessed

How it's assessed

Component	SL	HL
Paper 1 (no calculator)	1 h 30, 80 marks, 40%	2 h, 110 marks, 30%
Paper 2 (calculator)	1 h 30, 80 marks, 40%	2 h, 110 marks, 30%
Paper 3 (calculator)	—	1 h 15, 55 marks, 20%
Internal assessment: mathematical exploration	20%	20%

These figures are for the current course, examined up to November 2028. A revised course is first examined in May 2029, with fewer marks per paper, a 1-hour Paper 3, and financial applications and bivariate data no longer in AA.

Tap a topic to see what it asks, what you need to be able to do, the key terms and the trap. Topic titles follow the course structure; all explanations are ours.

Topic 1 — Number and algebra

Sequences, exponents and logarithms, simple proof and the binomial theorem, with counting, complex numbers, formal proof and linear systems at HL.

1.1Standard form and laws of exponentsSL

Writing numbers as a × 10ᵏ with 1 ≤ a < 10 and simplifying powers with the laws of exponents; questions ask you to calculate in standard form, simplify expressions and solve equations by matching bases.

You need to be able to

Multiply or divide in standard form by handling the front numbers and the powers of 10 separately, then readjust so that 1 ≤ a < 10.
Apply aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ, including to every factor inside a bracket.
Use a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(m/n) = (ⁿ√a)ᵐ, and solve equations such as 9ˣ = 27ˣ⁻² by rewriting both sides in base 3.

Key terms

standard form
exponent (index)
base
reciprocal
rational exponent
nth root

Common trap In (2x³)⁴ students raise only x³ to the power 4 and write 2x¹²; the power applies to every factor in the bracket, so the answer is 16x¹².

Practise 1.1 in Heptavo →

1.2Arithmetic sequences and seriesSL

Sequences that go up or down by a constant difference, and their sums; questions ask for a term, a sum or the number of terms, often from two given facts, and use sigma notation.

You need to be able to

Use uₙ = u₁ + (n − 1)d and Sₙ = (n/2)(2u₁ + (n − 1)d) = (n/2)(u₁ + uₙ), forming simultaneous equations when two facts are given.
Recover a single term from sums with uₙ = Sₙ − Sₙ₋₁.
Evaluate a sum written in sigma notation, counting the terms as top − bottom + 1.

Key terms

common difference
nth term
arithmetic series
sigma notation
number of terms

Common trap Using nd instead of (n − 1)d: the 10th term is nine steps from the first, so u₁₀ = u₁ + 9d.

Practise 1.2 in Heptavo →

1.3Geometric sequences and seriesSL

Sequences with a constant ratio and their finite sums; questions ask for terms, the ratio, a sum, or how many terms are needed to pass a given value.

You need to be able to

Use uₙ = u₁rⁿ⁻¹ and Sₙ = u₁(rⁿ − 1)/(r − 1) for r ≠ 1.
Find r by dividing a term by the one before it, or from two non-adjacent terms with uₘ ÷ uₖ = rᵐ⁻ᵏ, keeping both signs of r when the gap is even.
Solve for n when it sits in the exponent, using logarithms or a GDC, and round to the right whole number for the question.

Key terms

common ratio
geometric series
finite sum
multiplier
alternating sequence

Common trap Finding r by subtracting consecutive terms; the ratio comes from dividing a term by the one before it.

Practise 1.3 in Heptavo →

1.4Financial applications of sequencesSL

Compound interest and depreciation as geometric sequences; questions ask for a future value, an interest rate or the time taken to reach a target, usually with a GDC finance solver.

You need to be able to

Use FV = PV × (1 + r/(100k))ᵏⁿ, changing the rate per period and the number of periods together when interest is compounded k times a year.
Model depreciation at a fixed percentage as repeated multiplication by 1 − r/100.
Solve for the number of periods with logarithms or the GDC, and round up to the first whole period in which the target is reached.

Key terms

compound interest
present value
future value
depreciation
compounding period
interest rate

Common trap Rounding the number of years down: if the solution is n = 7.3, the target is first reached after 8 whole years, not 7.

Practise 1.4 in Heptavo →

1.5Exponents and logarithmsSL

The logarithm as an exponent and its link to exponentials; questions ask you to evaluate logarithms exactly and solve simple exponential equations.

You need to be able to

Switch freely between aˣ = b and x = logₐ b.
Evaluate logarithms such as log₈ 2 = 1/3 or ln(1/e) = −1 without a calculator by writing the number as a power of the base.
Solve aˣ = b by matching bases when possible, otherwise by taking logarithms of both sides.

Key terms

logarithm
base
natural logarithm (ln)
exponent
inverse function

Common trap Forgetting that logₐ x exists only for x > 0; any solution that makes the argument zero or negative has to be rejected.

Practise 1.5 in Heptavo →

1.6Simple deductive proofSL

Short proofs that an identity holds or that a statement is true for all integers; questions ask you to show LHS ≡ RHS or prove a property with general algebra.

You need to be able to

Represent even numbers as 2n, odd numbers as 2n + 1 and consecutive integers as n, n + 1, n + 2.
Set out a proof that starts from one side and transforms it line by line into the other side.
Explain why checking examples is not a proof, while one counterexample is enough to disprove a claim.

Key terms

identity (≡)
deductive proof
LHS and RHS
integer
counterexample
converse

Common trap Working on both sides at once until something true like 0 = 0 appears; that assumes the result, so start from one side and finish at the other.

Practise 1.6 in Heptavo →

1.7Laws of exponents and logarithmsSL

The product, quotient and power laws of logarithms and the change of base formula; questions ask you to combine or expand logarithms and solve logarithmic and exponential equations.

You need to be able to

Use log(xy) = log x + log y, log(x/y) = log x − log y and log xᵐ = m log x, all in one base.
Combine an expression into a single logarithm, then rewrite it in exponential form to solve.
Apply logₐ x = log x ÷ log a to evaluate or solve, for example log₉ 27 = 3/2.

Key terms

product law
quotient law
power law
change of base
single logarithm

Common trap Splitting log(x + y) into log x + log y; there is no law for the logarithm of a sum.

Practise 1.7 in Heptavo →

1.8Infinite convergent geometric sequencesSL

When an infinite geometric series has a finite sum, and what that sum is; questions ask for S∞, the condition on r, or turn a recurring decimal into a fraction.

You need to be able to

Use S∞ = u₁/(1 − r), which is valid only when |r| < 1.
Find the values of x for which a series with ratio in terms of x converges by solving |r| < 1.
Write a recurring decimal such as 0.272727… as a geometric series and sum it to get 3/11.

Key terms

convergent
divergent
sum to infinity
common ratio
|r| < 1

Common trap Applying S∞ = u₁/(1 − r) when |r| ≥ 1; the formula still produces a number, but the series has no sum.

Practise 1.8 in Heptavo →

1.9The binomial theoremSL

Expanding (a + b)ⁿ for a positive integer n with binomial coefficients; questions ask for a full expansion, one coefficient or the constant term.

You need to be able to

Find coefficients from Pascal's triangle or ⁿCᵣ = n!/(r!(n − r)!).
Write the general term ⁿCᵣ aⁿ⁻ʳ bʳ, collect the powers of x and solve for r to find the term you need.
Handle terms like 3/x by writing them as 3x⁻¹ before collecting powers, as in the constant term of (2x − 3/x)⁶.

