Syllabus map · 75 topics · 33 HL only
IB Maths AI (Applications and Interpretation) syllabus map
IB Mathematics: Applications and Interpretation is the Diploma maths course built around modelling, statistics and using technology on real-world problems. 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.
How it's assessed
| Component | SL | HL |
|---|---|---|
| Paper 1 | 1 h 30, 80 marks, 40% | 2 h, 110 marks, 30% |
| Paper 2 | 1 h 30, 80 marks, 40% | 2 h, 110 marks, 30% |
| Paper 3 | — | 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 and a 1-hour Paper 3; check your subject guide for the content changes.
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
Numbers and money in context, from standard form and sequences to loans and approximation, with logarithms, complex numbers and matrices at HL.
1.1Standard form / scientific notationSL
Writing very large and very small quantities as a × 10ᵏ with 1 ≤ a < 10; questions ask you to convert, calculate in context and give answers in the required form.
You need to be able to
- Convert to and from standard form by counting how many places the decimal point moves.
- Multiply and divide in standard form, then readjust so that 1 ≤ a < 10.
- Enter standard form on a GDC and rewrite its output (such as 4.5E−5) in correct notation.
Key terms
- standard form
- scientific notation
- exponent
- significant figures
Common trap Copying the GDC display 4.5E−5 as the final answer; write it as 4.5 × 10⁻⁵.
Practise 1.1 in Heptavo →1.2Arithmetic sequences and seriesSL
Sequences with a constant difference and their sums, in contexts such as savings plans and rows of seats; questions ask for a term, a sum or the number of terms.
You need to be able to
- Use uₙ = u₁ + (n − 1)d and both forms of Sₙ from the formula booklet.
- Find d from two terms by dividing their gap by the number of steps between them.
- Set up and solve an equation for n, giving a whole number that makes sense in the context.
Key terms
- arithmetic sequence
- common difference
- arithmetic series
- nth term
- sum of n terms
Common trap Using 10d for the step from u₁ to u₁₀; there are only nine steps, so u₁₀ = u₁ + 9d.
Practise 1.2 in Heptavo →1.3Geometric sequences and seriesSL
Sequences with a constant ratio and their sums, in contexts such as population growth; questions ask for terms, sums and when a value is first passed.
You need to be able to
- Use uₙ = u₁rⁿ⁻¹ and Sₙ = u₁(rⁿ − 1)/(r − 1).
- Turn a percentage change into a multiplier, r = 1 + p/100 for growth and 1 − p/100 for decay.
- Use a GDC table or solver to find the first n at which a target is reached.
Key terms
- geometric sequence
- common ratio
- geometric series
- multiplier
- percentage change
Common trap Writing a 5% decrease as r = 0.05; the multiplier is what remains, so r = 0.95.
Practise 1.3 in Heptavo →1.4Compound interest, depreciation, financial appsSL
Compound interest, depreciation and inflation, worked with formulae and the GDC finance solver; questions ask for future values, rates, the time to reach a target and real values.
You need to be able to
- Use FV = PV × (1 + r/(100k))ᵏⁿ or the TVM solver, setting N, I%, PV, FV and the compounding periods per year correctly.
- Calculate depreciation by repeated multiplication by 1 − r/100.
- Compare nominal and real values by allowing for the rate of inflation.
Key terms
- compound interest
- depreciation
- present value
- future value
- inflation
- TVM solver
Common trap Entering PV and FV with the same sign in the TVM solver; money paid in and money received carry opposite signs.
Practise 1.4 in Heptavo →1.5Exponents and logarithmsSL
The laws of exponents and an introduction to logarithms; questions ask you to simplify expressions with exponents, evaluate logarithms and interpret exponential expressions in context.
You need to be able to
- Apply the laws of exponents, including zero and negative exponents.
- Treat log₁₀ x and ln x as exponents, and evaluate them with a GDC.
- Interpret a and b in N = a × bᵗ as the starting value and the multiplier per unit of time.
Key terms
- exponent
- logarithm
- base
- natural logarithm
- inverse
Common trap Reading 2⁻³ as −8; a negative exponent means a reciprocal, so 2⁻³ = 1/8.
Practise 1.5 in Heptavo →1.6Simple deductive proofSL
Short algebraic arguments that show two expressions are identical or that a statement holds for every integer; questions ask you to expand and simplify, or to choose a general form to work with.
You need to be able to
- Expand and simplify expressions such as (x + 3)² − (x − 3)² = 12x.
- Represent consecutive integers as n, n + 1, n + 2 (or n − 1, n, n + 1), even numbers as 2n and odd numbers as 2n + 1.
- Distinguish an identity (≡, true for every x) from an equation (true for particular values).
Key terms
- identity
- equation
- deductive proof
- consecutive integers
- expand
Common trap Testing a few numbers and calling it a proof; examples only illustrate a claim, while the general algebra proves it.
Practise 1.6 in Heptavo →1.7Systems of linear equationsSL
Solving simultaneous linear equations, mostly with a GDC, in contexts such as costs and mixtures; questions ask you to form the equations from a description and solve them.
You need to be able to
- Write a system of up to three linear equations from a context, defining each variable.
- Solve with a GDC equation solver, or by elimination for two unknowns.
- Recognise equations that describe parallel lines (no solution) or the same line (infinitely many).
Key terms
- simultaneous equations
- unknown
- elimination
- substitution
- GDC solver
Common trap Skipping the definitions; state what each variable stands for, with units, before writing the equations.
Practise 1.7 in Heptavo →1.8Approximation, rounding, error, boundsSL
Rounding, significant figures, upper and lower bounds and percentage error; questions ask for the bounds of a measured value and the percentage error of an estimate.
You need to be able to
- Round to decimal places and significant figures, giving answers to 3 significant figures unless told otherwise.
- Find the bounds of a rounded value as half a unit either side of it.
- Calculate percentage error as |vₐ − vₑ| ÷ vₑ × 100%.
Key terms
- significant figures
- decimal places
- upper bound
- lower bound
- percentage error
- estimate
Common trap Dividing by the approximate value when finding percentage error; the exact value always goes in the denominator.
Practise 1.8 in Heptavo →1.9Amortization and annuities (GDC)SL
Loans, savings with regular payments and pensions, worked with the GDC finance solver; questions ask for the payment, the term, the total interest or the balance after some time.
You need to be able to
- Set N, I%, PV, PMT, FV, P/Y and C/Y correctly for a loan or an annuity.
- Calculate the total paid as N × PMT and the total interest as N × PMT − PV.
- Read an amortization table: the interest part of each payment falls while the principal part rises.