Key terms

binomial coefficient
ⁿCᵣ
Pascal's triangle
general term
constant term

Common trap Writing 2x² instead of (2x)² in a term: the number inside the bracket is raised to the power as well.

Practise 1.9 in Heptavo →

1.10Counting principles and the extended binomial theoremHL

Counting arrangements and selections, and the binomial expansion for negative or fractional n; questions ask you to count cases with restrictions, or give the first terms of a series and where it is valid.

You need to be able to

Decide whether order matters and use ⁿPᵣ or ⁿCᵣ, multiplying successive stages and adding separate cases.
Handle restrictions by treating items that must stay together as one block, or by subtracting the complement.
Expand (1 + x)ⁿ for rational n as an infinite series valid for |x| < 1, rewriting (a + bx)ⁿ as aⁿ(1 + bx/a)ⁿ first.

Key terms

permutation
combination
factorial
extended binomial theorem
interval of validity

Common trap Stating |x| < 1 for (2 + x)⁻¹; after rewriting it as ½(1 + x/2)⁻¹ the series is valid for |x| < 2.

Practise 1.10 in Heptavo →

1.11Partial fractionsHL

Splitting a rational expression into simpler fractions, usually so that it can be integrated; questions ask for the decomposition and then an integral.

You need to be able to

Factorise the denominator and write A/(x − a) + B/(x − b) for two distinct linear factors.
Find A and B by substituting the value that makes each bracket zero, or by comparing coefficients.
Divide first when the degree of the numerator is at least the degree of the denominator.

Key terms

proper fraction
improper fraction
linear factor
identity
comparing coefficients

Common trap Skipping the division for an improper fraction; the constants then cannot be found, so divide out first.

Practise 1.11 in Heptavo →

1.12Complex numbers: Cartesian formHL

Numbers a + bi with i² = −1, their arithmetic, conjugates and the Argand diagram; questions ask you to calculate, divide using the conjugate and solve equations by equating parts.

You need to be able to

Add, subtract and multiply complex numbers, replacing i² by −1.
Divide by multiplying top and bottom by the conjugate of the denominator.
Equate real and imaginary parts to find unknowns, and plot z on an Argand diagram.

Key terms

imaginary unit
real part
imaginary part
complex conjugate
Argand diagram
modulus

Common trap Giving the imaginary part of 3 − 4i as −4i; the imaginary part is the real number −4.

Practise 1.12 in Heptavo →

1.13Modulus–argument and Euler formHL

Writing complex numbers in modulus–argument form r cis θ and Euler form re^(iθ); questions ask you to convert between forms and multiply or divide with moduli and arguments.

You need to be able to

Find r = √(a² + b²) and the argument θ, choosing the quadrant from the signs of a and b.
Multiply by multiplying moduli and adding arguments; divide by dividing moduli and subtracting arguments.
Convert back to Cartesian form with a = r cos θ and b = r sin θ.

Key terms

modulus
argument
principal argument
cis θ
Euler form

Common trap Taking arctan(b/a) for −1 − i and getting π/4; the point is in the third quadrant, so its argument is −3π/4.

Practise 1.13 in Heptavo →

1.14De Moivre's theorem and roots of complex numbersHL

Powers and roots of complex numbers in polar form; questions ask for zⁿ, all the nth roots of a number, and identities for cos nθ or sin nθ.

You need to be able to

Apply (r cis θ)ⁿ = rⁿ cis nθ.
Find all n roots of a number: modulus ⁿ√r and arguments (θ + 2πk)/n for k = 0, 1, …, n − 1.
Expand (cos θ + i sin θ)ⁿ binomially and compare with De Moivre's theorem to write cos nθ or sin nθ in powers of cos θ and sin θ.

Key terms

De Moivre's theorem
nth root
roots of unity
regular polygon
principal argument

Common trap Giving a single root; a non-zero complex number has exactly n distinct nth roots, spaced 2π/n apart.

Practise 1.14 in Heptavo →

1.15Proof by induction, contradiction and counterexampleHL

Three formal methods of proof; questions ask for an induction proof (sums, divisibility, derivatives), a proof by contradiction, or a single counterexample.

You need to be able to

Write an induction proof with a base case, the assumption for n = k, a step to n = k + 1 that uses the assumption, and a conclusion.
Prove by contradiction by assuming the statement is false and reasoning to something impossible, as in the proof that √2 is irrational.
Disprove a general claim with one explicit counterexample, showing it meets the condition but not the conclusion.

Key terms

base case
inductive hypothesis
inductive step
contradiction
counterexample
irrational

Common trap Writing an inductive step that never uses the n = k assumption; the step must substitute it explicitly, and the conclusion must say the result holds for all n from the base value.

Practise 1.15 in Heptavo →

1.16Systems of linear equationsHL

Solving up to three linear equations in three unknowns and interpreting the result; questions ask for a unique solution, the condition for no or infinitely many solutions, and a general solution in a parameter.

You need to be able to

Use row reduction to solve a system of three equations by hand.
Recognise an inconsistent system (a row reading 0 = k with k ≠ 0) and one with infinitely many solutions (a row reading 0 = 0).
Write infinitely many solutions in terms of a parameter, and find the values of a constant that produce each case.

Key terms

row reduction
consistent
inconsistent
unique solution
parameter
general solution

Common trap Reading a row of 0 = 0 as 'no solution'; it means one equation is redundant, so there are infinitely many solutions.

Practise 1.16 in Heptavo →

Topic 2 — Functions

How functions behave and how to sketch, transform, combine and invert them, from straight lines and quadratics to polynomial and rational functions.

2.1Equations of a straight lineSL

Gradients, intercepts and the different forms of a straight line; questions ask for the equation through given points, parallel or perpendicular lines, and points of intersection.

You need to be able to

Find the gradient from two points and write the line as y = mx + c, y − y₁ = m(x − x₁) or ax + by + d = 0.
Use equal gradients for parallel lines and m₁m₂ = −1 for perpendicular lines.
Find where two lines meet by solving their equations simultaneously.

Key terms

gradient
y-intercept
point–gradient form
general form
perpendicular
negative reciprocal

Common trap For a line perpendicular to gradient 2/3, writing 3/2 or −2/3; you must invert and change the sign, giving −3/2.

Practise 2.1 in Heptavo →

2.2Functions, domain, range and inverseSL

The idea of a function, its notation, domain and range, and its inverse; questions ask you to state a domain or range, evaluate f(a), and find or sketch f⁻¹.

You need to be able to

Find the largest possible domain by excluding values that make a denominator zero, a square root's argument negative or a logarithm's argument non-positive.
State the range from the graph, including end values that are actually reached.
Find f⁻¹(x) by swapping x and y and rearranging, and sketch it as the reflection of f in y = x.

Key terms

domain
range
function notation
one-to-one
inverse function
f⁻¹

Common trap Reading f⁻¹(x) as 1/f(x); the inverse undoes f, it is not its reciprocal.

Practise 2.2 in Heptavo →

2.3The graph of a functionSL

Sketching and reading graphs of functions by hand and with a GDC; questions ask for intercepts, the shape implied by a factorised form, and a labelled sketch.

You need to be able to

Find x-intercepts by solving f(x) = 0 and the y-intercept from f(0).
Sketch a polynomial from its factors, crossing at single roots and touching at repeated ones.
Use a GDC to locate intercepts, turning points and asymptotes, then transfer them to a labelled sketch.

Key terms

sketch
intercept
root (zero)
repeated root
vertical asymptote
viewing window

Common trap Drawing a sketch with no values marked; a sketch earns marks only when intercepts, turning points and asymptotes are labelled.

Practise 2.3 in Heptavo →

2.4Key features of graphsSL

Identifying the key features of a graph: vertices, zeros, intercepts, symmetry, asymptotes and turning points; questions usually ask you to find them, often with a GDC, and use them in context.

You need to be able to

Find maxima, minima, zeros and intercepts with a GDC to 3 significant figures.
Identify vertical and horizontal asymptotes and describe what happens as x → ±∞.
Distinguish the x-coordinate of a turning point (where it occurs) from the y-coordinate (the maximum or minimum value).

Key terms

vertex
zero
axis of symmetry
horizontal asymptote
local maximum
local minimum

Common trap Giving the x-coordinate when asked for the maximum value; the maximum value is the y-coordinate.

Practise 2.4 in Heptavo →

2.5Composite and inverse functionsSL

Combining functions and reversing them; questions ask for f ∘ g, solving f(g(x)) = k, finding f⁻¹ and its domain, and using f(f⁻¹(x)) = x.

You need to be able to

Evaluate (f ∘ g)(x) = f(g(x)) by applying g first, checking that g's outputs are allowed into f.
Find an inverse algebraically and state its domain as the range of f.
Restrict the domain of a many-to-one function so that it has an inverse.

Key terms

composite function
inverse function
one-to-one
many-to-one
restricted domain
reflection in y = x

Common trap Working out (f ∘ g)(x) in the wrong order; the function written next to x acts first.

Practise 2.5 in Heptavo →

2.6The quadratic functionSL

The forms and graph of y = ax² + bx + c; questions ask for the vertex, the axis of symmetry and intercepts, and for converting between forms.

You need to be able to

Read the y-intercept from ax² + bx + c, the roots from a(x − p)(x − q) and the vertex from a(x − h)² + k.
Complete the square to find the vertex.
Use x = −b/(2a) for the axis of symmetry and substitute it to get the maximum or minimum value.

Key terms

parabola
vertex
axis of symmetry
completing the square
vertex form
factorised form

Common trap Reading the vertex of 2(x + 3)² − 5 as (3, −5); the sign inside the bracket flips, so the vertex is (−3, −5).

Practise 2.6 in Heptavo →

2.7Solving quadratic equations and the discriminantSL

Solving quadratic equations and using the discriminant Δ = b² − 4ac; questions ask for roots, the number of real roots, or the values of k that give equal, real or no roots.

You need to be able to

Solve by factorising or with x = (−b ± √Δ)/(2a).
Use Δ > 0, Δ = 0 and Δ < 0 to count real roots, and form and solve inequalities in k.
Set up a quadratic from a context such as an area or a projectile, and reject roots outside the sensible domain.

Key terms

discriminant
real roots
equal (repeated) roots
quadratic formula
factorise

Common trap Writing Δ ≥ 0 for 'two distinct real roots'; distinct roots need Δ > 0, while 'real roots' alone allows Δ ≥ 0.

Practise 2.7 in Heptavo →

2.8Reciprocal and rational functionsSL

The graphs of y = 1/x and y = (ax + b)/(cx + d); questions ask for asymptotes, intercepts and a sketch.

You need to be able to

State the vertical asymptote x = −d/c and the horizontal asymptote y = a/c.
Find the x-intercept where the numerator is zero and the y-intercept at y = b/d.
Sketch both branches, approaching the asymptotes in the correct regions.

Key terms

reciprocal function
rational function
vertical asymptote
horizontal asymptote
branch

Common trap Getting the sign of the vertical asymptote wrong in (2x + 1)/(x − 3); the denominator is zero at x = 3, not x = −3.

Practise 2.8 in Heptavo →

2.9Exponential and logarithmic functionsSL

Graphs and properties of aˣ, eˣ, logₐ x and ln x, and exponential models; questions ask for asymptotes, domains, inverses and parameters in context.

You need to be able to

Sketch y = aˣ (through (0, 1), asymptote y = 0) and y = logₐ x (through (1, 0), asymptote x = 0).
Use eˣ and ln x as inverse functions whose graphs reflect in y = x.
Interpret A and k in N = Ae^(kt) and solve for t with natural logarithms.

Key terms

exponential function
logarithmic function
asymptote
growth
decay
half-life

Common trap Giving x = −2 as the asymptote of y = ln(x − 2); the argument must be positive, so the asymptote is x = 2.

Practise 2.9 in Heptavo →

2.10Solving equations graphically and analyticallySL

Choosing between algebra and a GDC to solve equations; questions ask you to solve f(x) = g(x), count solutions, or use a discriminant to decide whether a line meets a curve.

You need to be able to

Solve f(x) = g(x) on a GDC as the x-coordinates of the intersection points.
Solve equations that are quadratic in form, such as e²ˣ − 5eˣ + 6 = 0, by substituting u = eˣ.
Use the discriminant of the resulting quadratic to decide whether a line is a tangent to a curve.

Key terms

intersection
analytic solution
graphical solution
substitution
tangent

Common trap Giving the y-coordinate of an intersection point as the solution; the solution of the equation is the x-value.

Practise 2.10 in Heptavo →

2.11Transformations of graphsSL

Translations, stretches and reflections of y = f(x); questions ask you to describe a transformation in words, find the image of a point, or write the new equation.

You need to be able to

Describe f(x − h) + k as a translation by the vector (h, k).
Describe af(x) as a vertical stretch with scale factor a and f(bx) as a horizontal stretch with scale factor 1/b.
Treat −f(x) and f(−x) as reflections in the x-axis and y-axis, and track a point through several transformations in order.

Key terms

translation
vertical stretch
horizontal stretch
reflection
scale factor
image

Common trap Describing f(2x) as a horizontal stretch with scale factor 2; it is scale factor ½, because every x-coordinate is halved.

Practise 2.11 in Heptavo →

2.12Polynomial functions and the factor theoremHL

Higher-degree polynomials, their roots, and the factor and remainder theorems; questions ask you to find unknown coefficients, factorise, and use the sum and product of roots.

You need to be able to

Use P(a) for the remainder on division by (x − a), and P(a) = 0 to show (x − a) is a factor.
Factorise a cubic or quartic once one root is known, by division or by comparing coefficients.
Use the sum of roots −aₙ₋₁/aₙ and the product of roots (−1)ⁿa₀/aₙ.

Key terms

factor theorem
remainder theorem
polynomial division
sum of roots
product of roots
degree

Common trap Substituting x = −2 to test the factor (x − 2); the factor (x − a) is tested with x = a.

Practise 2.12 in Heptavo →

2.13Rational functions and their asymptotesHL

Rational functions with linear or quadratic numerators and denominators; questions ask for vertical, horizontal and oblique asymptotes and a sketch.

You need to be able to

Find vertical asymptotes at zeros of the denominator that do not cancel with the numerator.
Find a horizontal asymptote by comparing the degrees of numerator and denominator.
Divide to find an oblique asymptote when the numerator's degree is one more than the denominator's.

Key terms

rational function
vertical asymptote
horizontal asymptote
oblique asymptote
polynomial division

Common trap Assuming a curve can never cross its horizontal asymptote; the asymptote describes behaviour as x → ±∞, and the curve may cross it for finite x.