Key terms
- amortization
- annuity
- loan
- payment (PMT)
- principal
- total interest
Common trap Setting FV equal to the amount borrowed; a fully repaid loan ends at FV = 0, and the amount borrowed is PV.
Practise 1.9 in Heptavo →1.10Laws of logarithms, exponential equationsHL
The laws of logarithms and solving exponential equations; questions ask you to simplify log expressions, solve for an unknown exponent and linearise exponential models.
You need to be able to
- Apply log(xy) = log x + log y, log(x/y) = log x − log y and log xᵐ = m log x.
- Solve aˣ = b by taking logarithms: x = log b ÷ log a.
- Linearise y = a × bˣ to log y = log a + x log b.
Key terms
- product law
- quotient law
- power law
- exponential equation
- change of base
- linearisation
Common trap Taking logarithms before clearing a coefficient; in 3 × 2ˣ = 48, divide by 3 first to get 2ˣ = 16, so x = 4.
Practise 1.10 in Heptavo →1.11Sum of infinite convergent geometric sequencesHL
When an infinite geometric series has a sum, and what that sum is; questions ask for S∞, the ratio from given terms, or the fraction equal to a recurring decimal.
You need to be able to
- Use S∞ = u₁/(1 − r), only when |r| < 1.
- Find r from two given terms, then the sum to infinity.
- Write a recurring decimal as a geometric series and convert it to a fraction.
Key terms
- infinite series
- convergent
- sum to infinity
- common ratio
- recurring decimal
Common trap Using r⁴ for the step from u₁ to u₄; it is three steps, so u₄ = u₁r³.
Practise 1.11 in Heptavo →1.12Complex numbers: Cartesian formHL
Complex numbers a + bi and their arithmetic; questions ask you to calculate, plot on an Argand diagram and solve quadratics with complex roots.
You need to be able to
- Add, subtract, multiply and divide in the form a + bi, using i² = −1.
- Use the conjugate to divide, and to recognise the conjugate pair of roots of a real quadratic.
- Plot points on an Argand diagram and find the modulus.
Key terms
- imaginary unit
- real part
- imaginary part
- complex conjugate
- Argand diagram
- modulus
Common trap Leaving i² in an answer; replace it with −1 so that the result is in the form a + bi.
Practise 1.12 in Heptavo →1.13Complex numbers: polar/Euler form, De MoivreHL
Modulus–argument and exponential forms, products, quotients and powers; questions ask you to convert between forms and use them for rotations and for adding sinusoids of the same frequency.
You need to be able to
- Convert a + bi to r(cos θ + i sin θ) and re^(iθ), checking the quadrant from a sketch.
- Multiply and divide by combining moduli and arguments, and raise to powers with De Moivre's theorem.
- Use complex numbers to add sinusoidal functions with the same frequency but different phase, as in AC circuits.
Key terms
- modulus
- argument
- polar form
- Euler form
- De Moivre's theorem
- phase shift
Common trap Taking the calculator's arctan(b/a) as the argument for a point in the second or third quadrant; sketch the point and adjust by π.
Practise 1.13 in Heptavo →1.14Matrices: algebra, determinants, inverses, eigenvaluesHL
Matrix operations, determinants, inverses and eigenvalues; questions ask you to multiply matrices, solve systems with an inverse, and find eigenvalues and eigenvectors of 2 × 2 matrices.
You need to be able to
- Check dimensions before multiplying: an m × n matrix times an n × p matrix gives m × p.
- Find the determinant and inverse of a 2 × 2 matrix by hand (larger ones on a GDC), and solve AX = B as X = A⁻¹B.
- Find eigenvalues from det(A − λI) = 0, then eigenvectors, and use them to diagonalise a matrix and find its powers.
Key terms
- determinant
- inverse matrix
- singular
- identity matrix
- eigenvalue
- eigenvector
Common trap Writing X = BA⁻¹ for AX = B; matrix multiplication is not commutative, so X = A⁻¹B.
Practise 1.14 in Heptavo →Topic 2 — Functions
Building, fitting and judging models of real situations with linear, quadratic, exponential, power and periodic functions.
2.1Straight lines: gradient, equationsSL
Gradients, intercepts and equations of straight lines; questions ask for the line through given points, parallel or perpendicular lines, and the meaning of a gradient in context.
You need to be able to
- Find the gradient as the change in y divided by the change in x.
- Write a line as y = mx + c, ax + by + d = 0 or y − y₁ = m(x − x₁).
- Use equal gradients for parallel lines and m₁m₂ = −1 for perpendicular lines.
Key terms
- gradient
- y-intercept
- general form
- point–gradient form
- perpendicular
Common trap Reading the gradient of ax + by + d = 0 as a; rearrange first, which gives m = −a/b.
Practise 2.1 in Heptavo →2.2Function concept, domain/range, notationSL
Function notation, domain and range, and reading graphs of functions; questions ask you to evaluate, give a sensible domain and range in context, and solve f(x) = k from a graph.
You need to be able to
- Evaluate f(a) and solve f(x) = k, from a formula and from a graph.
- State a domain and range that make sense for a model, including endpoints.
- Use the vertical line test to decide whether a graph shows a function.
Key terms
- function
- domain
- range
- function notation
- input
- output
Common trap Evaluating 5 − x² at x = −2 as 5 − (−4); (−2)² = 4, so f(−2) = 1.
Practise 2.2 in Heptavo →2.3Linear modelsSL
Using straight lines to model real situations; questions ask you to build a model from a description or data, interpret its parameters and make predictions.
You need to be able to
- Build y = mx + c from a fixed charge and a rate, or from two data points.
- Interpret m and c with units in the context.
- Find where two linear models give the same value, and say which is smaller on each side.
Key terms
- linear model
- gradient (rate of change)
- y-intercept (initial value)
- parameter
- piecewise linear
Common trap Finding the rate as y ÷ x from a single point when the model has a non-zero intercept; use the change between two points.
Practise 2.3 in Heptavo →2.4Quadratic modelsSL
Parabolas modelling projectiles, profit and areas; questions ask for the vertex, zeros and intercept, and what they mean in context.
You need to be able to
- Find the axis of symmetry x = −b/(2a), the vertex and the zeros, by algebra or a GDC.
- Interpret the vertex as a maximum or minimum and the zeros as start or end points.
- Restrict the domain to values that make sense, such as t ≥ 0.
Key terms
- quadratic model
- vertex
- axis of symmetry
- zeros
- maximum
- domain
Common trap Giving the time of the maximum when the question asks for the maximum height; substitute that time back into the model.