Practise 2.13 in Heptavo →

2.14Odd and even functions, self-inverse functionsHL

Symmetry of functions and functions that are their own inverse; questions ask you to prove a function is odd or even, show it is self-inverse, or restrict a domain so that an inverse exists.

You need to be able to

Test f(−x) = f(x) for even functions (symmetric in the y-axis) and f(−x) = −f(x) for odd functions (half-turn symmetry about the origin).
Show a function is self-inverse by proving f(f(x)) = x.
Choose a domain restriction that makes a function one-to-one, then find its inverse.

Key terms

even function
odd function
self-inverse function
symmetry
one-to-one
domain restriction

Common trap Concluding a function is odd after checking a single value; you have to show f(−x) = −f(x) algebraically for every x.

Practise 2.14 in Heptavo →

2.15Solving inequalitiesHL

Polynomial and rational inequalities solved by algebra or from a graph; questions ask for solution sets such as (x − 1)(x + 2) > 0 or g(x) ≥ f(x).

You need to be able to

Find the critical values and test the sign of the expression on each interval.
Read the solution intervals from a sketch of the graph.
Solve g(x) ≥ f(x) on a GDC from the intersection points and the graphs between them.

Key terms

critical value
sign diagram
interval
solution set
strict inequality

Common trap Multiplying both sides by an expression whose sign is unknown, such as x − 3; move everything to one side and use a sign diagram instead.

Practise 2.15 in Heptavo →

2.16Graphs of related functionsHL

Sketching graphs related to y = f(x), such as |f(x)|, f(|x|), 1/f(x) and [f(x)]²; questions ask for a sketch with key features carried over correctly.

You need to be able to

Reflect the parts of y = f(x) below the x-axis upwards to get y = |f(x)|.
Keep the part of the graph with x ≥ 0 and mirror it in the y-axis to get y = f(|x|).
Turn zeros of f into vertical asymptotes and maxima into minima when sketching y = 1/f(x).

Key terms

modulus function
reciprocal graph
vertical asymptote
reflection
invariant point

Common trap Confusing |f(x)| with f(|x|): the first reflects negative heights in the x-axis, the second discards the left half and mirrors the right half.

Practise 2.16 in Heptavo →

Topic 3 — Geometry and trigonometry

Measuring triangles, circles and solids, and working with trigonometric functions and identities, plus vectors, lines and planes in space at HL.

3.1Three-dimensional geometry: distance, midpoint, solidsSL

Distance and midpoint in three dimensions, volumes and surface areas of solids, and angles inside solids; questions ask for lengths, angles, volumes and areas.

You need to be able to

Use d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²) and average each coordinate for a midpoint.
Calculate volumes and surface areas of prisms, pyramids, cylinders, cones, spheres and hemispheres.
Find the angle between a line and a plane, or between two lines, inside a solid using right-angled triangles.

Key terms

space diagonal
midpoint
slant height
surface area
volume
right pyramid

Common trap Using the slant height in the volume of a cone; the volume uses the vertical height and the curved surface area uses the slant height.

Practise 3.1 in Heptavo →

3.2The sine rule, cosine rule and area of a triangleSL

Solving triangles that are not right-angled and finding their area; questions ask for missing sides and angles, including the ambiguous case.

You need to be able to

Use the sine rule when a side and its opposite angle are known, and the cosine rule for two sides with the included angle or three sides.
Find the area with ½ab sin C, where C is the angle between a and b.
Recognise the ambiguous case and find both possible angles, θ and 180° − θ.

Key terms

sine rule
cosine rule
included angle
ambiguous case
opposite side
obtuse angle

Common trap Missing the second triangle in the ambiguous case; check whether 180° − θ still leaves a positive third angle.

Practise 3.2 in Heptavo →

3.3Applications of trigonometry: bearings, elevation, depressionSL

Using trigonometry in real situations; questions ask for heights, distances and bearings, often from a diagram you have to draw yourself.

You need to be able to

Measure bearings clockwise from north as three figures, and find a back bearing by adding or subtracting 180°.
Use angles of elevation and depression, both measured from the horizontal.
Combine two right-angled triangles that share a height to find an unknown height or distance.

Key terms

bearing
angle of elevation
angle of depression
horizontal
three-figure bearing

Common trap Measuring the angle of depression from the vertical; it is always measured down from the horizontal.

Practise 3.3 in Heptavo →

3.4Radian measure, arc length and sector areaSL

Measuring angles in radians and finding arc lengths and sector areas; questions ask for perimeters of sectors, areas of segments and conversions.

You need to be able to

Convert between degrees and radians using π radians = 180°.
Use l = rθ and A = ½r²θ with θ in radians.
Find the area of a segment as sector minus triangle: ½r²θ − ½r² sin θ.

Key terms

radian
arc length
sector
segment
chord
perimeter

Common trap Using l = rθ with θ in degrees; convert to radians first, or the answer is out by a factor of π/180.

Practise 3.4 in Heptavo →

3.5The unit circle and exact trigonometric valuesSL

Defining sine, cosine and tangent for any angle with the unit circle; questions ask for exact values and signs, often on the non-calculator paper.

You need to be able to

Recall exact values at 0, π/6, π/4, π/3 and π/2, and extend them to other quadrants.
Fix the sign from the quadrant and the size from the reference angle.
Find the other ratios exactly from one given ratio and its quadrant.

Key terms

unit circle
reference angle
exact value
quadrant
tan θ = sin θ/cos θ

Common trap Giving cos(2π/3) as ½; the angle is in the second quadrant, where cosine is negative, so it is −½.

Practise 3.5 in Heptavo →

3.6Pythagorean identity and double angle formulaeSL

The identity sin²θ + cos²θ = 1 and the double angle formulae; questions ask for exact values, simplifications and solutions of equations.

You need to be able to

Use sin²θ + cos²θ = 1 to find cos θ from sin θ, taking the sign from the quadrant.
Apply sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
Choose the form of cos 2θ that turns an equation into a quadratic in a single function.

Key terms

Pythagorean identity
double angle formula
exact value
quadrant
identity

Common trap Writing sin 2θ = 2 sin θ; the double angle formula is sin 2θ = 2 sin θ cos θ.

Practise 3.6 in Heptavo →

3.7Circular functions and their graphsSL

The graphs of sine, cosine and tangent and their transformations; questions ask for amplitude, period and principal axis, and to fit y = a sin(b(x − c)) + d to a context.

You need to be able to

Read the amplitude |a|, the period 2π/b and the principal axis y = d from the equation.
Find a and d from the maximum and minimum: a = (max − min)/2 and d = (max + min)/2.
Sketch y = tan x with period π and vertical asymptotes where cos x = 0.

Key terms

amplitude
period
principal axis
phase shift
horizontal stretch

Common trap Writing the period of sin 3x as 6π; the period is 2π divided by b, so 2π/3.

Practise 3.7 in Heptavo →

3.8Solving trigonometric equationsSL

Finding every solution of a trigonometric equation in a given interval; questions include linear, quadratic and double-angle equations.

You need to be able to

Find the principal value, then use symmetry for the second solution in each cycle.
For an equation such as sin 2x = k, widen the interval for 2x first and halve the answers at the end.
Solve quadratics in sin x or cos x by factorising, rejecting any value outside −1 to 1.

Key terms

principal value
interval
quadratic in sin x
reject
symmetry

Common trap Dividing both sides by sin x and losing the solutions where sin x = 0; factorise instead.

Practise 3.8 in Heptavo →

3.9Reciprocal and inverse trigonometric functionsHL

The reciprocal functions sec, cosec and cot, and the inverse functions arcsin, arccos and arctan; questions ask for exact values, identities and solutions of equations.

You need to be able to

Use sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ.
Apply 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
State the domain and range of arcsin, arccos and arctan, and evaluate them exactly.

Key terms

secant
cosecant
cotangent
arcsin
arccos
arctan

Common trap Giving arcsin(−½) as 7π/6; the range of arcsin is −π/2 to π/2, so the answer is −π/6.

Practise 3.9 in Heptavo →

3.10Compound angle identitiesHL

The expansions of sin(A ± B), cos(A ± B) and tan(A ± B); questions ask for exact values such as sin 75°, proofs of identities and solutions of equations.

You need to be able to

Apply each expansion with the correct signs.
Write angles such as 15° or 5π/12 as sums or differences of standard angles.
Derive the double angle formulae by setting B = A, and use the expansions in proofs.

Key terms

compound angle
expansion
identity
exact value
double angle formula

Common trap Writing cos(A + B) = cos A cos B + sin A sin B; the cosine expansion reverses the sign, so it is a minus.

Practise 3.10 in Heptavo →

3.11Symmetry properties and further trigonometric graphsHL

Symmetry properties of the trigonometric functions and graphs of sec, cosec and cot; questions ask you to simplify expressions such as sin(π − θ) and sketch the reciprocal graphs.

You need to be able to

Use sin(π − θ) = sin θ, cos(−θ) = cos θ, sin(−θ) = −sin θ and sin(π/2 − θ) = cos θ.
Sketch sec x, cosec x and cot x, with asymptotes where the partner function is zero.
Classify sine and tangent as odd functions and cosine as even.

Key terms

odd function
even function
complementary angle
reflection
asymptote

Common trap Drawing sec x with values between −1 and 1; its range is y ≤ −1 or y ≥ 1.

Practise 3.11 in Heptavo →

3.12Vectors: definitions and algebraHL

Vectors in two and three dimensions, their components, magnitudes and algebra; questions ask for a displacement vector, a magnitude, a unit vector or a test for parallel vectors.

You need to be able to

Find AB = b − a and its magnitude using Pythagoras in three dimensions.
Find a unit vector by dividing a vector by its magnitude.
Show that vectors are parallel by writing one as a scalar multiple of the other.

Key terms

position vector
displacement vector
magnitude
unit vector
scalar multiple
collinear

Common trap Calculating AB as a − b; a displacement is finish minus start, b − a.

Practise 3.12 in Heptavo →

3.13The scalar product and the angle between vectorsHL

The scalar product and the angle between two vectors; questions ask for a · b, an angle between vectors or lines, and perpendicularity.

You need to be able to

Compute a · b = a₁b₁ + a₂b₂ + a₃b₃.
Find the angle with cos θ = (a · b)/(|a||b|).
Use a · b = 0 to show that vectors are perpendicular or to find an unknown component.

Key terms

scalar (dot) product
angle between vectors
perpendicular
magnitude
acute
obtuse

Common trap Finding the angle between two lines from the position vectors of points on them; use their direction vectors.

Practise 3.13 in Heptavo →

3.14Vector equation of a lineHL

Describing lines as r = a + λb in two and three dimensions; questions ask you to write the equation, test whether a point lies on the line, and convert between forms.

You need to be able to

Write r = a + λb from a point and a direction, or from two points.
Test whether a point lies on a line by finding one value of λ that fits every component.
Convert between vector, parametric and Cartesian forms, and read r = a + tv as motion with velocity v.

Key terms

direction vector
position vector
parameter
parametric equations
Cartesian equation
velocity vector

Common trap Using a point on the line as its direction vector; the direction is the difference between two points.

Practise 3.14 in Heptavo →

3.15Coincident, parallel, intersecting and skew linesHL

Classifying pairs of lines in space and finding where they meet; questions ask whether two lines are the same, parallel, intersecting or skew, and for the point of intersection.

You need to be able to

Check whether the direction vectors are scalar multiples to test for parallel lines.
Solve two component equations for the two parameters and check the result in the third.
State the conclusion (coincident, parallel, intersecting or skew) with the reason.

Key terms

coincident
parallel
intersecting
skew
parameter

Common trap Using the same parameter for both lines; give them different letters (λ and μ), because each line reaches the shared point at its own parameter value.

Practise 3.15 in Heptavo →

3.16The vector productHL

The vector product a × b and its uses; questions ask for a vector perpendicular to two others and for areas of parallelograms and triangles.

You need to be able to

Compute a × b with the component formula.
Use |a × b| = |a||b| sin θ for areas: |a × b| for a parallelogram and ½|a × b| for a triangle.
Use a × b as a normal vector to the plane containing a and b.

Key terms

vector product
cross product
normal vector
right-hand rule
area of a parallelogram

Common trap Assuming a × b = b × a; reversing the order reverses the direction, so b × a = −(a × b).

Practise 3.16 in Heptavo →

3.17Vector equations of a planeHL

Equations of planes in vector, parametric and Cartesian form; questions ask for the equation of a plane through three points or with a given normal.

You need to be able to

Write r · n = a · n and expand it to ax + by + cz = d.
Find a normal from the vector product of two direction vectors lying in the plane.
Use the parametric form r = a + λb + μc.

Key terms

normal vector
Cartesian equation of a plane
scalar product form
parametric form
coplanar

Common trap Using a point's position vector as the normal; the normal must be perpendicular to the plane, usually found with a vector product.

Practise 3.17 in Heptavo →

3.18Intersections of lines and planes, angles between themHL

Where lines and planes meet and the angles between them; questions ask for the intersection of a line and a plane, of two or three planes, and for angles.

You need to be able to

Substitute a line's parametric equations into a plane's equation to find their point of intersection.
Find the line where two planes meet, using the vector product of the normals for its direction.
Find the angle between a line and a plane with sin θ = |d · n|/(|d||n|), and between two planes from their normals.

Key terms

line of intersection
normal
angle between a line and a plane
angle between planes
intersection of three planes

Common trap Giving the angle between the line's direction and the normal as the answer; that is the complement, so subtract it from 90°.

Practise 3.18 in Heptavo →

Topic 4 — Statistics and probability

Collecting and summarising data, regression, probability, and the binomial and normal distributions, with Bayes' theorem and continuous random variables at HL.

4.1Concepts of population, sample and sampling techniquesSL

How data are collected and how samples are chosen; questions ask you to identify a sampling method, calculate a stratified sample and explain sources of bias.

You need to be able to

Describe simple random, systematic, stratified, quota and convenience sampling.
Calculate the number from each stratum as (stratum size ÷ population size) × sample size.
Explain how bias arises and why a bigger sample does not remove it.

Key terms

population
sample
sampling frame
stratified sample
bias
discrete and continuous data

Common trap Rounding stratum numbers so that they no longer add up to the sample size; check the total afterwards.

Practise 4.1 in Heptavo →

4.2Presentation of dataSL

Displaying data with histograms, cumulative frequency graphs and box plots; questions ask you to read medians and quartiles, identify outliers and compare distributions.