Practise 2.4 in Heptavo →2.5Exponential modelsSL
Growth and decay models of the form f(x) = k × aˣ + c and k × e^(rx) + c; questions ask you to find parameters, interpret the asymptote and make predictions.
You need to be able to
- Interpret k + c as the initial value, a as the multiplier per unit of time and c as the horizontal asymptote.
- Turn a percentage rate into a base, such as 6% growth giving a = 1.06.
- Find unknown parameters from data points and use the model to predict.
Key terms
- exponential growth
- exponential decay
- horizontal asymptote
- half-life
- initial value
- multiplier
Common trap Using k as the starting value when the model has + c; the value at x = 0 is k + c.
Practise 2.5 in Heptavo →2.6Direct/inverse variation, cubic modelsSL
Direct and inverse variation, power models y = kxⁿ and cubic models; questions ask for k from a data point, predictions, and features of a cubic.
You need to be able to
- Identify direct variation (y = kx) and inverse variation (y = k/x), and find k from a known pair.
- Use y = axⁿ for power models, remembering that a negative n means a reciprocal.
- Find the turning points and zeros of a cubic model with a GDC.
Key terms
- direct variation
- inverse variation
- constant of proportionality
- power model
- cubic model
Common trap Halving y when x doubles in y = k/x²; y is divided by 2² = 4.
Practise 2.6 in Heptavo →2.7Modelling skills: fitting and evaluating modelsSL
Choosing, fitting and evaluating models; questions ask you to pick a model type, find its parameters, test it against data and comment on how reasonable it is.
You need to be able to
- Choose a model from the data: equal differences suggest linear, equal ratios exponential, one turning point quadratic, a repeating pattern sinusoidal.
- Follow the modelling cycle: develop a model, test it, reflect on the fit and revise it.
- Judge predictions: interpolation is safer than extrapolation, and outputs impossible in context rule a model out.
Key terms
- model
- parameter
- interpolation
- extrapolation
- reasonableness
- modelling cycle
Common trap Trusting an extrapolated prediction because the model fits the data well; the data say nothing about values outside their range.
Practise 2.7 in Heptavo →2.8Composite and inverse functionsHL
Combining and reversing functions; questions ask for (f ∘ g)(x), inverse functions and their domains, and their meaning in context.
You need to be able to
- Evaluate f(g(x)) by applying g first, then f.
- Find f⁻¹ algebraically and sketch it as the reflection of f in y = x.
- State the domain of f⁻¹ as the range of f, and restrict domains to make a function one-to-one.
Key terms
- composite function
- inverse function
- one-to-one
- domain
- range
- reflection in y = x
Common trap Treating f⁻¹(x) as 1/f(x); the inverse reverses f, it is not its reciprocal.
Practise 2.8 in Heptavo →2.9Transformations of graphsHL
Translations, stretches and reflections of graphs; questions ask you to describe a transformation, write the new equation or find the image of a point.
You need to be able to
- Describe f(x) + k as a move up by k and f(x − h) as a move right by h.
- Describe af(x) as a vertical stretch with scale factor a and f(bx) as a horizontal stretch with scale factor 1/b.
- Apply combined transformations in order and track a point through each one.
Key terms
- translation
- vertical stretch
- horizontal stretch
- reflection
- scale factor
- image
Common trap Moving f(x + 3) to the right; a change inside the bracket works the opposite way, so it moves 3 to the left.
Practise 2.9 in Heptavo →2.10Rational functionsHL
Rational functions of the form (ax + b)/(cx + d) used as models; questions ask for asymptotes, intercepts and a sketch, and what the asymptotes mean in context.
You need to be able to
- State the vertical asymptote x = −d/c and the horizontal asymptote y = a/c.
- Find the intercepts and sketch both branches.
- Interpret a horizontal asymptote in a model as the long-run value.
Key terms
- rational function
- vertical asymptote
- horizontal asymptote
- intercept
- reciprocal function
Common trap Getting the sign of the vertical asymptote wrong: (3x − 2)/(x + 5) has x = −5, because that is where the denominator is zero.
Practise 2.10 in Heptavo →2.11Polynomial functions, factor/remainder theoremHL
Polynomials, their factors and their end behaviour; questions ask you to test a factor, factorise, and match a graph to its equation.
You need to be able to
- Use P(a) = 0 to show (x − a) is a factor, and P(a) for the remainder on division by (x − a).
- Match a sketch to its factors: a single factor crosses the x-axis, a repeated factor touches it.
- Describe the end behaviour from the degree and the sign of the leading coefficient.
Key terms
- factor theorem
- remainder theorem
- root
- repeated root
- leading coefficient
- degree
Common trap Testing the factor (x − 2) by substituting x = −2; the factor (x − a) is tested with x = a.
Practise 2.11 in Heptavo →2.12Sinusoidal modelsHL
Modelling repeating patterns with f(x) = a sin(b(x − c)) + d; questions ask you to find the parameters from maximum, minimum and period, and make predictions.
You need to be able to
- Find a = (max − min)/2 and d = (max + min)/2.
- Find b from the period: b = 360° ÷ period in degrees.
- Find c from the time of a known point, such as the first maximum or the start of a cycle.
Key terms
- amplitude
- period
- principal axis
- phase shift
- sinusoidal model
Common trap Taking the amplitude as max − min; it is half of that, the distance from the principal axis to a peak.
Practise 2.12 in Heptavo →2.13Log, logistic and piecewise modelsHL
Logarithmic, logistic and piecewise models; questions ask you to interpret parameters such as a carrying capacity, evaluate the right piece of a function and make a model continuous.
You need to be able to
- Interpret a and b in y = a + b ln x, defined only for x > 0.
- Interpret L in f(x) = L/(1 + Ce^(−kx)) as the carrying capacity, the level the model approaches.
- Evaluate a piecewise model on the correct interval, and find a parameter that makes it continuous at a join.
Key terms
- logarithmic model
- logistic model
- carrying capacity
- piecewise function
- continuity
Common trap Using the wrong rule at a boundary value; check which interval includes the endpoint before substituting.
Practise 2.13 in Heptavo →Topic 3 — Geometry and trigonometry
Measurement and trigonometry in two and three dimensions and Voronoi diagrams, with vectors and graph theory at HL.
3.13D solids: volume and surface areaSL
Volumes and surface areas of solids and distances in three dimensions; questions ask for capacities, amounts of material and lengths inside boxes and pyramids.
You need to be able to
- Use the formulae for prisms, cylinders, cones, pyramids, spheres and hemispheres.
- Use Pythagoras twice to find space diagonals and slant heights.
- Find the distance and the midpoint between two points in 3D.
Key terms
- volume
- surface area
- slant height
- hemisphere
- space diagonal
- midpoint
Common trap Leaving out the flat face of a solid hemisphere; its total surface area is 2πr² + πr² = 3πr².
Practise 3.1 in Heptavo →3.2Angles of elevation/depression, perpendicular distanceSL
Right-angled trigonometry in context; questions ask for heights and distances from angles of elevation and depression, and for shortest distances.
You need to be able to
- Use sin, cos and tan with the sides named from the angle you are using.
- Measure elevation and depression from the horizontal, and use the fact that they are equal for the same line of sight.
- Find the shortest distance from a point to a line along the perpendicular.
Key terms
- hypotenuse
- opposite
- adjacent
- angle of elevation
- angle of depression
- perpendicular distance
Common trap Measuring the angle of depression from the vertical; it is measured down from the horizontal.
Practise 3.2 in Heptavo →3.3Sine rule, cosine rule, area of a triangleSL
Triangles that are not right-angled, in context; questions ask for sides, angles and areas from survey-style measurements.
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.
- Spot the ambiguous case, where 180° − θ may give a second triangle.
Key terms
- sine rule
- cosine rule
- included angle
- ambiguous case
- area of a triangle
Common trap Leaving the GDC in radian mode; these questions are in degrees, so check the mode before using sin or cos.
Practise 3.3 in Heptavo →3.4Bearings and geometric transformationsSL
Bearings in navigation and transformations of shapes; questions ask for distances and bearings between points, and the images of shapes.
You need to be able to
- Write bearings as three figures clockwise from north, and find a back bearing by adding or subtracting 180°.
- Split a journey of length d on bearing θ into an east part d sin θ and a north part d cos θ.
- Describe translations, reflections, rotations and enlargements, using k² as the area scale factor of an enlargement.
Key terms
- bearing
- back bearing
- isometry
- enlargement
- scale factor
- congruent
Common trap Using cosine for the east component; because a bearing is measured from north, east is d sin θ.
Practise 3.4 in Heptavo →3.5The unit circle, trig ratiosHL
Radians, arcs and sectors, and the unit-circle definition of the trigonometric ratios; questions ask for conversions, arc lengths, sector areas and exact values.
You need to be able to
- Convert between degrees and radians with π radians = 180°.
- Use l = rθ and A = ½r²θ with θ in radians.
- Use the unit circle point (cos θ, sin θ) and the quadrant to find signs and exact values.
Key terms
- radian
- arc length
- sector
- unit circle
- exact value
- reference angle
Common trap Using l = rθ with θ in degrees; convert to radians first, or use the fraction θ/360 of the circumference.
Practise 3.5 in Heptavo →3.6Graphs of sin/cos/tan and transformationsHL
The graphs of sine, cosine and tangent and their transformations; questions ask for amplitude, period and principal axis, and for sketches.
You need to be able to
- Sketch sin x and cos x with period 360° and amplitude 1, and tan x with period 180° and asymptotes at 90°, 270°, ….
- Read the amplitude a, the period 360° ÷ b and the principal axis y = d from a sin(bx) + d.
- Apply vertical and horizontal stretches and translations to the basic graphs.
Key terms
- amplitude
- period
- principal axis
- asymptote
- horizontal stretch
Common trap Writing the period of sin 2x as 720°; multiplying x by 2 halves the period, giving 180°.
Practise 3.6 in Heptavo →3.7Real-life periodic phenomena modelled by trig functionsSL
Using trigonometric functions to model tides, Ferris wheels and temperatures; questions ask you to build the model and use it to predict.
You need to be able to
- Find d as the mean level, a as half the range and b from the period.
- Choose sine, cosine or negative cosine to match where the quantity is at the start.
- Use a GDC to find when the model reaches a given value.
Key terms
- periodic function
- amplitude
- period
- principal axis
- phase shift
Common trap Modelling a Ferris wheel rider who boards at the bottom with d + a cos(bt); that starts at the top, so use d − a cos(bt).
Practise 3.7 in Heptavo →3.8Voronoi diagramsSL
Dividing a region into cells nearest each site; questions ask for perpendicular bisectors, how a diagram changes when a site is added, and the point furthest from every site.
You need to be able to
- Find the equation of a perpendicular bisector from the midpoint and the negative reciprocal gradient.
- Complete a Voronoi diagram, or update it after adding a site.
- Solve the largest empty circle (toxic waste dump) problem by comparing the vertices and the boundary.
Key terms
- site
- cell
- edge
- vertex
- perpendicular bisector
- nearest neighbour interpolation
Common trap Using the gradient of the line joining the two sites for the bisector; the bisector's gradient is its negative reciprocal.
Practise 3.8 in Heptavo →3.9Vectors in two and three dimensionsHL
Vectors in two and three dimensions, their components, magnitudes and algebra; questions ask for displacement vectors, magnitudes and tests for parallel vectors.
You need to be able to
- Find a magnitude with Pythagoras on the components, including the third square in 3D.
- Add vectors and multiply them by scalars component by component.
- Find AB = b − a and show vectors are parallel by writing one as a multiple of the other.
Key terms
- vector
- component
- magnitude
- unit vector
- position vector
- scalar multiple
Common trap Writing AB as a − b; a displacement is end minus start, b − a.
Practise 3.9 in Heptavo →3.10Vector equation of a lineHL
Describing a line by r = a + tb; questions ask you to write the equation, check whether a point is on the line, and compare two lines.
You need to be able to
- Write r = a + tb from a point and a direction, or from two points.
- Check whether a point lies on the line by finding one value of t that works for every component.
- Compare directions first, then decide whether lines are parallel, intersecting or, in 3D, skew.
Key terms
- vector equation of a line
- direction vector
- parameter
- position vector
- skew lines
Common trap Using the same parameter for both lines when testing for an intersection; each line needs its own parameter.
Practise 3.10 in Heptavo →3.11Vector applications to motionHL
Position, velocity and speed of objects moving with constant velocity; questions ask where an object is at time t, how fast it moves and whether two objects collide.
You need to be able to
- Use r = r₀ + tv for position at time t.
- Find speed as |v| and distance travelled as speed × time.
- Test for a collision by finding a single value of t that gives the same position for both objects.
Key terms
- position vector
- velocity vector
- speed
- displacement
- collision
- closest approach
Common trap Concluding that two objects collide because their paths cross; they must be at the crossing point at the same time.
Practise 3.11 in Heptavo →3.12Scalar (dot) productHL
The scalar product and the angle between vectors; questions ask for a · b, the angle between two vectors or lines, and perpendicularity.
You need to be able to
- Compute a · b by multiplying matching components and adding.