You need to be able to

Read the median and quartiles from a cumulative frequency curve at n/2, n/4 and 3n/4.
Draw a box-and-whisker diagram and identify outliers more than 1.5 × IQR beyond the quartiles.
Compare two distributions using a measure of centre and a measure of spread, in context.

Key terms

cumulative frequency
box-and-whisker diagram
interquartile range
outlier
histogram
percentile

Common trap Comparing distributions with numbers alone; a full answer names a centre and a spread for both groups and says what the difference means.

Practise 4.2 in Heptavo →

4.3Measures of central tendency and dispersionSL

Measures of centre and spread: mean, median, mode, range, IQR, variance and standard deviation; questions ask for these from raw or grouped data and for the effect of changing the data.

You need to be able to

Find statistics on a GDC, estimating the mean of grouped data from mid-interval values.
Predict the effect of adding a constant (the mean moves, the spread does not) or multiplying by a (the mean and standard deviation scale by a, the variance by a²).
Use Σx = n × x̄ when values are added to or removed from a data set.

Key terms

mean
median
standard deviation
variance
interquartile range
mid-interval value

Common trap Changing the standard deviation after adding 5 to every value; a shift moves the mean but leaves the spread unchanged.

Practise 4.3 in Heptavo →

4.4Linear correlation and regressionSL

Scatter diagrams, Pearson's correlation coefficient r and the regression line of y on x; questions ask you to find r and the line on a GDC, interpret them and judge predictions.

You need to be able to

Find r and the regression line of y on x on a GDC, and describe the strength and direction of the correlation.
Interpret the gradient and intercept of the regression line in context.
Decide whether a prediction is reliable: strong correlation, interpolation, and y predicted from x.

Key terms

Pearson's correlation coefficient
regression line
interpolation
extrapolation
causation
mean point

Common trap Claiming that one variable causes the other because r is close to 1; correlation alone does not show causation.

Practise 4.4 in Heptavo →

4.5Probability concepts and complementary eventsSL

Basic probability, sample spaces and complementary events; questions ask for probabilities of equally likely outcomes and expected numbers of occurrences.

You need to be able to

Use P(A) = n(A)/n(U) for equally likely outcomes.
Use P(A′) = 1 − P(A), especially for 'at least one' questions.
Find the expected number of occurrences as n × P(A).

Key terms

sample space
event
complementary event
equally likely
expected number

Common trap Counting 'at least one six' on two dice as 12 of 36 outcomes; the double six has been counted twice, so it is 11/36.

Practise 4.5 in Heptavo →

4.6Combined events, Venn diagrams and tree diagramsSL

Unions, intersections, Venn diagrams and tree diagrams; questions ask for P(A ∪ B), probabilities with and without replacement, and probabilities read from regions.

You need to be able to

Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Fill a Venn diagram from the intersection outwards.
Draw tree diagrams, multiplying along branches and adding paths, and change the second-stage fractions when items are not replaced.

Key terms

union
intersection
mutually exclusive
Venn diagram
tree diagram
without replacement

Common trap Using the same fraction at both stages when sampling without replacement; the second draw has one item fewer.

Practise 4.6 in Heptavo →

4.7Discrete random variables and expected valueSL

Probability distributions of discrete random variables and their expected value; questions ask you to find a missing probability, E(X), and whether a game is fair.

You need to be able to

Use ΣP(X = x) = 1 to find an unknown probability.
Calculate E(X) = Σ x P(X = x).
Decide that a game is fair when the expected gain is zero.

Key terms

discrete random variable
probability distribution
expected value
fair game
expected gain

Common trap Ignoring the cost to play when judging fairness; compare the expected winnings with the stake.

Practise 4.7 in Heptavo →

4.8The binomial distributionSL

Modelling the number of successes in n independent trials; questions ask you to justify the model and find P(X = r), cumulative probabilities, the mean and the variance.

You need to be able to

State the conditions: a fixed number of trials, two outcomes, constant p and independent trials.
Use the GDC binomial pdf and cdf for exact and cumulative probabilities.
Use E(X) = np and Var(X) = np(1 − p).

Key terms

binomial distribution
trial
success
independent
cumulative probability
B(n, p)

Common trap Entering P(X ≥ 3) as 1 − P(X ≤ 3); it is 1 − P(X ≤ 2), because X = 3 belongs to the event.

Practise 4.8 in Heptavo →

4.9The normal distributionSL

Continuous data modelled by the normal distribution; questions ask for probabilities, inverse normal values and the proportion in a range.

You need to be able to

Find P(a < X < b) with the GDC normal cdf.
Use the inverse normal to find the value with a given probability to its left.
Use symmetry and the 68–95–99.7% guide to sketch and sense-check answers.

Key terms

normal distribution
mean
standard deviation
inverse normal
symmetry
area under the curve

Common trap Entering the variance instead of the standard deviation; X ~ N(μ, σ²) gives the variance, so take its square root first.

Practise 4.9 in Heptavo →

4.10The regression line of x on ySL

The regression line of x on y, used to estimate x from a known y; questions ask you to choose the right line for a prediction.

You need to be able to

Find the x-on-y line on a GDC by treating y as the independent variable.
Use the y-on-x line to predict y and the x-on-y line to predict x.
State that both lines pass through the mean point (x̄, ȳ).

Key terms

regression line of x on y
regression line of y on x
mean point
prediction
least squares

Common trap Rearranging the y-on-x line to estimate x; that is a different line and gives a different, wrong estimate.

Practise 4.10 in Heptavo →

4.11Conditional probability and independenceSL

Probability given that an event has happened, and testing for independence; questions ask for P(A | B) from tables, Venn diagrams and trees, and whether events are independent.

You need to be able to

Use P(A | B) = P(A ∩ B)/P(B).
Test independence with P(A ∩ B) = P(A)P(B), or P(A | B) = P(A).
Read conditional probabilities from a two-way table using the total of the given row or column.

Key terms

conditional probability
independent events
given that
two-way table
mutually exclusive

Common trap Treating mutually exclusive events as independent; events with non-zero probabilities that cannot happen together are dependent.

Practise 4.11 in Heptavo →

4.12Standardization of normal variablesSL

Converting normal values to z-scores; questions ask you to find an unknown mean or standard deviation from given probabilities.

You need to be able to

Calculate z = (x − μ)/σ.
Use the inverse normal on N(0, 1) to turn a probability into z, then solve for μ or σ.
Set up and solve simultaneous equations when both μ and σ are unknown.

Key terms

standardised value
z-score
standard normal distribution
inverse normal
simultaneous equations

Common trap Dropping the minus sign of z for a value below the mean, which puts the mean on the wrong side.

Practise 4.12 in Heptavo →

4.13Bayes' theoremHL

Reversing conditional probabilities with Bayes' theorem; questions ask for the probability of a cause given the evidence, with two or three possible causes.

You need to be able to

Find the total probability of the evidence by adding every branch that leads to it.
Calculate P(A | B) = P(A)P(B | A)/P(B).
Explain why a rare condition gives a low P(condition | positive) even for an accurate test.

Key terms

Bayes' theorem
prior probability
posterior probability
law of total probability
false positive

Common trap Confusing P(positive | condition) with P(condition | positive); they are different, and Bayes' theorem converts one into the other.