- Find the angle with cos θ = (a · b)/(|a||b|).
- Use a · b = 0 as the test for perpendicular vectors.
Key terms
- scalar (dot) product
- angle between vectors
- perpendicular
- magnitude
- obtuse angle
Common trap Finding the angle between two lines from position vectors of points on them; use the direction vectors.
Practise 3.12 in Heptavo →3.13Vector (cross) product, areasHL
The vector product and the areas it gives; 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 from its components.
- Find the area of a parallelogram as |a × b| and of a triangle as ½|a × b|.
- Use a × b as a normal to the plane containing a and b.
Key terms
- vector product
- cross product
- normal vector
- right-hand rule
- area
Common trap Assuming a × b = b × a; swapping the order reverses the direction.
Practise 3.13 in Heptavo →3.14Vector equation of a plane, lines and planesHL
Equations of planes and how lines and planes meet; questions ask for a plane's equation, where a line meets a plane, intercepts, and the angle between planes.
You need to be able to
- Write a plane as (r − a) · n = 0 and expand it to ax + by + cz = d.
- Find where a line meets a plane by substituting its parametric equations into the plane's equation.
- Find the angle between two planes from the angle between their normals.
Key terms
- normal vector
- Cartesian equation of a plane
- intercept
- line of intersection
- angle between planes
Common trap Finding where a plane crosses the z-axis without setting x and y to zero; on the z-axis only z can be non-zero.
Practise 3.14 in Heptavo →3.15Graphs: vertices, edges, degree, kinds of graphHL
The language of graph theory: vertices, edges, degrees and kinds of graph; questions ask you to classify graphs and use the link between degrees and edges.
You need to be able to
- Use the fact that the degrees add up to twice the number of edges.
- Recognise simple, complete, connected, directed and weighted graphs, and that Kₙ has n(n − 1)/2 edges.
- Use the fact that a tree on n vertices is connected, has no cycles and has n − 1 edges.
Key terms
- vertex
- edge
- degree
- simple graph
- complete graph
- tree
Common trap Counting a loop as adding 1 to a vertex's degree; both ends meet that vertex, so a loop adds 2.
Practise 3.15 in Heptavo →3.16Adjacency matrices, walks and transition matricesHL
Adjacency matrices, counting walks and transition matrices; questions ask for the number of walks of a given length and for long-run probabilities in a Markov chain.
You need to be able to
- Write the adjacency matrix of a graph and use Aᵏ to count walks of length k.
- Build a transition matrix whose columns each add to 1, and find the state after n steps as Tⁿs₀.
- Find the steady state as the vector s with Ts = s whose entries add to 1.
Key terms
- adjacency matrix
- walk
- transition matrix
- Markov chain
- steady state
- eigenvector
Common trap Using A³ alone for walks of length up to 3; A³ counts walks of length exactly 3, so add A + A² + A³.
Practise 3.16 in Heptavo →3.17Trees, cycles, minimum spanning trees, postman and salesmanHL
Routes and networks: spanning trees, the Chinese postman problem and the travelling salesman problem; questions ask you to apply an algorithm and state the total weight.
You need to be able to
- Find a minimum spanning tree with Kruskal's or Prim's algorithm, listing edges in the order chosen.
- Solve the Chinese postman problem by pairing odd vertices and repeating the shortest pairing.
- Bound the travelling salesman tour: nearest neighbour gives an upper bound and deleted vertex a lower bound.
Key terms
- minimum spanning tree
- Eulerian circuit
- Hamiltonian cycle
- Chinese postman problem
- travelling salesman problem
- upper and lower bound
Common trap With four odd vertices, checking only one pairing; compare all three pairings and repeat the one with the smallest total.
Practise 3.17 in Heptavo →Topic 4 — Statistics and probability
Sampling, statistics, probability distributions and hypothesis tests, with Poisson models, confidence intervals and further tests at HL.
4.1Sampling techniquesSL
How data are collected and how samples are chosen; questions ask you to identify a sampling method, calculate a stratified sample and explain bias.
You need to be able to
- Describe simple random, systematic, stratified, quota and convenience sampling.
- Calculate each stratum's share as (stratum size ÷ N) × sample size, checking that the shares add to the sample size.
- Explain that bias comes from how the sample is chosen, so a bigger sample does not fix it.
Key terms
- population
- sample
- stratified sample
- systematic sample
- bias
- discrete and continuous data
Common trap Saying a larger sample removes bias; it only reduces random variation, and a biased method stays biased.
Practise 4.1 in Heptavo →4.2Presentation of data: histograms, box-and-whiskerSL
Displaying data with histograms, cumulative frequency graphs and box-and-whisker diagrams; questions ask you to read quartiles and percentiles, spot outliers and compare data sets.
You need to be able to
- Draw and read a box plot from the five-number summary.
- Read the median, quartiles and percentiles from a cumulative frequency curve.
- Identify outliers more than 1.5 × IQR beyond the quartiles.
Key terms
- histogram
- cumulative frequency
- box-and-whisker diagram
- interquartile range
- outlier
- percentile
Common trap Reading a 'more than' value straight off a cumulative frequency curve; the curve counts values up to a point, so subtract from the total.
Practise 4.2 in Heptavo →4.3Measures of central tendency and dispersionSL
Measures of centre and spread for raw and grouped data; questions ask you to calculate them with a GDC and to choose and interpret the right measure.
You need to be able to
- Find the mean, median, mode, range, IQR and standard deviation on a GDC.
- Estimate the mean of grouped data using class midpoints.
- Choose the median for skewed data, and predict how adding or multiplying by a constant changes the statistics.
Key terms
- mean
- median
- mode
- standard deviation
- interquartile range
- midpoint
Common trap Using a class boundary instead of the midpoint when estimating a grouped mean.
Practise 4.3 in Heptavo →4.4Correlation, scatter diagrams, Pearson’s rSL
Scatter diagrams and Pearson's correlation coefficient r; questions ask you to find r with a GDC, describe the correlation and comment on causation.
You need to be able to
- Find r on a GDC and describe its strength and direction.
- Remember that r measures only how close the points are to a straight line.
- Explain a correlation without claiming cause, for example by naming a third variable.
Key terms
- scatter diagram
- Pearson's correlation coefficient
- positive correlation
- negative correlation
- causation
- outlier
Common trap Taking r ≈ 0 to mean there is no relationship; a clear curved pattern can still give r close to 0.
Practise 4.4 in Heptavo →4.5Linear regressionSL
The regression line of y on x and its use for predictions; questions ask you to find the line on a GDC, interpret its gradient and intercept, and judge a prediction.
You need to be able to
- Find the regression line y = ax + b of y on x with a GDC.
- Interpret a as the change in y per unit of x and b as the predicted y when x = 0, with units.
- Decide whether a prediction is reliable: interpolation with strong correlation, not extrapolation.
Key terms
- regression line
- least squares
- gradient
- intercept
- interpolation
- extrapolation
Common trap Trusting a prediction far outside the data because r is close to 1; extrapolation is unreliable however strong the correlation.
Practise 4.5 in Heptavo →4.6Probability concepts, Venn/tree/sample space diagramsSL
Basic probability and the diagrams used to organise it; questions ask for probabilities from sample space diagrams, Venn diagrams and tree diagrams, and for expected numbers.
You need to be able to
- Use P(A) = n(A)/n(U) for equally likely outcomes, and the 36-cell grid for two dice.
- Use P(A′) = 1 − P(A), especially for 'at least one'.
- Draw tree diagrams, multiplying along branches and adding the paths that give the event.
Key terms
- sample space
- event
- complement
- Venn diagram
- tree diagram
- expected number
Common trap Counting 'at least one six' with two dice as 12 of 36; the double six is counted twice, so it is 11/36.
Practise 4.6 in Heptavo →4.7Combined events, conditional probability, independenceSL
Combined events, conditional probability and independence; questions ask for P(A ∪ B), P(A | B) and whether events are independent.
You need to be able to
- Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Use P(A | B) = P(A ∩ B)/P(B), with the given event as the denominator.
- Test independence with P(A ∩ B) = P(A) × P(B).
Key terms
- union
- intersection
- conditional probability
- independent events
- mutually exclusive
Common trap Treating mutually exclusive events as independent; if they cannot happen together, knowing one has happened changes the other's probability to 0.
Practise 4.7 in Heptavo →4.8Discrete random variables and their distributionsSL
Discrete random variables and their probability distributions; questions ask you to complete a distribution, find E(X) and judge a game.
You need to be able to
- Use the fact that the probabilities add to 1 to find an unknown.
- Calculate E(X) = Σ x P(X = x).
- Find the expected gain of a game and decide whether it is fair.
Key terms
- discrete random variable
- probability distribution
- expected value
- fair game
- expected gain
Common trap Treating E(X) as the most likely value; it is the long-run average, and need not be a value X can take.
Practise 4.8 in Heptavo →4.9Binomial distributionSL
Modelling the number of successes in n independent trials; questions ask you to justify the model and find probabilities, the mean and the variance with a GDC.
You need to be able to
- State the four conditions: fixed n, two outcomes, constant p, independent trials.
- Use the binomial pdf for P(X = r) and the cdf for P(X ≤ r).
- 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 is part of the event.
Practise 4.9 in Heptavo →4.10Normal distributionSL
Continuous data modelled by the normal distribution; questions ask for probabilities and for values with a given probability, using a GDC.
You need to be able to
- Find probabilities with the normal cdf, using lower and upper bounds.
- Use the inverse normal to find the value with a given area to its left.
- Use symmetry and the 68–95–99.7% guide to sketch and check answers.
Key terms
- normal distribution
- mean
- standard deviation
- normal cdf
- inverse normal
- symmetry
Common trap Entering the variance instead of the standard deviation; X ~ N(μ, σ²) states the variance, so take its square root.
Practise 4.10 in Heptavo →4.11Spearman’s rank correlation coefficientSL
Spearman's rank correlation coefficient, which measures whether one variable consistently rises or falls with another; questions ask you to rank data, find rₛ and compare it with Pearson's r.
You need to be able to
- Rank both variables the same way round, giving tied values the average of the ranks they share.
- Calculate rₛ as Pearson's r of the ranks on a GDC; with no ties, 1 − 6Σd²/(n(n² − 1)) gives the same value.
- Explain when rₛ suits the data better than r, such as for a curved but consistently rising pattern or data with outliers.
Key terms
- Spearman's rank correlation coefficient
- rank
- tied ranks
- monotonic relationship
- Pearson's r
Common trap Ranking one variable from highest and the other from lowest, which reverses the sign of rₛ.
Practise 4.11 in Heptavo →4.12Poisson distributionHL
Modelling counts of random events in a fixed interval of time or space; questions ask you to justify the model, scale the mean and find probabilities.
You need to be able to
- State the conditions: events occur independently, at a constant average rate, one at a time.
- Use E(X) = Var(X) = λ and find probabilities with the Poisson pdf and cdf on a GDC.
- Scale λ to the interval in the question, and add the means of independent Poisson variables.
Key terms
- Poisson distribution
- mean rate
- λ
- independent events
- fixed interval
Common trap Using the rate per hour for a three-hour question; scale the mean first, so λ becomes 3 times as large.
Practise 4.12 in Heptavo →4.13Expected value, games of chanceSL
Expected value applied to games of chance and other repeated situations; questions ask for E(X), the expected gain and whether a game is fair.
You need to be able to
- Calculate E(X) = Σ x P(X = x).
- Find the expected gain as expected winnings minus the cost to play, and call the game fair when it is zero.
- Use E(aX + b) = aE(X) + b, and multiply by N for the expected total over N games.
Key terms
- expected value
- expected gain
- fair game
- probability distribution
- long-run average
Common trap Forgetting to subtract the cost to play; a game is fair when the expected winnings equal the stake.
Practise 4.13 in Heptavo →4.14Chi-squared and t-test hypothesis testingSL
Hypothesis testing with χ² tests and the t-test; questions ask you to state hypotheses, find expected frequencies and degrees of freedom, obtain a p-value and conclude in context.
You need to be able to
- Write H₀ and H₁ in words for the context.
- Find expected frequencies as row total × column total ÷ grand total, with (r − 1)(c − 1) degrees of freedom for independence and k − 1 for a goodness of fit test.
- Compare the p-value with the significance level and conclude in context, without claiming H₀ is proved.
Key terms
- null hypothesis
- alternative hypothesis
- p-value
- significance level
- degrees of freedom
- expected frequency
Common trap Writing that H₀ is accepted or proved; you either reject H₀ or have insufficient evidence to reject it.
Practise 4.14 in Heptavo →4.15Standardization of normal variables (z-values)HL
Converting normal values to z-values; questions ask how many standard deviations a value is from the mean, and for an unknown mean or standard deviation.
You need to be able to
- Calculate z = (x − μ)/σ.
- Use the inverse normal on N(0, 1) to find z from a probability, then solve for μ or σ.
- Compare values from different normal distributions using their z-values.