Practise 4.13 in Heptavo →

4.14Variance and expectation of discrete and continuous variablesHL

Expectation and variance for discrete and continuous random variables; questions ask for E(X), Var(X), the median or mode of a pdf, and the effect of linear transformations.

You need to be able to

Use E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
Calculate Var(X) = E(X²) − [E(X)]².
For a pdf f(x), use a total area of 1, E(X) = ∫x f(x) dx, and find the median m where the area up to m is 0.5.

Key terms

probability density function
expectation
variance
median
mode
linear transformation

Common trap Giving E(X²) as the variance; subtract [E(X)]² to get Var(X).

Practise 4.14 in Heptavo →

Topic 5 — Calculus

Differentiation and integration and their uses, with limits, further techniques, volumes, differential equations and Maclaurin series at HL.

5.1Limits and the derivative from first principlesSL

The derivative as the limit of chord gradients and as a rate of change; questions ask you to interpret f′(x), estimate a limit from a table, or link chord and tangent gradients.

You need to be able to

Explain f′(x) as the gradient of the tangent and the rate of change of y with respect to x.
Estimate a limit from a table of values or a graph as x approaches a value.
Calculate the chord gradient [f(x + h) − f(x)]/h and see it approach f′(x) as h → 0.

Key terms

limit
tangent
chord
gradient function
rate of change
dy/dx

Common trap Reading f′(3) as the value of f at x = 3; it is the gradient of the curve there.

Practise 5.1 in Heptavo →

5.2Increasing and decreasing functionsSL

Using the sign of f′(x) to find where a function increases or decreases; questions ask for intervals and for reading f's behaviour from a graph of f′.

You need to be able to

Solve f′(x) > 0 and f′(x) < 0 to find where f increases and decreases.
Read where f is increasing from where the graph of f′ lies above the x-axis.
Identify turning points of f where f′ changes sign.

Key terms

increasing
decreasing
stationary point
gradient
sign of f′(x)

Common trap Using where f(x) is positive instead of where f′(x) is positive; the sign of f itself says nothing about direction.

Practise 5.2 in Heptavo →

5.3Derivatives of polynomialsSL

Differentiating with the power rule; questions ask for f′(x) for polynomials and powers of x, including negative and fractional powers, and gradients at points.

You need to be able to

Differentiate axⁿ to naxⁿ⁻¹ for any real n.
Rewrite roots and fractions as powers first, such as 3/x² = 3x⁻² and √x = x^(1/2).
Find a gradient by substituting into f′(x), or a point with a given gradient by solving f′(x) = k.

Key terms

derivative
power rule
gradient function
negative exponent
fractional exponent

Common trap Differentiating 1/x³ as 1/(3x²); rewrite it as x⁻³ first, giving −3x⁻⁴.

Practise 5.3 in Heptavo →

5.4Tangents and normalsSL

Equations of the tangent and normal to a curve at a point; questions ask for these lines or for where a tangent meets an axis.

You need to be able to

Find the point (a, f(a)) and the gradient f′(a).
Write the tangent as y − f(a) = f′(a)(x − a), and the normal with gradient −1/f′(a).
Find points where the tangent has a given gradient by solving f′(x) = m.

Key terms

tangent
normal
gradient
point of contact
negative reciprocal

Common trap Putting f′(x), with x still in it, into the line equation; substitute x = a to get a number first.

Practise 5.4 in Heptavo →

5.5Introduction to integrationSL

Integration as reversing differentiation and as area under a curve; questions ask for indefinite integrals of powers, the constant from a known point, and simple definite integrals.

You need to be able to

Integrate xⁿ to xⁿ⁺¹/(n + 1) + C for n ≠ −1.
Use a point on the curve to find the constant of integration.
Find the area under a curve with a definite integral, using a GDC where it is allowed.

Key terms

anti-derivative
indefinite integral
constant of integration
definite integral
area under a curve

Common trap Leaving out + C, or not finding it when a point on the curve is given.

Practise 5.5 in Heptavo →

5.6Derivatives of elementary functions, chain, product and quotient rulesSL

Differentiating eˣ, ln x, sin x and cos x, and combining functions with the chain, product and quotient rules; questions ask for derivatives of composite, product and quotient functions.

You need to be able to

Recall the derivatives of sin x, cos x, eˣ and ln x.
Apply the chain rule, for example d/dx e^(3x²) = 6x e^(3x²).
Choose the product rule u′v + uv′ or the quotient rule (u′v − uv′)/v² from how the function is built.

Key terms

chain rule
product rule
quotient rule
composite function
derivative

Common trap Forgetting the derivative of the inside: d/dx sin(2x) is 2 cos(2x), not cos(2x).

Practise 5.6 in Heptavo →

5.7The second derivative and concavitySL

The second derivative and concavity; questions ask you to find f″(x), classify stationary points and locate points of inflexion.

You need to be able to

Find f″(x) by differentiating twice.
Link f″ > 0 to concave up and f″ < 0 to concave down.
Find points of inflexion where f″(x) = 0 and changes sign.

Key terms

second derivative
concave up
concave down
point of inflexion
rate of change of gradient

Common trap Calling every point with f″(x) = 0 a point of inflexion; f″ must also change sign, as y = x⁴ at x = 0 shows.

Practise 5.7 in Heptavo →

5.8Local maxima, minima and optimizationSL

Finding maximum and minimum points and applying them to real problems; questions ask for stationary points, their nature, and the optimal value in a context.

You need to be able to

Solve f′(x) = 0 and classify each point with the second derivative or a sign test.
Use the constraint to write the quantity in one variable before differentiating.
Check the endpoints of a closed interval and give the quantity the question asks for.

Key terms

stationary point
local maximum
local minimum
optimisation
constraint
endpoint

Common trap Stopping at the x-value; if the question asks for the maximum area, substitute back and give the area.

Practise 5.8 in Heptavo →

5.9Kinematics: displacement, velocity and accelerationSL

Displacement, velocity and acceleration linked by derivatives and integrals; questions ask when a particle is at rest, when it changes direction, and how far it travels.

You need to be able to

Differentiate s to get v and v to get a; integrate the other way, using initial conditions.
Find when the particle is at rest (v = 0) and when it changes direction (v changes sign).
Distinguish displacement, ∫v dt, from distance travelled, ∫|v| dt.

Key terms

displacement
velocity
acceleration
speed
distance travelled
at rest

Common trap Giving the displacement when the total distance is asked for; integrate |v| on a GDC, or split the integral where v = 0.

Practise 5.9 in Heptavo →

5.10Indefinite integration of standard functionsSL

Integrating the standard functions and simple composites; questions ask for integrals of eˣ, 1/x, sin x, cos x, functions of ax + b, and simple substitutions.

You need to be able to

Integrate eˣ, 1/x (to ln|x|), sin x and cos x.
Integrate f(ax + b) by integrating as usual and dividing by a.
Use substitution when the derivative of an inner function appears as a factor.

Key terms

indefinite integral
standard integral
ln|x|
linear composite
substitution
constant of integration

Common trap Integrating sin x to cos x; the integral is −cos x + C.

Practise 5.10 in Heptavo →

5.11Definite integrals and areasSL

Evaluating definite integrals and finding areas between a curve and the x-axis or between two curves; questions ask for exact values and areas, often with a GDC.

You need to be able to

Evaluate F(b) − F(a) exactly.
Find areas below the x-axis by splitting at the roots and taking the size of each part.
Find the area between two curves with ∫(upper − lower) dx between their intersections.