Key terms
- z-value
- standardisation
- standard normal distribution
- inverse normal
- standard deviation
Common trap Losing the minus sign for a value below the mean; z = (85 − 100)/15 = −1, not 1.
Practise 4.15 in Heptavo →4.16Non-linear regression, choosing a modelHL
Fitting non-linear models to data and choosing between them; questions ask you to find a regression model on a GDC, compare fits with R² and use residuals.
You need to be able to
- Fit exponential, power, quadratic, cubic and sine regression models with a GDC.
- Use R² to compare fits, and linearise y = axᵇ as log y = log a + b log x.
- Calculate residuals (observed − predicted) and read their pattern to judge the model's shape.
Key terms
- coefficient of determination (R²)
- residual
- power model
- exponential regression
- linearisation
- sum of squared residuals
Common trap Choosing the model with the highest R² without thinking about context; a cubic may fit the data best but predict nonsense outside them.
Practise 4.16 in Heptavo →4.17Estimation, sample means and confidence intervalsHL
Estimating a population mean from a sample and combining random variables; questions ask for the distribution of a sample mean, confidence intervals and the mean and variance of sums.
You need to be able to
- Use the fact that X̄ has mean μ and standard deviation σ/√n, and is approximately normal for large n by the central limit theorem.
- Find a confidence interval for a mean on a GDC, using z when σ is known and t with n − 1 degrees of freedom when it is estimated.
- Use E(X ± Y) = E(X) ± E(Y) and, for independent variables, Var(X ± Y) = Var(X) + Var(Y).
Key terms
- unbiased estimate
- sample mean
- central limit theorem
- confidence interval
- t-distribution
- linear combination
Common trap Subtracting variances for X − Y; for independent variables the variances add, even for a difference.
Practise 4.17 in Heptavo →4.18Hypothesis testing: critical regions, errors, further testsHL
Hypothesis tests with critical regions and the errors they can make; questions ask you to find a critical region, carry out a test, and calculate Type I and Type II error probabilities.
You need to be able to
- Find the critical region for a test at a given significance level, one-tailed or two-tailed as the wording requires.
- Carry out tests for a population mean, a binomial proportion or a Poisson mean, and conclude in context.
- Calculate P(Type I error) as the probability of the critical region under H₀, and P(Type II error) from the true distribution outside it.
Key terms
- critical region
- critical value
- Type I error
- Type II error
- one-tailed test
- two-tailed test
Common trap Quoting the nominal 5% as the Type I probability of a discrete test; it is the actual tail probability of the critical region, which is usually smaller.
Practise 4.18 in Heptavo →Topic 5 — Calculus
Derivatives and integrals for rates, optimisation and areas, with differential equations and their systems at HL.
5.1Limits, derivative as gradient / rate of changeSL
The derivative as the gradient of a curve and as a rate of change; questions ask you to interpret f′(a), give its units, and link chord gradients to the tangent.
You need to be able to
- Distinguish f(a), the height of the curve, from f′(a), its gradient.
- Give dy/dx in units of y per unit of x and interpret it in context.
- Estimate the gradient at a point from chords that get shorter, and connect this to a limit.
Key terms
- derivative
- gradient
- tangent
- rate of change
- limit
- chord
Common trap Giving a rate of change without units; a rate is in units of y per unit of x, such as litres per minute.
Practise 5.1 in Heptavo →5.2Increasing/decreasing functions, sign of f′(x)SL
Using the sign of f′(x) to see 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
- Find where f′(x) > 0 (increasing) and f′(x) < 0 (decreasing).
- Locate stationary points where f′(x) = 0, then get their heights from f.
- Read a graph of f′: above the axis f rises, below it f falls, and a crossing is a turning point.
Key terms
- increasing
- decreasing
- stationary point
- sign of f′(x)
- turning point
Common trap Solving f(x) = 0 to find stationary points; that gives the roots, while stationary points come from f′(x) = 0.
Practise 5.2 in Heptavo →5.3Derivatives of power functions, sum/difference ruleSL
Differentiating powers of x and sums of them; questions ask for derivatives, gradients at points and the x-values where the gradient takes a given value.
You need to be able to
- Differentiate axⁿ to naxⁿ⁻¹, term by term.
- Rewrite terms such as 4/x² as 4x⁻² before differentiating.
- Substitute into f′(x) for a gradient, or solve f′(x) = k for x.
Key terms
- derivative
- power rule
- gradient function
- negative exponent
- sum rule
Common trap Differentiating a product of brackets factor by factor and multiplying; expand first, then differentiate each term.
Practise 5.3 in Heptavo →5.4Tangent and normal linesSL
Equations of tangent and normal lines to a curve; questions ask for these lines at a given point, often using the GDC to find the gradient.
You need to be able to
- Find the point from f(a) and the gradient from f′(a), by hand or with a GDC.
- Write the tangent as y − y₁ = m(x − x₁) and the normal with gradient −1/m.
- Recognise that a horizontal tangent y = f(a) has a vertical normal x = a.
Key terms
- tangent
- normal
- gradient
- point of contact
- negative reciprocal
Common trap Using the point's y-coordinate as the intercept c; c is where the line meets the y-axis, y₁ − mx₁.
Practise 5.4 in Heptavo →5.5Integration as anti-differentiation, boundary conditionsSL
Integration as reversing differentiation, boundary conditions, and definite integrals as areas; questions ask for f(x) from f′(x) and a point, and for areas with a GDC.
You need to be able to
- Integrate axⁿ to axⁿ⁺¹/(n + 1) + c for n ≠ −1.
- Find c by substituting a known point into y = F(x) + c.
- Find the area under a curve above the x-axis with a definite integral on a GDC.
Key terms
- anti-differentiation
- indefinite integral
- constant of integration
- boundary condition
- definite integral
Common trap Leaving out + c, or assuming c equals the given y-value when the known point is not at x = 0.
Practise 5.5 in Heptavo →5.6Trapezoidal rule, approximating areaSL
Estimating the area under a curve or a table of data with trapeziums; questions ask for the estimate and whether it is too large or too small.
You need to be able to
- Find the strip width h = (b − a)/n.
- Apply ½h[y₀ + yₙ + 2(y₁ + … + yₙ₋₁)], doubling only the interior ordinates.
- Decide from the concavity whether the estimate is too large (concave up) or too small (concave down).
Key terms
- trapezoidal rule
- strip width
- ordinate
- overestimate
- underestimate
- concavity
Common trap Confusing strips with ordinates; n strips need n + 1 ordinates.
Practise 5.6 in Heptavo →5.7Derivatives of exp/log/trig, product/quotient/chain ruleHL
Derivatives of eˣ, ln x, sin x and cos x, and the chain, product and quotient rules; questions ask you to differentiate combinations of functions.