Key terms

definite integral
limits of integration
area between curves
signed area
intersection

Common trap Integrating straight across a root, so that areas above and below the axis cancel; split the integral or integrate |f(x)|.

Practise 5.11 in Heptavo →

5.12Continuity, differentiability and limits from first principlesHL

Continuity, differentiability and differentiating from first principles; questions ask for a first-principles proof or for testing a piecewise function where its pieces join.

You need to be able to

Differentiate from first principles: expand f(x + h) − f(x), divide by h, then let h → 0.
Test continuity at a point: the limit exists and equals f(a).
Test a piecewise function for differentiability by matching both values and gradients at the join.

Key terms

continuity
differentiability
first principles
limit
piecewise function
left and right limits

Common trap Setting h = 0 before cancelling, which gives 0/0; simplify first, then take the limit.

Practise 5.12 in Heptavo →

5.13Evaluation of limits and L'Hôpital's ruleHL

Evaluating limits of indeterminate forms; questions ask for limits as x → 0 or x → ∞, using L'Hôpital's rule or series.

You need to be able to

Apply L'Hôpital's rule to a 0/0 or ∞/∞ form, repeating it if the form persists.
Evaluate limits as x → ∞ by comparing how fast powers, exponentials and logarithms grow.
Use a Maclaurin series to evaluate a limit as x → 0.

Key terms

indeterminate form
L'Hôpital's rule
limit
growth rate
Maclaurin series

Common trap Using the quotient rule on the fraction; L'Hôpital's rule differentiates the numerator and the denominator separately.

Practise 5.13 in Heptavo →

5.14Implicit differentiation and related ratesHL

Differentiating equations that are not solved for y, and linking rates of change; questions ask for dy/dx at a point and for rates such as dV/dt.

You need to be able to

Differentiate every term with respect to x, multiplying each y-term's derivative by dy/dx.
Find gradients and tangent lines for implicitly defined curves.
Solve related-rates problems with the chain rule, substituting values only after differentiating.

Key terms

implicit differentiation
related rates
chain rule
rate of change
dy/dx

Common trap Substituting the value at the instant (say r = 3) before differentiating, which turns a variable into a constant.

Practise 5.14 in Heptavo →

5.15Derivatives of further functionsHL

Derivatives of tan, sec, cosec, cot, aˣ, logₐ x, arcsin, arccos and arctan; questions combine these with the chain, product and quotient rules.

You need to be able to

Use d/dx tan x = sec²x, d/dx sec x = sec x tan x, d/dx cosec x = −cosec x cot x and d/dx cot x = −cosec²x.
Use d/dx aˣ = aˣ ln a and d/dx logₐ x = 1/(x ln a).
Use d/dx arcsin x = 1/√(1 − x²), d/dx arccos x = −1/√(1 − x²) and d/dx arctan x = 1/(1 + x²).

Key terms

reciprocal trigonometric functions
inverse trigonometric functions
exponential function
chain rule
ln a

Common trap Differentiating 2ˣ as x·2ˣ⁻¹; the power rule needs the variable in the base, so d/dx 2ˣ = 2ˣ ln 2.

Practise 5.15 in Heptavo →

5.16Integration by substitution and by partsHL

Integrating composite functions and products; questions ask for integrals such as ∫x e^(x²) dx and ∫x ln x dx, often definite.

You need to be able to

Choose u so that du appears in the integrand, and change the limits for a definite integral.
Use ∫u dv = uv − ∫v du, choosing as u the factor that becomes simpler when differentiated.
Handle repeated integration by parts, and integrals such as ∫eˣ sin x dx that return to the start, by solving for the integral.

Key terms

substitution
integration by parts
change of limits
repeated integration by parts
definite integral

Common trap Keeping the x-limits after substituting u; convert the limits to values of u, or return to x before using them.

Practise 5.16 in Heptavo →

5.17Areas and volumes of revolutionHL

Areas between a curve and the y-axis, and volumes of solids formed by rotating curves; questions ask for V = π∫y² dx or V = π∫x² dy.

You need to be able to

Find areas with respect to y using ∫x dy.
Calculate volumes of revolution about the x-axis with V = π∫y² dx.
Calculate volumes about the y-axis with V = π∫x² dy, after rearranging for x² in terms of y.

Key terms

volume of revolution
disc
axis of rotation
area with respect to y
limits of integration

Common trap Forgetting to square, or squaring after integrating; square y (or x) first, then integrate.

Practise 5.17 in Heptavo →

5.18Differential equationsHL

Solving first-order differential equations exactly and numerically; questions ask for general and particular solutions by separation, substitution y = vx or an integrating factor, and Euler's method estimates.

You need to be able to

Separate the variables and integrate, using an initial condition for the particular solution.
Use y = vx for homogeneous equations and the integrating factor e^(∫P dx) for dy/dx + P(x)y = Q(x).
Apply Euler's method, yₙ₊₁ = yₙ + h × f(xₙ, yₙ), and comment on how the step size affects accuracy.

Key terms

separable
homogeneous
integrating factor
general solution
particular solution
Euler's method

Common trap Writing the solution of dy/dx = y as y = eˣ + C; integrating gives ln|y| = x + C, so y = Aeˣ.

Practise 5.18 in Heptavo →

5.19Maclaurin seriesHL

Approximating functions with Maclaurin series; questions ask for the first terms of a series, building new series from standard ones, and using series for approximations or limits.

You need to be able to

Build a Maclaurin series from f(0), f′(0), f″(0), …, with coefficients f⁽ⁿ⁾(0)/n!.
Derive new series by substituting into, multiplying, differentiating or integrating standard series.
Use a series to approximate a value, evaluate a limit or find a series solution to a differential equation.

Key terms

Maclaurin series
power series
coefficient
standard series
approximation

Common trap Substituting 2x into the series for sin x but raising only x to the power; the whole bracket takes the power, so the second term is −(2x)³/3! = −4x³/3, not −2x³/6.

Practise 5.19 in Heptavo →

Try 4 Heptavo questions

Real questions from the app, with the explanation after each answer.

1.1 · Standard form and laws of exponents

Solve 9^x = 27^x−2.

−6
2
3
6

Show the answer and why

Answer: D

Write both sides to base 3: 9^x = 3^2x and 27^x−2 = 3^3(x−2). Equal bases give 2x = 3x − 6, so x = 6. Subtracting 2 from the index without multiplying it by 3 first gives x = 2, and reading 9 and 27 as ordinary coefficients rather than powers of 3 gives x = 3.

2.9 · Exponential and logarithmic functions

The vertical asymptote of y = ln(x − 2) is

y = 2
x = 0
x = −2
x = 2

Show the answer and why

Answer: D

The logarithm needs a positive argument, so the graph exists only for x > 2 and the asymptote is at the boundary. The sign inside the bracket reverses, as with any horizontal translation.

4.6 · Combined events, Venn diagrams and tree diagrams

A fair coin is tossed three times. What is P(at least one head)?

Show the answer and why

Answer: D

1 − P(no heads) = 1 − (1/2)³ = 7/8. Working through the complement is far quicker than listing the seven favourable outcomes.

5.2 · Increasing and decreasing functions

For f(x) = x² − 6x, the function is decreasing for

x < 3
x > 3
all x
x < 0

Show the answer and why

Answer: A

f′(x) = 2x − 6, which is negative when x < 3.

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