You need to be able to
- Recall the derivatives of eˣ, ln x, sin x and cos x.
- Apply the chain rule, multiplying by the derivative of the inside function.
- Use the product rule u′v + uv′ and the quotient rule (u′v − uv′)/v², choosing from how the function is built.
Key terms
- chain rule
- product rule
- quotient rule
- composite function
- derivative
Common trap Differentiating cos x as sin x; the derivative is −sin x.
Practise 5.7 in Heptavo →5.8Second derivative, concavity, inflection pointsHL
The second derivative, concavity and points of inflexion; questions ask you to find f″(x), classify stationary points and locate inflexion points.
You need to be able to
- Find f″(x) and link its sign to concavity: positive concave up, negative concave down.
- Classify a stationary point with the sign of f″ there.
- Find points of inflexion where f″(x) = 0 and changes sign.
Key terms
- second derivative
- concave up
- concave down
- point of inflexion
- stationary point
Common trap Calling every point with f″(x) = 0 a point of inflexion; f″ must change sign, and y = x⁴ at x = 0 is a minimum.
Practise 5.8 in Heptavo →5.9Optimization and kinematics with derivativesSL
Optimisation in real contexts and the link between displacement, velocity and acceleration; questions ask you to set up a function, find the optimum and interpret motion.
You need to be able to
- Write the quantity to optimise in one variable using the constraint.
- Solve f′(x) = 0 (by hand or with a GDC) and justify that the point is the maximum or minimum wanted.
- Differentiate displacement for velocity and velocity for acceleration, and find when the object is at rest.
Key terms
- optimisation
- constraint
- stationary point
- velocity
- acceleration
- at rest
Common trap Ignoring the endpoints on a closed interval; the greatest value may occur at an end rather than at a stationary point.
Practise 5.9 in Heptavo →5.10Integration by substitutionHL
Integrating standard functions and using substitution; questions ask for indefinite and definite integrals of composite functions.
You need to be able to
- Integrate 1/x to ln|x|, cos x to sin x and sin x to −cos x.
- Integrate a function of ax + b by integrating as usual and dividing by a.
- Use substitution when the derivative of the inner function appears in the integrand, converting the limits for definite integrals.
Key terms
- integration by substitution
- standard integral
- ln|x|
- inner function
- limits of integration
Common trap Integrating sin x to cos x; the integral is −cos x + c.
Practise 5.10 in Heptavo →5.11Area between curves, kinematics with integralsHL
Areas between curves and kinematics with integrals; questions ask for enclosed areas, displacements and total distances travelled.
You need to be able to
- Find the area between two curves as ∫(top − bottom) dx between their intersections.
- Split integrals at roots so that areas below the x-axis count positively.
- Find displacement as ∫v dt and distance travelled as ∫|v| dt.
Key terms
- area between curves
- definite integral
- displacement
- distance travelled
- velocity
Common trap Giving ∫v dt as the distance travelled when the velocity changes sign; integrate |v| instead.
Practise 5.11 in Heptavo →5.12Setting up and solving differential equations, Euler’s methodHL
Forming differential equations from descriptions, solving separable ones and approximating solutions with Euler's method; questions ask for models, exact solutions and numerical estimates.
You need to be able to
- Translate 'the rate of change of P is proportional to P' into dP/dt = kP.
- Solve separable equations, using an initial condition to find the constant.
- Apply Euler's method with a GDC or spreadsheet, and explain how a smaller step improves accuracy.
Key terms
- differential equation
- proportional to
- separation of variables
- initial condition
- Euler's method
- step size
Common trap Writing the solution of dy/dx = ky as y = e^(kx) + C; integrating gives ln|y| = kx + C, so y = Ae^(kx).
Practise 5.12 in Heptavo →5.13Slope fields, separable / coupled / 2nd-order DEsHL
Slope fields, coupled linear systems and second-order equations; questions ask you to read slope fields, use eigenvalues to describe phase portraits, and apply Euler's method to systems.
You need to be able to
- Draw and read a slope field, sketching a solution curve that follows the segments.
- Use the eigenvalues of a coupled system's matrix to classify the equilibrium: sink, source, saddle or spiral.
- Rewrite a second-order equation as two coupled first-order equations and apply Euler's method.
Key terms
- slope field
- coupled differential equations
- phase portrait
- eigenvalue
- equilibrium point
- saddle point
Common trap Calling the origin stable when only one eigenvalue is negative; real eigenvalues of opposite signs give a saddle, and most trajectories move away.
Practise 5.13 in Heptavo →Try 4 Heptavo questions
Real questions from the app, with the explanation after each answer.
1.1 · Standard form / scientific notation
A country has national debt 3.4 × 1012 dollars and population 6.8 × 107. Find the debt per person.
- 5 × 10⁵ dollars
- 2 × 10⁻⁵ dollars
- 5 × 10¹⁹ dollars
- 5 × 10⁴ dollars
Show the answer and why
Answer: D
3.4 ÷ 6.8 = 0.5 and 12 − 7 = 5, giving 0.5 × 105, which tidies to 5 × 104. Option A is the answer of someone who tidied the front number but not the exponent; option B divided the wrong way round.
3.4 · Bearings and geometric transformations
A ship sails 8 km on a bearing of 120°. How far east of its starting point is it?
- 4.00 km
- 6.93 km
- 8.00 km
- 13.9 km
Show the answer and why
Answer: B
The bearing is measured from north, so the east–west component uses sine: 8 sin 120° = 6.93. The value 4.00 km is the north–south component (and the ship has actually gone 4 km south). Which of sine and cosine gives "east" depends on the angle being measured from north, not from the horizontal.
4.4 · Correlation, scatter diagrams, Pearson’s r
A data set has r = 0.78. Every y-value is then doubled. The new value of r is
- 0.39
- 0.61
- 0.78
- 1.56
Show the answer and why
Answer: C
Multiplying by a positive constant rescales the axis without changing the pattern, so r is unchanged. The value 1.56 is the giveaway answer — it is greater than 1 and therefore impossible before you even think about the mathematics.
5.2 · Increasing/decreasing functions, sign of f′(x)
A curve has f′(x) > 0 for every x. The curve
- can cross the x-axis at most once
- has at least one stationary point
- never crosses the x-axis
- has exactly one maximum
Show the answer and why
Answer: A
An always-rising curve can only pass a given height once, so it can meet the x-axis at most one time. That is why a strictly increasing model has a unique solution to f(x) = k — the fact behind every single-intersection answer you find on a GDC.
